comment {

Calculation formulas developed by Kerry Mitchell
for version 3 of Ultra Fractal.  Due to UF3's new
features and syntax, these formulas will not work 
with version 2.

Compilation updated 27 November 2021

Includes:
  Rotated Mandelbrot & Julia
  Rational Newton Mandelbrot & Julia (updated from v2)
  Caramel Popcorn
  Caramel Popcorn 3 Axis
  Embossed Mandelbrot & Julia Rose Range Lite
  Relativistic Addition Mandelbrot & Julia
  Burning Ship Mandelbrot & Julia
  Relaxed Newton/Halley
  Unit Vector Tweak Mandelbrot & Julia
  Partial cz Mandelbrot & Julia
  Alternating Functions
  Interpolating Functions
  Fourier Popcorn
  Breaking Newton - Real
  Breaking Newton - Complex
  Compound Mandelbrot
  Compound Julia
  Mod Mandelbrot
  Mod Julia
  Relaxed Householder's Method
  Extra Foam
  Round and Transform
}

rotated-mandelbrot { ; Kerry Mitchell 06oct2002
;
; Rotates the z variable in the standard Mandelbrot
; calculation each iteration.
;
; Use the 'rotation type' parameter to rotate before or 
; after raising z to the desired power, or both before and 
; after.  The '+before, -after' setting rotates the opposite
; way after raising z to the power.
;
; The rotation angle is the 'rotation factor' times the angle
; of the pixel or z.  The 'constant' setting in 'rotation type'
; makes the rotational angle a constant 'rotation factor'
; degrees.
;
init:
  c=#pixel
  z=c+@manparam
  float t=0.0
  float trad=@rotfac*#pi/180
  rot=(0,0)
loop:
;
; set up the rotation angle
;
  if(@angletype==1)          ; pixel
    t=@rotfac*atan2(#pixel)
  elseif(@angletype==2)      ; z
    t=@rotfac*atan2(z)
  else                       ; constant
    t=trad
  endif
  rot=cos(t)+flip(sin(t))
;
; iterate z, taking into account the rotation type
;
  if(@rottype==1)            ; before
    z=z*rot
    z=z^@power+c
  elseif(@rottype==2)        ; after
    z=z^@power+c
    if(@angletype==2)
      t=@rotfac*atan2(z)
      rot=cos(t)+flip(sin(t))
    endif
    z=z*rot
  elseif(@rottype==3)        ; before & after
    z=z*rot
    z=z^@power+c
    if(@angletype==2)
      t=@rotfac*atan2(z)
      rot=cos(t)+flip(sin(t))
    endif
    z=z*rot
  elseif(@rottype==4)        ; +before, -after
    z=z*rot
    z=z^@power+c
    if(@angletype==2)
      t=@rotfac*atan2(z)
      rot=cos(t)+flip(sin(t))
    endif
    z=z/rot
  else                       ; none
    z=z^@power+c
  endif
bailout:
  |z|<@bailout
default:
  title="Rotated Mandelbrot"
  periodicity=0
  param manparam
    caption="perturbation"
    default=(0,0)
  endparam
  param power
    caption="power"
    default=(2,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000.0
  endparam
  param rottype
    caption="rotation type"
    default=1
    enum="none" "before iterating" "after iterating" \
      "before & after" "+before, -after"
    hint="How z is rotated each iteration."
  endparam
  float param rotfac
    caption="rotation factor"
    default=1.0
    enabled=@rottype!="none"
    hint="If 'angle type' is 'constant', then this is the \
      rotation angle in degrees. Otherwise, it is the factor \
      that multiples the pixel or z angle to make the rotation \
      angle."
  endparam
  param angletype
    caption="angle type"
    default=2
    enum="constant" "pixel" "z"
    enabled=@rottype!="none"
    hint="Use 'constant' to specify a constant angle in degrees. \
      Otherwise, the rotation is based on the angle of the pixel \
      or the angle of z."
  endparam
switch:
  type="rotated-julia"
  julparam=#pixel
  bailout=bailout
  power=power
  rottype=rottype
  rotfac=rotfac
  angletype=angletype
}

rotated-julia { ; Kerry Mitchell 06oct2002
;
; Rotates the z variable in the standard Julia set
; calculation each iteration.
;
; Use the 'rotation type' parameter to rotate before or 
; after raising z to the desired power, or both before and 
; after.  The '+before, -after' setting rotates the opposite
; way after raising z to the power.
;
; The rotation angle is the 'rotation factor' times the angle
; of the pixel or z.  The 'constant' setting in 'rotation type'
; makes the rotational angle a constant 'rotation factor'
; degrees.
;
init:
  z=#pixel
  c=@julparam
  float t=0.0
  float trad=@rotfac*#pi/180
  rot=(0,0)
loop:
;
; set up the rotation angle
;
  if(@angletype==1)          ; pixel
    t=@rotfac*atan2(#pixel)
  elseif(@angletype==2)      ; z
    t=@rotfac*atan2(z)
  else                       ; constant
    t=trad
  endif
  rot=cos(t)+flip(sin(t))
;
; iterate z, taking into account the rotation type
;
  if(@rottype==1)            ; before
    z=z*rot
    z=z^@power+c
  elseif(@rottype==2)        ; after
    z=z^@power+c
    if(@angletype==2)
      t=@rotfac*atan2(z)
      rot=cos(t)+flip(sin(t))
    endif
    z=z*rot
  elseif(@rottype==3)        ; before & after
    z=z*rot
    z=z^@power+c
    if(@angletype==2)
      t=@rotfac*atan2(z)
      rot=cos(t)+flip(sin(t))
    endif
    z=z*rot
  elseif(@rottype==4)        ; +before, -after
    z=z*rot
    z=z^@power+c
    if(@angletype==2)
      t=@rotfac*atan2(z)
      rot=cos(t)+flip(sin(t))
    endif
    z=z/rot
  else                       ; none
    z=z^@power+c
  endif
bailout:
  |z|<@bailout
default:
  title="Rotated Julia"
  periodicity=0
  param julparam
    caption="Julia parameter"
    default=(0,0)
  endparam
  param power
    caption="power"
    default=(2,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000.0
  endparam
  param rottype
    caption="rotation type"
    default=1
    enum="none" "before iterating" "after iterating" \
      "before & after" "+before, -after"
    hint="How z is rotated each iteration."
  endparam
  float param rotfac
    caption="rotation factor"
    default=1.0
    enabled=@rottype!="none"
    hint="If 'angle type' is 'constant', then this is the \
      rotation angle in degrees. Otherwise, it is the factor \
      that multiples the pixel or z angle to make the rotation \
      angle."
  endparam
  param angletype
    caption="angle type"
    default=2
    enum="constant" "pixel" "z"
    enabled=@rottype!="none"
    hint="Use 'constant' to specify a constant angle in degrees. \
      Otherwise, the rotation is based on the angle of the pixel \
      or the angle of z."
  endparam
switch:
  type="rotated-mandelbrot"
  bailout=bailout
  power=power
  rottype=rottype
  rotfac=rotfac
  angletype=angletype
}

comment { ; narrative copyright Kerry Mitchell 28apr2001
Updated 03jan2003

Mandelbrot-Newton Formulas

Mandelbrot and Newton are two of the more popular fractal types, and 
have been studied for quite a while.  Each has its own unique style, 
due to the nature of the underlying formulas.  

Mandelbrot sets

Mandelbrot fractals come from iterating z = f(z), where f is a function 
of the variable z and a parameter c.  The initial value of z is the 
critical point of f:  the value z0 such that the derivative of f at z0 
is 0, or f'(z0) = 0.  Typically, c comes from the pixel value.  The 
image shows which c values are such that f(z) never goes to infinity.  
Thus, the "bailout" condition is generally some variant of stopping the 
iteration when z becomes large.  For polynomials, this means that the 
fractal structure is located around the origin of the plane (around the 
point (0,0)), because as c gets large, z gets large.

Julia sets can be thought of as variants of the Mandelbrot set for a 
given formula.  The major difference is in the initialization:  z is 
initialized to the pixel value and c is a user-input constant.  The 
Mandelbrot set acts as a catalog of the Julia sets:  the structure of 
the Mandelbrot set around a given point c is similar to the structure 
of the Julia set for that c.  Consequently, Julia sets typically do 
not exhibit the diversity of structure of the Mandelbrot set.  In 
particular, Julia sets do not contain "midgets," small copies of 
the Mandelbrot set.

Newton fractals

Newton fractals are the result of iterating z = z - g(z)/g'(z), where 
g is a function of the variable z and a parameter c, and g'(z) is the 
first derivative of g(z) with respect to z.  Newton's method is a 
technique for finding roots to an equation, specifically, finding those 
z values such that g(z) = 0.  Consequently, the initial z value is 
typically taken as the pixel (like a Julia set), and the "bailout" 
condition is when two consecutive z values are close enough, indicating 
convergence to a root.

Most Newton images have a characteristic structure of strings of loops.  
This lace-like form extends to infinity, since the iteration results in 
z settling down to a final value for almost any initial guess.  Newton 
fractals are generally created by taking the initial z value from the 
pixel and having c be constant; an example is solving z^4 - 1 = 0.  
This is essentially a Julia set formulation, where c takes the role of 
1.  Thus, the structure of the fractal does not show a great deal of 
variation, and Newton fractals typically do not contain midgets.

Mandelbrot-Newton formula

What is missing is a Mandelbrot implementation of Newton's method.  This 
image (presumably) would have a great diversity of structure, as it would 
catalog the infinity of Julia-Newton images available from the same 
formula.  If done correctly, a Mandelbrot-Newton image would also extend 
to infinity, instead of being constrained to a relatively small region 
around the origin.

The problem is in determining the critical point of the formula, so that 
the appropriate initial z can be found.  To do this, compare the 
Mandelbrot and Newton formulas:

Mandelbrot:  z = f(z)
Newton:  z = z - g(z)/g'(z), thus f(z) = z - g(z)/g'(z).

The Mandelbrot initial z is z0 such that f'(z0) = 0:

f = z - g/g',
f' = 1 - (g' g' - g g'')/(g'g'), or
f' = 1 - 1 + (g g'') / (g' g'), resulting in
f' = (g g'') / (g' g').

So f'(z) will be 0 whenever g(z) = 0 or g''(z) = 0.  The first condition 
is the solution that the formula is trying to find, so using that for the 
initial z will not be productive.  Thus, the critical point comes from 
having the second derivative of g be 0:  g''(z0) = 0.  (Strictly speaking, 
there are critical points where f'(z) fails to exist, which would occur 
when g'(z) = 0.  However, this would cause the iteration formula to fail 
by dividing by zero, so it is not a valid beginning point.)

Finding z such that g''(z) = 0 is not a trivial task for a general function 
g.  However, this can be simplified by constructing g in a special way:

g(z) = h(z) + az^2 + bz + c,

where h(z) is any general function of z, and a, b, and c are constants.  
Differentiating, 

g'(z) = h'(z) + 2az + b, and
g''(z) = h''(z) + 2a.

The advantage of this is that any z can be the critical point, through 
a judicious choice of the constant a.  In practice, the user inputs the 
initial point z0.  Then, a is determined by forcing g''(z0) to be 0:

g''(z0) = 0 = h''(z0) + 2a, or, solving for a:
a = -1/2 h''(z0).

Once the function h(z) is chosen, it is reasonably straightforward to 
determine h''(z0), and thus a.  The user inputs b, and the bz term is 
just to add more flexibility.  The constant term c is supplied by the 
pixel value, just as in typical Mandelbrot formulas.

To recap, a Mandelbrot version of Newton's method can be created by 
starting with any function h(z) whose second derivative is non-zero.  
The user specifies the critical point and initial z value z0.  Then, 
a is determined, a = -0.5 h''(z0).  The constant b is another user 
input, and c comes from the pixel value. The formulas are then:

Newton:  g(z) = h(z) + az^2 + bz + c, g'(z) = h'(z) + 2az + b
Mandelbrot:  f(z) = z - g(z)/g'(z)

The bailout condition is when the next z value is close enough to the 
previous z, indicating that g(z) = close enough to 0, or that z is a 
root of g(z).

Rational implementation

The initial implementation of this idea was with a rational h(z).  
Rational functions are ratios of polynomial functions.  Polynomials 
themselves are rational functions, as a polynomial p(z) can be written 
as p(z)/1, and 1 is a (constant) polynomial function.  The function h(z) 
is composed of 2 pieces, p(z) and q(z), both of which are polynomials.  
To balance between simplicity and flexibility,

p(z) = z^m + b, and
q(z) = z^n - b,

where b is the constant from above (this version does not use the linear 
bz term).  Strictly speaking, m and n should both be non-negative integers 
(e.g., 0, 1, 2, 17, etc.), but they are allowed to be real numbers (e.g., 
1.234, -7.89) for flexibility.  Arithmetic combinations of p and q yields 
h(z); h can be any of the following:  p, q, p+q, p-q, q-p, p*q, p/q, or q/p.

Using the formula

The formula has 7 parameters:  "critical point", "bailout", "m", "n", "b", 
"h(z) type", and "z type":

"critical point":  This is the initial z value (complex).  Whatever value 
is entered will be the critical point for the function f(z).  Smaller 
values work better; larger values may require magnifications much, much 
less than 1 (extreme zooming out) to see any detail.

"bailout":  Use a large (real) value.  Since this formula is looking for 
z to converge to a final value instead of z to get very large, the 
condition that is tested is 1/|znew - zold|.  As znew gets very close 
to zold, the reciprocal of their difference goes to infinity.

"m":  The z exponent in the p function:  p(z) = z^m + b.  Can be any 
real value (not complex).

"n":  The z exponent in the q function:  q(z) = z^n - b.  Can be any 
real value (not complex).

"b":  The (complex) constant in the p and q functions.  The nature of 
the fractal may or may not be sensitive to b, depending on the critical 
point and the h(z) type.  In particular, b has no effect when the h(z) 
type is set to "p+q", as the b values s in the 2 functions cancel out.

"h(z) type":  How p(z) and q(z) are combined into h(z).  Choose from:  p, 
q, p+q, p-q, q-p, p*q, p/q, and q/p.

"z type":  This formula offers the opportunity to generate an "embossed"
image, based on the number of iterations until bailout.  If this option is 
not exercised, then the "z type" parameter determines what value of z is sent 
to the coloring routine.  The choices are "iterate" (the z value that converges 
to a root of g(z)); "1/change in z", the value checked for bailout; and 
"derivative", the derivative of g with respect to z.  If the embossed image 
is chosen, the "z type" parameter is hidden.

For any set m, n, b, and h(z) type, the entered "critical point" value 
should indeed be a critical point for f(z).  Consequently, the rendered 
Mandelbrot set should serve as a catalog for Julia sets from the same 
formula.  To keep the same formula, all 7 parameters (plus the embossing 
parameters) are passed with the "switch" function to the Julia version of 
the formula.  The c constant, which took on the pixel value in the Mandelbrot 
version, is an input ("Julia parameter") in the Julia version (or passed from 
the pixel value during the "switch").  Note that the "critical point" is the 
initial z value only for the Mandelbrot case.  It is used in the Julia case to 
determine the a constant, but the initial z comes from the pixel 
value.

}

rational-newton-mandelbrot { ; Kerry Mitchell 28apr2001
;
; Mandelbrot and Newton set in one
; iterates z = f(z), beginning at zc = critical point of f(z)
; f(z) = Newton algorithm = z - g(z)/g'(z)
; g(z) = h(z) + a*z^2 + c
; h(z) = rational combination of 2 polynomial functions p(z) & q(z)
; p(z) = z^m + b
; q(z) = z^n - b
; a = -h"(zc)/2, so that g"(zc) = 0 and zc is a critical point of f(z)
; b = user input
; c = pixel
;
; Updated 02jan2003:
;   added optional embossing capability
;   added 'ztype' parameter
;   updated syntax to version 3
;
init:
;
; embossing initializations
;
  float theta=@lightangle*#pi/180.0
  float size=@sizefac*0.0065/#magn
  dr=size*(cos(theta)+flip(sin(theta)))
  c1=#pixel-dr
  c2=#pixel+dr
  int iter=0
  int iter1=0
  int iter2=0
  if(@emboss==false)         ; no embossing
    iter2=1
  endif
  int iter=0
;
; rational newton initializations
;
  z=@zc
  z1=@zc
  z2=@zc
  g=(0,0)
  gp=(0,0)
  h=(0,0)
  hp=(0,0)
  hpp=(0,0)
  dz=(0,0)
  temp=(0,0)
;
; first polynomial, p(z) = z^m + b
;
  temp=z^(@m-1)
  p=z*temp+@b
  pp=@m*temp
  ppp=(@m-1)*pp/z
;
; second polynomial, q(z) = z^n - b
;
  temp=z^(@n-1)
  q=z*temp-@b
  qp=@n*temp
  qpp=(@n-1)*qp/z
;
; combine p & q to find a
;
  if(@htype==1)              ; q
    hpp=qpp
  elseif(@htype==2)          ; p+q
    hpp=ppp+qpp
  elseif(@htype==3)          ; p-q
    hpp=ppp-qpp
  elseif(@htype==4)          ; q-p
    hpp=qpp-ppp
  elseif(@htype==5)          ; p*q
    hpp=p*qpp+2*pp*qp+ppp*q
  elseif(@htype==6)          ; p/q
    temp=sqr(q)
    hpp=temp*(q*ppp-p*qpp)-2*q*qp*(q*pp-p*qp)
    hpp=hpp/sqr(temp)
  elseif(@htype==7)          ; q/p
    temp=sqr(p)
    hpp=temp*(p*qpp-q*ppp)-2*p*pp*(p*qp-q*pp)
    hpp=hpp/sqr(temp)
  else                       ; p
    hpp=ppp
  endif
  a=-0.5*hpp
loop:
  iter=iter+1
;
;    first test point
;
  if(iter1==0)
;
; first polynomial, p(z) = z^m + b
;
    temp=z1^(@m-1)
    p=z1*temp+@b
    pp=@m*temp
;
; second polynomial, q(z) = z^n - b
;
    temp=z1^(@n-1)
    q=z1*temp-@b
    qp=@n*temp
;
; combine p & q to make h(z)
;
    if(@htype==1)            ; q
      h=q
      hp=qp
    elseif(@htype==2)        ; p+q
      h=p+q
      hp=pp+qp
    elseif(@htype==3)        ; p-q
      h=p-q
      hp=pp-qp
    elseif(@htype==4)        ; q-p
      h=q-p
      hp=qp-pp
    elseif(@htype==5)        ; p*q
      h=p*q
      hp=p*qp+pp*q
    elseif(@htype==6)        ; p/q
      h=p/q
      hp=(q*pp-p*qp)/sqr(q)
    elseif(@htype==7)        ; q/p
      h=q/p
      hp=(p*qp-q*pp)/sqr(p)
    else                     ; p
      h=p
      hp=pp
    endif
;
; add quadratic to make g(z)
;
    g=h+a*sqr(z1)+c1
    gp=hp+2*a*z1
;
; Newton iteration; check 1/(change in z) for bailout
;
    dz=g/gp
    z1=z1-dz
    temp=1/dz
    if(|temp|>@bailout)
      iter1=iter
    endif
  endif
;
;    second test point
;
  if(iter2==0)
;
; first polynomial, p(z) = z^m + b
;
    temp=z2^(@m-1)
    p=z2*temp+@b
    pp=@m*temp
;
; second polynomial, q(z) = z^n - b
;
    temp=z2^(@n-1)
    q=z2*temp-@b
    qp=@n*temp
;
; combine p & q to make h(z)
;
    if(@htype==1)            ; q
      h=q
      hp=qp
    elseif(@htype==2)        ; p+q
      h=p+q
      hp=pp+qp
    elseif(@htype==3)        ; p-q
      h=p-q
      hp=pp-qp
    elseif(@htype==4)        ; q-p
      h=q-p
      hp=qp-pp
    elseif(@htype==5)        ; p*q
      h=p*q
      hp=p*qp+pp*q
    elseif(@htype==6)        ; p/q
      h=p/q
      hp=(q*pp-p*qp)/sqr(q)
    elseif(@htype==7)        ; q/p
      h=q/p
      hp=(p*qp-q*pp)/sqr(p)
    else                     ; p
      h=p
      hp=pp
    endif
;
; add quadratic to make g(z)
;
    g=h+a*sqr(z2)+c2
    gp=hp+2*a*z2
;
; Newton iteration; check 1/(change in z) for bailout
;
    dz=g/gp
    z2=z2-dz
    temp=1/dz
    if(|temp|>@bailout)
      iter2=iter
    endif
  endif
  if(@emboss==false)         ; no embossing--set z to:
    if(@ztype==1)            ;   1/dz
      z=temp
    elseif(@ztype==2)        ;   derivative
      z=gp
    else                     ;   iterate
      z=z1
    endif
  else                       ; embossing--z = iter1 & iter2
    z=iter1+flip(iter2)
  endif
bailout:
  (iter1==0)||(iter2==0)
default:
  title="Rational Newton Mandelbrot"
  method=multipass
  maxiter=100
  periodicity=0
  center=(0,0)
  magn=0.01
;
; rational newton mandelbrot stuff
;
  heading
    caption="Rational Newton parameters"
  endheading
  param zc
    caption="critical point"
    default=(1,0)
    hint="Keep to a fairly small value."
  endparam
  param bailout
    caption="bailout"
    default=1000.0
    hint="Set to a large value."
  endparam
  param m
    caption="m"
    default=8.0
    hint="Exponent in p(z) function; p(z) = z^m + b."
  endparam
  param n
    caption="n"
    default=4.0
    hint="Exponent in q(z) function; q(z) = z^n - b."
  endparam
  param b
    caption="b"
    default=(-1,0)
    hint="Constant in p(z) and q(z) functions.  Doesn't do anything when \
      'h(z) type' is set to 'p+q'."
  endparam
  param htype
    caption="h(z) type"
    default=2
    enum="p" "q" "p+q" "p-q" "q-p" "p*q" "p/q" "q/p"
    hint="How the p(z) and q(z) functions are combined to make h(z)."
  endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "1/change in z" "derivative"
    hint="Determines what value to send to coloring routine, \
      if the image is not embossed."
    visible=(@emboss==false)
  endparam
;
; embossing stuff
;
  heading
    caption="Embossing parameters"
  endheading
  bool param emboss
    caption="embossed image"
    default=false
  endparam
  float param sizefac
    caption="contour size"
    default=1.0
    hint="Specifies the relative size of the contour bands. Larger values \
          give thicker bands."
    visible=(@emboss==true)
  endparam
  float param lightangle
    caption="light angle"
    default=0.0
    hint="Angle of apparent light source, in degrees."
    visible=(@emboss==true)
  endparam
switch:
  type="rational-newton-julia"
  zc=zc
  julparam=#pixel
  bailout=bailout
  m=m
  n=n
  b=b
  htype=htype
  ztype=ztype
  emboss=emboss
  sizefac=sizefac
  lightangle=lightangle
}

rational-newton-julia { ; Kerry Mitchell 28apr2001
;
; Julia and Newton set in one
; iterates z = f(z), beginning at zc = critical point of f(z)
; f(z) = Newton algorithm = z - g(z)/g'(z)
; g(z) = h(z) + a*z^2 + c
; h(z) = rational combination of 2 polynomial functions p(z) & q(z)
; p(z) = z^m + b
; q(z) = z^n - b
; a = -h"(zc)/2, so that g"(zc) = 0 and zc is a critical point of f(z)
; b = user input
; c = pixel
;
; Updated 02jan2003:
;   added optional embossing capability
;   added 'ztype' parameter
;   updated syntax to version 3
;
init:
;
; embossing initializations
;
  float theta=@lightangle*#pi/180.0
  float size=@sizefac*0.0065/#magn
  dr=size*(cos(theta)+flip(sin(theta)))
  z1=#pixel-dr
  z2=#pixel+dr
  int iter=0
  int iter1=0
  int iter2=0
  if(@emboss==false)         ; no embossing
    iter2=1
  endif
  int iter=0
;
; rational newton initializations
;
  z=@zc
  g=(0,0)
  gp=(0,0)
  h=(0,0)
  hp=(0,0)
  hpp=(0,0)
  dz=(0,0)
  temp=(0,0)
;
; first polynomial, p(z) = z^m + b
;
  temp=z^(@m-1)
  p=z*temp+@b
  pp=@m*temp
  ppp=(@m-1)*pp/z
;
; second polynomial, q(z) = z^n - b
;
  temp=z^(@n-1)
  q=z*temp-@b
  qp=@n*temp
  qpp=(@n-1)*qp/z
;
; combine p & q to find a
;
  if(@htype==1)              ; q
    hpp=qpp
  elseif(@htype==2)          ; p+q
    hpp=ppp+qpp
  elseif(@htype==3)          ; p-q
    hpp=ppp-qpp
  elseif(@htype==4)          ; q-p
    hpp=qpp-ppp
  elseif(@htype==5)          ; p*q
    hpp=p*qpp+2*pp*qp+ppp*q
  elseif(@htype==6)          ; p/q
    temp=sqr(q)
    hpp=temp*(q*ppp-p*qpp)-2*q*qp*(q*pp-p*qp)
    hpp=hpp/sqr(temp)
  elseif(@htype==7)          ; q/p
    temp=sqr(p)
    hpp=temp*(p*qpp-q*ppp)-2*p*pp*(p*qp-q*pp)
    hpp=hpp/sqr(temp)
  else                       ; p
    hpp=ppp
  endif
  a=-0.5*hpp
  c1=@julparam
  c2=@julparam
loop:
  iter=iter+1
;
;    first test point
;
  if(iter1==0)
;
; first polynomial, p(z) = z^m + b
;
    temp=z1^(@m-1)
    p=z1*temp+@b
    pp=@m*temp
;
; second polynomial, q(z) = z^n - b
;
    temp=z1^(@n-1)
    q=z1*temp-@b
    qp=@n*temp
;
; combine p & q to make h(z)
;
    if(@htype==1)            ; q
      h=q
      hp=qp
    elseif(@htype==2)        ; p+q
      h=p+q
      hp=pp+qp
    elseif(@htype==3)        ; p-q
      h=p-q
      hp=pp-qp
    elseif(@htype==4)        ; q-p
      h=q-p
      hp=qp-pp
    elseif(@htype==5)        ; p*q
      h=p*q
      hp=p*qp+pp*q
    elseif(@htype==6)        ; p/q
      h=p/q
      hp=(q*pp-p*qp)/sqr(q)
    elseif(@htype==7)        ; q/p
      h=q/p
      hp=(p*qp-q*pp)/sqr(p)
    else                     ; p
      h=p
      hp=pp
    endif
;
; add quadratic to make g(z)
;
    g=h+a*sqr(z1)+c1
    gp=hp+2*a*z1
;
; Newton iteration; check 1/(change in z) for bailout
;
    dz=g/gp
    z1=z1-dz
    temp=1/dz
    if(|temp|>@bailout)
      iter1=iter
    endif
  endif
;
;    second test point
;
  if(iter2==0)
;
; first polynomial, p(z) = z^m + b
;
    temp=z2^(@m-1)
    p=z2*temp+@b
    pp=@m*temp
;
; second polynomial, q(z) = z^n - b
;
    temp=z2^(@n-1)
    q=z2*temp-@b
    qp=@n*temp
;
; combine p & q to make h(z)
;
    if(@htype==1)            ; q
      h=q
      hp=qp
    elseif(@htype==2)        ; p+q
      h=p+q
      hp=pp+qp
    elseif(@htype==3)        ; p-q
      h=p-q
      hp=pp-qp
    elseif(@htype==4)        ; q-p
      h=q-p
      hp=qp-pp
    elseif(@htype==5)        ; p*q
      h=p*q
      hp=p*qp+pp*q
    elseif(@htype==6)        ; p/q
      h=p/q
      hp=(q*pp-p*qp)/sqr(q)
    elseif(@htype==7)        ; q/p
      h=q/p
      hp=(p*qp-q*pp)/sqr(p)
    else                     ; p
      h=p
      hp=pp
    endif
;
; add quadratic to make g(z)
;
    g=h+a*sqr(z2)+c2
    gp=hp+2*a*z2
;
; Newton iteration; check 1/(change in z) for bailout
;
    dz=g/gp
    z2=z2-dz
    temp=1/dz
    if(|temp|>@bailout)
      iter2=iter
    endif
  endif
  if(@emboss==false)         ; no embossing--set z to:
    if(@ztype==1)            ;   1/dz
      z=temp
    elseif(@ztype==2)        ;   derivative
      z=gp
    else                     ;   iterate
      z=z1
    endif
  else                       ; embossing--z = iter1 & iter2
    z=iter1+flip(iter2)
  endif
bailout:
  (iter1==0)||(iter2==0)
default:
  title="Rational Newton Julia"
  method=multipass
  maxiter=100
  periodicity=0
  center=(0,0)
  magn=1
;
; rational newton julia stuff
;
  heading
    caption="Rational Newton parameters"
  endheading
  param julparam
    caption="Julia parameter"
    default=(0,0)
  endparam
  param zc
    caption="critical point"
    default=(1,0)
    hint="Keep to a fairly small value."
  endparam
  param bailout
    caption="bailout"
    default=1000.0
    hint="Set to a large value."
  endparam
  param m
    caption="m"
    default=8.0
    hint="Exponent in p(z) function; p(z) = z^m + b."
  endparam
  param n
    caption="n"
    default=4.0
    hint="Exponent in q(z) function; q(z) = z^n - b."
  endparam
  param b
    caption="b"
    default=(-1,0)
    hint="Constant in p(z) and q(z) functions.  Doesn't do anything when \
      'h(z) type' is set to 'p+q'."
  endparam
  param htype
    caption="h(z) type"
    default=2
    enum="p" "q" "p+q" "p-q" "q-p" "p*q" "p/q" "q/p"
    hint="How the p(z) and q(z) functions are combined to make h(z)."
  endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "1/change in z" "derivative"
    hint="Determines what value to send to coloring routine, \
      if the image is not embossed."
    visible=(@emboss==false)
  endparam
;
; embossing stuff
;
  heading
    caption="Embossing parameters"
  endheading
  bool param emboss
    caption="embossed image"
    default=false
  endparam
  float param sizefac
    caption="contour size"
    default=1.0
    hint="Specifies the relative size of the contour bands. Larger values \
          give thicker bands."
    visible=(@emboss==true)
  endparam
  float param lightangle
    caption="light angle"
    default=0.0
    hint="Angle of apparent light source, in degrees."
    visible=(@emboss==true)
  endparam
switch:
  type="rational-newton-mandelbrot"
  zc=zc
  bailout=bailout
  m=m
  n=n
  b=b
  htype=htype
  ztype=ztype
  emboss=emboss
  sizefac=sizefac
  lightangle=lightangle
}

caramel-popcorn { ; Kerry Mitchell 09Aug2003
;
; This is a complex-valued implementation and generalization
; of Cliff Pickover's "popcorn" algorithm.  His method iterates
; 2 real variables like this:
;
; xnew = xold - h * sin(a*y + tan(b*y))
; ynew = yold - h * sin(a*x + tan(b*x))
;
; where a = 1. The current method generalizes the above in 
; several ways:
;   - The variables x and y are complex.
;   - The parameters a, b, and h are all complex.
;   - The h coefficient has been changed to +1 in both equations.
;   - The outer function (sin) and the inner function (tan)
;     are user-specified.
;   - Instead of using y in the x equation or x in the y
;     equation, the user can choose to use a function of x,
;     a function of y, or a function of z (x+i*y) for each.
;
; In addition, several bailout methods have been included:
;   - if x gets large
;   - if y gets large
;   - if z gets large
;   - if either x or y get large
;   - if both x and y get large
;   - if the argument to the x outer function gets large
;   - if the argument to the y outer function gets large
;   - if either arguments get large
;   - if both arguments get large
;
; The latter 4 methods were added because, in the standard
; implementation using the tan() function as the inner function,
; large values of tan() can give visual noise in the image.
; Using these bailout methods can reduce or eliminate those
; features.
;
; To prevent the user from being overwhelmed with all this,
; there is a "controls" parameter.  Set it to "less" (default)
; to see a greatly limited set of choices, or set it to "more"
; to have access to all the parameters.
;
; Update history:
; First published 09 August 2003
; 
init:
  complex temp=(0,0)
  complex temp2=(0,0)
  complex x=real(#pixel)
  complex y=imag(#pixel)
  complex xnew=(0,0)
  complex ynew=(0,0)
  complex z=#pixel
  float r=0.0
  float rx=0.0
  float ry=0.0
loop:
;
; x equation
;
  if(@control=="more")
;
;   choose what variable to use
;
    if(@vartypex==1)           ; f(y)
      temp=@varfnx(y)
    elseif(@vartypex==2)       ; f(z)
      temp=@varfnx(z)
    else                       ; f(x)
      temp=@varfnx(x)
    endif
;
;   compute the inner function and its magnitude for the
;   "argument" bailout types
;
    temp2=@innerfnx(@bx*temp)
    rx=|temp2|
    xnew=x+@hx*@outerfnx(@ax*temp+temp2)
  else
    temp2=@innerfn(@b*y)
    xnew=x+@h*@outerfn(y+temp2)
  endif
;
; y equation
;
  if(@control=="more")
;
;   choose what variable to use
;
    if(@vartypey==1)           ; f(y)
      temp=@varfny(y)
    elseif(@vartypey==2)       ; f(z)
      temp=@varfny(z)
    else                       ; f(x)
      temp=@varfny(x)
    endif
;
;   compute the inner function and its magnitude for the
;   "argument" bailout types
;
    temp2=@innerfny(@by*temp)
    ry=|temp2|
    ynew=y+@hy*@outerfny(@ay*temp+temp2)
  else
    temp2=@innerfn(@b*x)
    ynew=y+@h*@outerfn(x+temp2)
  endif
;
; set z
;
  x=xnew
  y=ynew
  z=x+flip(y)
;
; check for bailout
;
  if(@control=="more")
    if(@bailtype==1)           ; x 
      r=|x|
    elseif(@bailtype==2)       ; y
      r=|y|
    elseif(@bailtype==3)       ; x or y
      r=|x|
      if(|y|>r)
        r=|y|
      endif
    elseif(@bailtype==4)       ; x and y
      r=|x|
      if(|y|<r)
        r=|y|
      endif
    elseif(@bailtype==5)       ; x argument
      r=rx
    elseif(@bailtype==6)       ; y argument
      r=ry
    elseif(@bailtype==7)       ; either
      r=rx
      if(ry>r)
        r=ry
      endif
    elseif(@bailtype==8)       ; both
      r=rx
      if(ry<r)
        r=ry
      endif
    else                       ; z
      r=|z|
    endif
  else
    r=|z|
  endif
bailout:
  r<@bailout
default:
  title="Caramel Popcorn"
  maxiter=16
  method=multipass
  periodicity=0
;
; general parameters
;
  heading
    caption="General parameters"
  endheading
  param control
    caption="controls"
    default=0
    enum="less" "more"
  endparam
  param bailtype
    caption="bailout type"
    default=0
    enum="z" "x" "y" "x or y" "x and y" "x argument"\
      "y argument" "either argument" "both arguments"
    visible=(@control=="more")
  endparam
  float param bailout
    caption="bailout value"
    default=1000.0
  endparam
;
; less controls
;
  heading
    caption="Popcorn parameters"
    visible=(@control=="less")
  endheading
  complex param h
    caption="step size"
    default=(0.1,0)
    visible=(@control=="less")
  endparam
  complex param b
    caption="frequency"
    default=(3,0)
    visible=(@control=="less")
  endparam
  func outerfn
    caption="outer function"
    default=sin()
    visible=(@control=="less")
  endfunc
  func innerfn
    caption="inner function"
    default=tan()
    visible=(@control=="less")
  endfunc
;
; x parameters
;
  heading
    caption="x parameters"
    visible=(@control=="more")
  endheading
  complex param hx
    caption="step size"
    default=(0.1,0)
    visible=(@control=="more")
  endparam
  complex param ax
    caption="slope"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param bx
    caption="frequency"
    default=(3.1415926535897932384626433832795,0)
    visible=(@control=="more")
  endparam
  param vartypex
    caption="variable type"
    default=1
    enum="f(x)" "f(y)" "f(x+iy)"
    visible=(@control=="more")
  endparam
  func varfnx
    caption="variable function"
    default=ident()
    visible=(@control=="more")
  endfunc
  func outerfnx
    caption="outer function"
    default=sin()
    visible=(@control=="more")
  endfunc
  func innerfnx
    caption="inner function"
    default=tan()
    visible=(@control=="more")
  endfunc
;
; y parameters
;
  heading
    caption="y parameters"
    visible=(@control=="more")
  endheading
  complex param hy
    caption="step size"
    default=(0.1,0)
    visible=(@control=="more")
  endparam
  complex param ay
    caption="slope"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param by
    caption="frequency"
    default=(3.1415926535897932384626433832795,0)
    visible=(@control=="more")
  endparam
  param vartypey
    caption="variable type"
    default=0
    enum="f(x)" "f(y)" "f(x+iy)"
    visible=(@control=="more")
  endparam
  func varfny
    caption="variable function"
    default=ident()
    visible=(@control=="more")
  endfunc
  func outerfny
    caption="outer function"
    default=sin()
    visible=(@control=="more")
  endfunc
  func innerfny
    caption="inner function"
    default=tan()
    visible=(@control=="more")
  endfunc
}

caramel-popcorn-3axis { ; Kerry Mitchell 26oct2003
;
; This is a 3-axis, complex-valued generalization of Cliff 
; Pickover's "popcorn" algorithm.  His method iterates 2 real 
; variables like this:
;
; xnew = xold - h * sin(a*y + tan(b*y))
; ynew = yold - h * sin(a*x + tan(b*x))
;
; where a = 1. The current method generalizes the above in 
; several ways:
;   - There are now three variables: x1, x2, and x3.
;   - The variables are complex.
;   - The parameters a, b, and h are all complex.
;   - The h coefficient has been changed to +1 in both equations.
;   - The outer function (sin) and the inner function (tan)
;     are user-specified.
;   - The user can choose which variable to use inside the
;     outer function.  The choices are:  x1, x2, x3, x1 & x2,
;     x2 & x3, x3 & x1, and all 3.  In the 2 variable choices,
;     the new inside variable is variable1 + flip(variable2).
;     In the "all 3" choice, x1, x2 and x3 are all offset 120
;     degrees from each other and added together.  Whatever the
;     variable choice, it can be modified by a user-specified
;     function.
;
; The z variable, that what gets passed to the coloring routine,
; can be specified in the same ways that the inside variable can.
;
; In addition, several bailout methods have been included:
;   - if the z variable gets large
;   - if x1 gets large
;   - if x2 gets large
;   - if x3 gets large
;   - if either x1, x2 or x3 gets large
;   - if all three get large
;   - if the argument to the x1 outer function gets large
;   - if the argument to the x2 outer function gets large
;   - if the argument to the x3 outer function gets large
;   - if either argument gets large
;   - if all 3 arguments get large
;
; The latter 5 methods were added because, in the standard
; implementation using the tan() function as the inner function,
; large values of tan() can give visual noise in the image.
; Using these bailout methods can reduce or eliminate those
; features.
;
; To prevent the user from being overwhelmed with all this,
; there is a "controls" parameter.  Set it to "less" (default)
; to see a greatly limited set of choices, or set it to "more"
; to have access to all the parameters.
;
; Update history:
; First published 26 October 2003
; 
init:
  complex temp=(0,0)
  complex temp2=(0,0)
  complex x1=(0,0)
  complex x2=(0,0)
  complex x3=(0,0)
  complex x1new=(0,0)
  complex x2new=(0,0)
  complex x3new=(0,0)
  complex z=#pixel
  float r=0.0
  float r1=0.0
  float r2=0.0
  float r3=0.0
  float t=#pi/3
  float cost=cos(t)
  float sint=sin(t)
  if(@control=="more")
    if(@ztype=="x1")
      x1=#pixel
      x2=@x2init
      x3=@x3init
    elseif(@ztype=="x2")
      x2=#pixel
      x3=@x3init
      x1=@x1init
    elseif(@ztype=="x3")
      x3=#pixel
      x1=@x1init
      x2=@x2init
    elseif(@ztype=="x1 and x2")
      x1=real(#pixel)
      x2=imag(#pixel)
      x3=@x3init
    elseif(@ztype=="x2 and x3")
      x2=real(#pixel)
      x3=imag(#pixel)
      x1=@x1init
    elseif(@ztype=="x3 and x1")
      x3=real(#pixel)
      x1=imag(#pixel)
      x2=@x2init
    else                     ; phasor diagram
      x1=4*real(#pixel)/3
      x2=(real(#pixel)/3/cost+imag(#pixel)/sint)/2
      x3=(real(#pixel)/3/cost-imag(#pixel)/sint)/2
    endif
  else                       ; phasor diagram
    x1=4*real(#pixel)/3
    x2=(real(#pixel)/3/cost+imag(#pixel)/sint)/2
    x3=(real(#pixel)/3/cost-imag(#pixel)/sint)/2
  endif
loop:
;
; x1 equation
;
  if(@control=="more")
;
;   choose what variable to use
;
    if(@vartype1=="x1")
      temp=x1
    elseif(@vartype1=="x2")
      temp=x2
    elseif(@vartype1=="x3")
      temp=x3
    elseif(@vartype1=="x1 and x2")
      temp=x1+flip(x2)
    elseif(@vartype1=="x2 and x3")
      temp=x2+flip(x3)
    elseif(@vartype1=="x3 and x1")
      temp=x3+flip(x1)
    else                     ; all 3
      z=x1-cost*(x2+x3)+flip(sint*(x2-x3))
    endif
;
;   compute the inner function and its magnitude for the
;   "argument" bailout types
;
    temp2=@innerfn1(@b1*temp)
    r1=|temp2|
    x1new=x1+@h1*@outerfn1(@a1*x2+temp2)
  else
    temp2=@innerfn(@b*x3)
    x1new=x1+@h*@outerfn(x2+temp2)
  endif
;
; x2 equation
;
  if(@control=="more")
;
;   choose what variable to use
;
    if(@vartype2=="x1")
      temp=x1
    elseif(@vartype2=="x2")
      temp=x2
    elseif(@vartype2=="x3")
      temp=x3
    elseif(@vartype2=="x1 and x2")
      temp=x1+flip(x2)
    elseif(@vartype2=="x2 and x3")
      temp=x2+flip(x3)
    elseif(@vartype2=="x3 and x1")
      temp=x3+flip(x1)
    else                     ; all 3
      z=x1-cost*(x2+x3)+flip(sint*(x2-x3))
    endif
;
;   compute the inner function and its magnitude for the
;   "argument" bailout types
;
    temp2=@innerfn2(@b2*temp)
    r2=|temp2|
    x2new=x2+@h2*@outerfn2(@a2*x3+temp2)
  else
    temp2=@innerfn(@b*x1)
    x2new=x2+@h*@outerfn(x3+temp2)
  endif
;
; x3 equation
;
  if(@control=="more")
;
;   choose what variable to use
;
    if(@vartype3=="x1")
      temp=x1
    elseif(@vartype3=="x2")
      temp=x2
    elseif(@vartype3=="x3")
      temp=x3
    elseif(@vartype3=="x1 and x2")
      temp=x1+flip(x2)
    elseif(@vartype3=="x2 and x3")
      temp=x2+flip(x3)
    elseif(@vartype3=="x3 and x1")
      temp=x3+flip(x1)
    else                     ; all 3
      z=x1-cost*(x2+x3)+flip(sint*(x2-x3))
    endif
;
;   compute the inner function and its magnitude for the
;   "argument" bailout types
;
    temp2=@innerfn3(@b3*temp)
    r3=|temp2|
    x3new=x3+@h3*@outerfn3(@a3*x1+temp2)
  else
    temp2=@innerfn(@b*x2)
    x3new=x3+@h*@outerfn(x1+temp2)
  endif
;
; set z
;
  x1=x1new
  x2=x2new
  x3=x3new
  if(@control=="more")
    if(@ztype=="x1")
      z=x1
    elseif(@ztype=="x2")
      z=x2
    elseif(@ztype=="x3")
      z=x3
    elseif(@ztype=="x1 and x2")
      z=x1+flip(x2)
    elseif(@ztype=="x2 and x3")
      z=x2+flip(x3)
    elseif(@ztype=="x3 and x1")
      z=x3+flip(x1)
    else
      z=x1-cost*(x2+x3)+flip(sint*(x2-x3))
    endif
  else
    z=x1-cost*(x2+x3)+flip(sint*(x2-x3))
  endif
;
; check for bailout
;
  if(@control=="more")
    if(@bailtype==1)           ; x1 
      r=|x1|
    elseif(@bailtype==2)       ; x2
      r=|x2|
    elseif(@bailtype==3)       ; x3
      r=|x3|
    elseif(@bailtype==4)       ; any
      r=|x1|
      if(|x2|>r)
        r=|x2|
      endif
      if(|x3|>r)
        r=|x3|
      endif
    elseif(@bailtype==5)       ; all
      r=|x1|
      if(|x2|<r)
        r=|x2|
      endif
      if(|x3|<r)
        r=|x3|
      endif
    elseif(@bailtype==6)       ; x1 argument
      r=r1
    elseif(@bailtype==7)       ; x2 argument
      r=r2
    elseif(@bailtype==8)       ; x3 argument
      r=r3
    elseif(@bailtype==9)       ; any
      r=r1
      if(r2>r)
        r=r2
      endif
      if(r3>r)
        r=r3
      endif
    elseif(@bailtype==10)       ; all
      r=r1
      if(r2<r)
        r=r2
      endif
      if(r3<r)
        r=r3
      endif
    else                       ; z
      r=|z|
    endif
  else
    r=|z|
  endif
bailout:
  r<@bailout
default:
  title="Caramel Popcorn 3 Axis"
  maxiter=16
  method=multipass
  periodicity=0
;
; general parameters
;
  heading
    caption="General parameters"
  endheading
  param control
    caption="controls"
    default=0
    enum="less" "more"
  endparam
  float param bailout
    caption="bailout value"
    default=1000.0
  endparam
;
;   more controls
;
  param bailtype
    caption="bailout type"
    default=0
    enum="z" "x1" "x2" "x3" "x1, x2, or x3" "x1, x2, and x3"\
    "x1 argument" "x2 argument" "x3 argument" \
    "any argument" "all arguments"
    visible=(@control=="more")
  endparam
  param ztype
    caption="z type"
    default=0
    enum="all 3" "x1" "x2" "x3" "x1 and x2" "x2 and x3" "x3 and x1"
    visible=(@control=="more")
  endparam
;
; popcorn parameters
;
  heading
    caption="Popcorn parameters"
  endheading
;
;   less controls
;
  complex param h
    caption="step size"
    default=(1,0)
    visible=(@control=="less")
  endparam
  complex param b
    caption="frequency"
    default=(1,0)
    visible=(@control=="less")
  endparam
  func outerfn
    caption="outer function"
    default=sin()
    visible=(@control=="less")
  endfunc
  func innerfn
    caption="inner function"
    default=ident()
    visible=(@control=="less")
  endfunc
;
;   more controls
;
;     x1 parameters
;
  heading
    caption="x1 parameters"
    visible=(@control=="more")
  endheading
  complex param x1init
    caption="initial value"
    default=(0,0)
    visible=((@control=="more")&&((@ztype=="x2")||\
      (@ztype=="x3")||(@ztype=="x2 and x3")))
  endparam
  complex param h1
    caption="step size"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param a1
    caption="slope"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param b1
    caption="frequency"
    default=(1,0)
    visible=(@control=="more")
  endparam
  param vartype1
    caption="variable type"
    default=3
    enum="all 3" "x1" "x2" "x3" "x1 and x2" "x2 and x3" "x3 and x1"
    visible=(@control=="more")
  endparam
  func varfn1
    caption="variable function"
    default=ident()
    visible=(@control=="more")
  endfunc
  func outerfn1
    caption="outer function"
    default=sin()
    visible=(@control=="more")
  endfunc
  func innerfn1
    caption="inner function"
    default=ident()
    visible=(@control=="more")
  endfunc
;
;     x2 parameters
;
  heading
    caption="x2 parameters"
    visible=(@control=="more")
  endheading
  complex param x2init
    caption="initial value"
    default=(0,0)
    visible=((@control=="more")&&((@ztype=="x1")||\
      (@ztype=="x3")||(@ztype=="x3 and x1")))
  endparam
  complex param h2
    caption="step size"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param a2
    caption="slope"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param b2
    caption="frequency"
    default=(1,0)
    visible=(@control=="more")
  endparam
  param vartype2
    caption="variable type"
    default=1
    enum="all 3" "x1" "x2" "x3" "x1 and x2" "x2 and x3" "x3 and x1"
    visible=(@control=="more")
  endparam
  func varfn2
    caption="variable function"
    default=ident()
    visible=(@control=="more")
  endfunc
  func outerfn2
    caption="outer function"
    default=sin()
    visible=(@control=="more")
  endfunc
  func innerfn2
    caption="inner function"
    default=ident()
    visible=(@control=="more")
  endfunc
;
;     x3 parameters
;
  heading
    caption="x3 parameters"
    visible=(@control=="more")
  endheading
  complex param x3init
    caption="initial value"
    default=(0,0)
    visible=((@control=="more")&&((@ztype=="x1")||\
      (@ztype=="x2")||(@ztype=="x1 and x2")))
  endparam
  complex param h3
    caption="step size"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param a3
    caption="slope"
    default=(1,0)
    visible=(@control=="more")
  endparam
  complex param b3
    caption="frequency"
    default=(1,0)
    visible=(@control=="more")
  endparam
  param vartype3
    caption="variable type"
    default=2
    enum="all 3" "x1" "x2" "x3" "x1 and x2" "x2 and x3" "x3 and x1"
    visible=(@control=="more")
  endparam
  func varfn3
    caption="variable function"
    default=ident()
    visible=(@control=="more")
  endfunc
  func outerfn3
    caption="outer function"
    default=sin()
    visible=(@control=="more")
  endfunc
  func innerfn3
    caption="inner function"
    default=ident()
    visible=(@control=="more")
  endfunc
}

embossed-mandelbrot-rrl { ; Kerry Mitchell 10March2007
;
; This combines the embossed Mandelbrot set with the geometry of the Rose Range
; Lite coloring.  The image can be embossed along the edges of the rose curves
; or along internal sections.  The parameters are very similar to those in the
; embossed Mandelbrot formula and the Rose Range Lite coloring.
;
global:
  bool do_this[round(#maxiter+1)]
  bool fib[round(#maxiter+1)]
  bool tri[round(#maxiter+1)]
  int do_last=0
  int fib1=0
  int fib2=0
  int gfac=0
  int gi=0
  int gj=0
;
; set up selected iterations
;
; clear do_this array
;
  gi=-1
  while(gi<#maxiter)
    gi=gi+1
    do_this[gi]=false
  endwhile
;
; set up counter for final iteration
;
  if((@dolast==0)||(@dolast>#maxiter))
    do_last=#maxiter
  else
    do_last=@dolast
  endif
;
; create array of Fibonacci numbers
;
  gi=0
  while(gi<#maxiter)
    gi=gi+1
    fib[gi]=false
  endwhile
  fib1=0
  fib2=1
  gi=1
  while(gi<#maxiter)
    fib[gi]=true
    fib1=fib2
    fib2=gi
    gi=fib1+fib2
  endwhile
;
; create array of triangular numbers
;
  gi=0
  while(gi<#maxiter)
    gi=gi+1
    tri[gi]=false
  endwhile
  gi=0
  gj=1
  while(gi<#maxiter)
    tri[gi]=true
    gi=gi+gj
    gj=gj+1
  endwhile
;
; set do_this flag if that iteration is to be used
;
  gi=@do_first-1
  while(gi<(do_last-1))
    gi=gi+1
    if(@modtype=="do all") ; do each iteration
      do_this[gi]=true
    elseif(@modtype=="do every n") ; do every nth
      gj=(gi-@do_first+#maxiter*@modbase)%@modbase
      if(gj==0)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip every n") ; skip every nth
      gj=(gi-@do_first+#maxiter*@modbase)%@modbase
      if(gj>0)
        do_this[gi]=true
      endif
    elseif(@modtype=="do primes") ; do if iteration # is prime
      gj=1
      gfac=2
      while(gj<(gi-1))
        gj=gj+1
        if((gi%gj)==0)
          gfac=gfac+1
        endif
      endwhile
      if(gfac==2)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip primes") ; do if iteration # is not prime
      if(gi==1)
        do_this[gi]=true
      endif
      gj=1
      gfac=2
      while(gj<(gi-1))
        gj=gj+1
        if((gi%gj)==0)
          gfac=gfac+1
        endif
      endwhile
      if(gfac>2)
        do_this[gi]=true
      endif
    elseif(@modtype=="do Fibonacci") ; do if iteration is a Fibonacci #
      if(fib[gi]==true)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip Fibonacci") ; do if iteration is not a Fibonacci #
      if(fib[gi]==false)
        do_this[gi]=true
      endif
    elseif(@modtype=="do triangular") ; do if iteration is a triangular #
      if(tri[gi]==true)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip triangular") ; do if iteration is not a triangular #
      if(tri[gi]==false)
        do_this[gi]=true
      endif
    elseif(@modtype=="blocks") ; blocks of trap some, then skip some
      gj=((gi-@do_first)%@block_size)+1
      if(gj<=@block_trap)
        do_this[gi]=true
      endif
    endif
  endwhile
;
; print do_this
;
  if(@print_do_this==true)
    gi=0
    while(gi<#maxiter)
      gi=gi+1
      if(do_this[gi]==true)
        print(gi)
      endif
    endwhile
  endif
init:
  bool keepgoing1=true
  bool keepgoing2=true
  complex c1=(0,0)
  complex c2=(0,0)
  complex dr=(0,0)
  complex z=(0,0)
  complex z1=@manparam
  complex z2=@manparam
  float part1=0
  float part2=0
  float r11=0
  float r12=0
  float rfaze=@rphase/180*#pi
  float rho=0
  float rmax=0
  float rmaxfac=@scale*(1+0.5*@rangewidth)
  float rmin=0
  float rminfac=@scale*(1-0.5*@rangewidth)
  float rn1=0
  float rn2=0
  float rotangle=-@rotd/180*#pi
  float rw=0
  float rz=0
  float size=0
  float t11=0
  float t12=0
  float theta=0
  float tn1=0
  float tn2=0
  float tz=0
  int iter=0
  int iter11=0
  int iter12=0
  int itern1=0
  int itern2=0
  int rangeiter1=0
  int rangeiter2=0
;
; set up embossing
;
  theta=@lightangle*#pi/180
  size=@sizefac*0.0065/#magn
  dr=size*(cos(theta)+flip(sin(theta)))
  c1=#pixel-dr
  c2=#pixel+dr
loop:
  iter=iter+1
;
; iterate first point
;
  if(keepgoing1==true)
    z1=z1^@power+c1
    if(|z1|>@bailout)
      keepgoing1=false
    endif
;
;   check iterate against rose curve
;
    if(do_this[iter]==true)
      tz=atan2(z1)+rotangle
      rz=@ac*cos(@bc*tz)+@as*sin(@bs*tz)
      if(@vary_width==true)
        rho=(1-cos(@rfreq*tz+rfaze))/2
        rw=rho*(@maxwidth-@minwidth)+@minwidth
        rminfac=@scale*(1-0.5*rw)
        rmaxfac=@scale*(1+0.5*rw)
      endif
      rmin=rz*rminfac
      rmax=rz*rmaxfac
      rz=cabs(z1-@curvecenter)
      if((rz>=rmin)&&(rz<=rmax))
        rangeiter1=rangeiter1+1
        rn1=rz
        tn1=tz
        itern1=iter
        if(rangeiter1==1)
          r11=rn1
          t11=tn1
          iter11=iter
        endif
      endif
    endif
  endif
;
; iterate second point
;
  if(keepgoing2==true)
    z2=z2^@power+c2
    if(|z2|>@bailout)
      keepgoing2=false
    endif
;
;   check iterate against rose curve
;
    if(do_this[iter]==true)
      tz=atan2(z2)+rotangle
      rz=@ac*cos(@bc*tz)+@as*sin(@bs*tz)
      rmin=rz*rminfac
      rmax=rz*rmaxfac
      rz=cabs(z2-@curvecenter)
      if((rz>=rmin)&&(rz<=rmax))
        rangeiter2=rangeiter2+1
        rn2=rz
        tn2=tz
        itern2=iter
        if(rangeiter2==1)
          r12=rn2
          t12=tn2
          iter12=iter
        endif
      endif
    endif
  endif
;
; assign parts of z
;
  if(@parttype==0)
    part1=rangeiter1
    part2=rangeiter2
  elseif(@parttype==1)
    part1=iter11
    part2=iter12
  elseif(@parttype==2)
    part1=itern1
    part2=itern2
  elseif(@parttype==3)
    part1=trunc(r11*@sections)
    part2=trunc(r12*@sections)
  elseif(@parttype==4)
    part1=trunc(rn1*@sections)
    part2=trunc(rn2*@sections)
  elseif(@parttype==5)
    part1=trunc(t11*@sections)
    part2=trunc(t12*@sections)
  elseif(@parttype==6)
    part1=trunc(tn1*@sections)
    part2=trunc(tn2*@sections)
  endif
  z=part1+flip(part2)
bailout:
  (keepgoing1==true)||(keepgoing2==true)
default:
  title="Embossed Mandelbrot - Rose Range Lite"
  method=multipass
  periodicity=0
;
; Mandelbrot parameters
;
  heading
    caption="Mandelbrot parameters"
  endheading
  complex param manparam
    caption="initial z"
    default=(0,0)
  endparam
  complex param power
    caption="power"
    default=(2,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; embossing parameters
;
  heading
    caption="Embossing parameters"
  endheading
  param parttype
    caption="emboss type"
    default=0
    enum="in range" "first iteration" "last iteration" "first magnitude" \
      "last magnitude" "first angle" "last angle"
  endparam
  float param sections
    caption="# sections"
    default=2
    visible=(@parttype>2)
  endparam
  float param lightangle
    caption="light angle"
    default=0
    hint="Angle of apparant light source, in degrees"
  endparam
  float param sizefac
    caption="contour size"
    default=1
    hint="relative size of contours bands"
  endparam
;
; rose range lite parameters
;
  heading
    caption="Rose Range Lite parameters"
  endheading
  float param scale
    caption="range scale"
    default=1
  endparam
  bool param vary_width
    caption="vary width"
    default=false
  endparam
  float param rangewidth
    caption="range width"
    default=0.1
    visible=(@vary_width==false)
  endparam
  float param minwidth
    caption="min range width"
    default=0.1
    visible=(@vary_width==true)
  endparam
  float param maxwidth
    caption="max range width"
    default=0.4
    visible=(@vary_width==true)
  endparam
  float param rfreq
    caption="range frequency"
    default=1
    visible=(@vary_width==true)
  endparam
  float param rphase
    caption="range phase, deg"
    default=0
    visible=(@vary_width==true)
  endparam
  float param ac
    caption="cos amplitude"
    default=1
  endparam
  float param bc
    caption="cos frequency"
    default=3
  endparam
  float param as
    caption="sin amplitude"
    default=0
  endparam
  float param bs
    caption="sin frequency"
    default=0
  endparam
  complex param curvecenter
    caption="curve center"
    default=(0,0)
  endparam
  float param rotd
    caption="rotation angle"
    default=0
    hint="degrees"
  endparam
;
; selected iterations paramaters
;
  heading
    caption="Selected Iterations parameters"
  endheading
  param modtype
    caption="mod type"
    default=0
    enum="do all" "do every n" "skip every n" "do primes" "skip primes"\
      "do Fibonacci" "skip Fibonacci" "do triangular" "skip triangular" \
      "blocks"
  endparam
  int param modbase
    caption="n"
    default=2
    min=2
    hint="The 'n' for type 'every n'; at least 2."
    visible=((@modtype=="do every n")||(@modtype=="skip every n"))
  endparam
  int param block_size
    caption="block size"
    default=4
    min=3
    visible=(@modtype=="blocks")
  endparam
  int param block_trap
    caption="block trap"
    default=2
    min=1
    visible=(@modtype=="blocks")
  endparam
  int param do_first
    caption="first iteration"
    default=1
    min=1
    hint="Start with this iteration."
  endparam
  int param dolast
    caption="last iteration"
    default=0
    min=0
    hint="End on this iteration; use 0 for maximum iterations."
  endparam
  bool param print_do_this
    caption="print iterations"
    default=false
    hint="Print those iterations that will be calculated"
  endparam
switch:
  type="embossed-julia-rrl"
  julparam=#pixel
  ac=ac
  as=as
  bailout=bailout
  bc=bc
  block_size=block_size
  block_trap=block_trap
  bs=bs
  curvecenter=curvecenter
  do_first=do_first
  dolast=dolast
  julparam=julparam
  lightangle=lightangle
  maxwidth=maxwidth
  minwidth=minwidth
  modbase=modbase
  modtype=modtype
  parttype=parttype
  power=power
  print_do_this=print_do_this
  rangewidth=rangewidth
  rfreq=rfreq
  rotd=rotd
  rphase=rphase
  scale=scale
  sections=sections
  sizefac=sizefac
  vary_width=vary_width
}

embossed-julia-rrl { ; Kerry Mitchell 10March2007
;
; This combines the embossed Julia set with the geometry of the Rose Range
; Lite coloring.  The image can be embossed along the edges of the rose curves
; or along internal sections.  The parameters are very similar to those in the
; embossed Julia formula and the Rose Range Lite coloring.
;
global:
  bool do_this[round(#maxiter+1)]
  bool fib[round(#maxiter+1)]
  bool tri[round(#maxiter+1)]
  int do_last=0
  int fib1=0
  int fib2=0
  int gfac=0
  int gi=0
  int gj=0
;
; set up selected iterations
;
; clear do_this array
;
  gi=-1
  while(gi<#maxiter)
    gi=gi+1
    do_this[gi]=false
  endwhile
;
; set up counter for final iteration
;
  if((@dolast==0)||(@dolast>#maxiter))
    do_last=#maxiter
  else
    do_last=@dolast
  endif
;
; create array of Fibonacci numbers
;
  gi=0
  while(gi<#maxiter)
    gi=gi+1
    fib[gi]=false
  endwhile
  fib1=0
  fib2=1
  gi=1
  while(gi<#maxiter)
    fib[gi]=true
    fib1=fib2
    fib2=gi
    gi=fib1+fib2
  endwhile
;
; create array of triangular numbers
;
  gi=0
  while(gi<#maxiter)
    gi=gi+1
    tri[gi]=false
  endwhile
  gi=0
  gj=1
  while(gi<#maxiter)
    tri[gi]=true
    gi=gi+gj
    gj=gj+1
  endwhile
;
; set do_this flag if that iteration is to be used
;
  gi=@do_first-1
  while(gi<(do_last-1))
    gi=gi+1
    if(@modtype=="do all") ; do each iteration
      do_this[gi]=true
    elseif(@modtype=="do every n") ; do every nth
      gj=(gi-@do_first+#maxiter*@modbase)%@modbase
      if(gj==0)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip every n") ; skip every nth
      gj=(gi-@do_first+#maxiter*@modbase)%@modbase
      if(gj>0)
        do_this[gi]=true
      endif
    elseif(@modtype=="do primes") ; do if iteration # is prime
      gj=1
      gfac=2
      while(gj<(gi-1))
        gj=gj+1
        if((gi%gj)==0)
          gfac=gfac+1
        endif
      endwhile
      if(gfac==2)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip primes") ; do if iteration # is not prime
      if(gi==1)
        do_this[gi]=true
      endif
      gj=1
      gfac=2
      while(gj<(gi-1))
        gj=gj+1
        if((gi%gj)==0)
          gfac=gfac+1
        endif
      endwhile
      if(gfac>2)
        do_this[gi]=true
      endif
    elseif(@modtype=="do Fibonacci") ; do if iteration is a Fibonacci #
      if(fib[gi]==true)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip Fibonacci") ; do if iteration is not a Fibonacci #
      if(fib[gi]==false)
        do_this[gi]=true
      endif
    elseif(@modtype=="do triangular") ; do if iteration is a triangular #
      if(tri[gi]==true)
        do_this[gi]=true
      endif
    elseif(@modtype=="skip triangular") ; do if iteration is not a triangular #
      if(tri[gi]==false)
        do_this[gi]=true
      endif
    elseif(@modtype=="blocks") ; blocks of trap some, then skip some
      gj=((gi-@do_first)%@block_size)+1
      if(gj<=@block_trap)
        do_this[gi]=true
      endif
    endif
  endwhile
;
; print do_this
;
  if(@print_do_this==true)
    gi=0
    while(gi<#maxiter)
      gi=gi+1
      if(do_this[gi]==true)
        print(gi)
      endif
    endwhile
  endif
init:
  bool keepgoing1=true
  bool keepgoing2=true
  complex c1=@julparam
  complex c2=@julparam
  complex dr=(0,0)
  complex z=(0,0)
  complex z1=(0,0)
  complex z2=(0,0)
  float part1=0
  float part2=0
  float r11=0
  float r12=0
  float rfaze=@rphase/180*#pi
  float rho=0
  float rmax=0
  float rmaxfac=@scale*(1+0.5*@rangewidth)
  float rmin=0
  float rminfac=@scale*(1-0.5*@rangewidth)
  float rn1=0
  float rn2=0
  float rotangle=-@rotd/180*#pi
  float rw=0
  float rz=0
  float size=0
  float t11=0
  float t12=0
  float theta=0
  float tn1=0
  float tn2=0
  float tz=0
  int iter=0
  int iter11=0
  int iter12=0
  int itern1=0
  int itern2=0
  int rangeiter1=0
  int rangeiter2=0
;
; set up embossing
;
  theta=@lightangle*#pi/180
  size=@sizefac*0.0065/#magn
  dr=size*(cos(theta)+flip(sin(theta)))
  z1=#pixel-dr
  z2=#pixel+dr
loop:
  iter=iter+1
;
; iterate first point
;
  if(keepgoing1==true)
    z1=z1^@power+c1
    if(|z1|>@bailout)
      keepgoing1=false
    endif
;
;   check iterate against rose curve
;
    if(do_this[iter]==true)
      tz=atan2(z1)+rotangle
      rz=@ac*cos(@bc*tz)+@as*sin(@bs*tz)
      if(@vary_width==true)
        rho=(1-cos(@rfreq*tz+rfaze))/2
        rw=rho*(@maxwidth-@minwidth)+@minwidth
        rminfac=@scale*(1-0.5*rw)
        rmaxfac=@scale*(1+0.5*rw)
      endif
      rmin=rz*rminfac
      rmax=rz*rmaxfac
      rz=cabs(z1-@curvecenter)
      if((rz>=rmin)&&(rz<=rmax))
        rangeiter1=rangeiter1+1
        rn1=rz
        tn1=tz
        itern1=iter
        if(rangeiter1==1)
          r11=rn1
          t11=tn1
          iter11=iter
        endif
      endif
    endif
  endif
;
; iterate second point
;
  if(keepgoing2==true)
    z2=z2^@power+c2
    if(|z2|>@bailout)
      keepgoing2=false
    endif
;
;   check iterate against rose curve
;
    if(do_this[iter]==true)
      tz=atan2(z2)+rotangle
      rz=@ac*cos(@bc*tz)+@as*sin(@bs*tz)
      rmin=rz*rminfac
      rmax=rz*rmaxfac
      rz=cabs(z2-@curvecenter)
      if((rz>=rmin)&&(rz<=rmax))
        rangeiter2=rangeiter2+1
        rn2=rz
        tn2=tz
        itern2=iter
        if(rangeiter2==1)
          r12=rn2
          t12=tn2
          iter12=iter
        endif
      endif
    endif
  endif
;
; assign parts of z
;
  if(@parttype==0)
    part1=rangeiter1
    part2=rangeiter2
  elseif(@parttype==1)
    part1=iter11
    part2=iter12
  elseif(@parttype==2)
    part1=itern1
    part2=itern2
  elseif(@parttype==3)
    part1=trunc(r11*@sections)
    part2=trunc(r12*@sections)
  elseif(@parttype==4)
    part1=trunc(rn1*@sections)
    part2=trunc(rn2*@sections)
  elseif(@parttype==5)
    part1=trunc(t11*@sections)
    part2=trunc(t12*@sections)
  elseif(@parttype==6)
    part1=trunc(tn1*@sections)
    part2=trunc(tn2*@sections)
  endif
  z=part1+flip(part2)
bailout:
  (keepgoing1==true)||(keepgoing2==true)
default:
  title="Embossed Julia - Rose Range Lite"
  method=multipass
  periodicity=0
;
; Julia parameters
;
  heading
    caption="Julia parameters"
  endheading
  complex param julparam
    caption="Julia parameter"
    default=(0,1)
  endparam
  complex param power
    caption="power"
    default=(2,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; embossing parameters
;
  heading
    caption="Embossing parameters"
  endheading
  param parttype
    caption="emboss type"
    default=0
    enum="in range" "first iteration" "last iteration" "first magnitude" \
      "last magnitude" "first angle" "last angle"
  endparam
  float param sections
    caption="# sections"
    default=2
    visible=(@parttype>2)
  endparam
  float param lightangle
    caption="light angle"
    default=0
    hint="Angle of apparant light source, in degrees"
  endparam
  float param sizefac
    caption="contour size"
    default=1
    hint="relative size of contours bands"
  endparam
;
; rose range lite parameters
;
  heading
    caption="Rose Range Lite parameters"
  endheading
  float param scale
    caption="range scale"
    default=1
  endparam
  bool param vary_width
    caption="vary width"
    default=false
  endparam
  float param rangewidth
    caption="range width"
    default=0.1
    visible=(@vary_width==false)
  endparam
  float param minwidth
    caption="min range width"
    default=0.1
    visible=(@vary_width==true)
  endparam
  float param maxwidth
    caption="max range width"
    default=0.4
    visible=(@vary_width==true)
  endparam
  float param rfreq
    caption="range frequency"
    default=1
    visible=(@vary_width==true)
  endparam
  float param rphase
    caption="range phase, deg"
    default=0
    visible=(@vary_width==true)
  endparam
  float param ac
    caption="cos amplitude"
    default=1
  endparam
  float param bc
    caption="cos frequency"
    default=3
  endparam
  float param as
    caption="sin amplitude"
    default=0
  endparam
  float param bs
    caption="sin frequency"
    default=0
  endparam
  complex param curvecenter
    caption="curve center"
    default=(0,0)
  endparam
  float param rotd
    caption="rotation angle"
    default=0
    hint="degrees"
  endparam
;
; selected iterations paramaters
;
  heading
    caption="Selected Iterations parameters"
  endheading
  param modtype
    caption="mod type"
    default=0
    enum="do all" "do every n" "skip every n" "do primes" "skip primes"\
      "do Fibonacci" "skip Fibonacci" "do triangular" "skip triangular" \
      "blocks"
  endparam
  int param modbase
    caption="n"
    default=2
    min=2
    hint="The 'n' for type 'every n'; at least 2."
    visible=((@modtype=="do every n")||(@modtype=="skip every n"))
  endparam
  int param block_size
    caption="block size"
    default=4
    min=3
    visible=(@modtype=="blocks")
  endparam
  int param block_trap
    caption="block trap"
    default=2
    min=1
    visible=(@modtype=="blocks")
  endparam
  int param do_first
    caption="first iteration"
    default=1
    min=1
    hint="Start with this iteration."
  endparam
  int param dolast
    caption="last iteration"
    default=0
    min=0
    hint="End on this iteration; use 0 for maximum iterations."
  endparam
  bool param print_do_this
    caption="print iterations"
    default=false
    hint="Print those iterations that will be calculated"
  endparam
switch:
  type="embossed-mandelbrot-rrl"
  ac=ac
  as=as
  bailout=bailout
  bc=bc
  block_size=block_size
  block_trap=block_trap
  bs=bs
  curvecenter=curvecenter
  do_first=do_first
  dolast=dolast
  julparam=julparam
  lightangle=lightangle
  maxwidth=maxwidth
  minwidth=minwidth
  modbase=modbase
  modtype=modtype
  parttype=parttype
  power=power
  print_do_this=print_do_this
  rangewidth=rangewidth
  rfreq=rfreq
  rotd=rotd
  rphase=rphase
  scale=scale
  sections=sections
  sizefac=sizefac
  vary_width=vary_width
}

comment {

Relativistic Addition

These formulas are based on the addition of velocities according to special
relativity.

If you took physics, you probably learned the Galilean method of adding
velocities.  Say you're riding in a bus at 60 mph and you roll a bowling ball
forward at 20 mph relative to you.  Relative to an observer outside of the bus,
the ball has its own velocity plus that of the bus, so would be moving at 80 mph.
This is fine for most situations, but falls apart when things move very quickly.
If that same bus were moving along at a speedy 600,000,000 mph (about 90% of the
speed of light) and you managed to roll the ball at 200,000,000 mph relative to
you, then the Galilean method would have the ball moving at 800,000,000 mph r
elative to an observer outside of the bus, or about 20% faster than the speed of
light.  According to Einstein's theory of relativity, nothing can move faster
than light.  To accommodate this, velocities don't add regularly, but through
what is known as a Lorentzian transformation.  Using such a transformation here,
the bowling ball would be moving at 94% of the speed of light, relative to the
outside observer.

Under relativistic addition, if two motions are added and both are slower than
the speed of light, then the sum will always be slower as well.  If one motion
is at the speed of the light, then the sum will be, as well.  This can be
applied to the standard Mandelbrot/Julia formula z = z^n + c by considering
each of the terms on the right side of the equation to be a velocity and adding
them relativistically.  Then, if they are both smaller in magnitude than the
"speed of light" parameter, the result will be as well.  This results in some
very interesting warping of the fractal for small light speeds.  As the speed
parameter increases, the warping decreases.

The other major aspect of this is the choice of observer.  In the case of two
independent motions, the observer can be moving with either particle or
independent of both.  The sum of the two velocities will depend on the frame
of reference of the observer.  In these formulas, there are choices for that
frame of reference.  There is also a simplification of the relativistic equation
that is used in the case of collinear motion (both motions in the same
direction).  If you choose that method, then the assumed motion can be in the
direction of c, of z^n, or in some independent direction.

}

relativistic-addition-julia { ; Kerry Mitchell 03aug2007
;
; Replace the standard addition with relativistic addition
; both addends are treated like velocities
; use the full-blown equation found in wikipedia
;
; http://en.wikipedia.org/wiki/Velocity-addition_formula
;
init:
  complex c=@julparam
  complex v1=(0,0)
  complex v2=(0,0)
  complex w=(0,0)
  complex z=#pixel
  float fac1=0
  float fac2=0
  float fac3=0
  float recip_speed2=1/sqr(@speed)
loop:
  w=z^@power
  if(@vtype=="c frame")
    v1=c
    v2=w
    fac1=1+(real(v1)*real(v2)+imag(v1)*imag(v2))*recip_speed2
    fac2=recip_speed2*(1+sqrt(1-|v1|*recip_speed2))
    fac3=real(v1)*imag(v2)-real(v2)*imag(v1)
    z=(v1+v2)/fac1+fac2*(imag(v1)*fac3-flip(real(v1)*fac3))/fac1
  elseif(@vtype=="z frame")
    v2=c
    v1=w
    fac1=1+(real(v1)*real(v2)+imag(v1)*imag(v2))*recip_speed2
    fac2=recip_speed2*(1+sqrt(1-|v1|*recip_speed2))
    fac3=real(v1)*imag(v2)-real(v2)*imag(v1)
    z=(v1+v2)/fac1+fac2*(imag(v1)*fac3-flip(real(v1)*fac3))/fac1
  elseif(@vtype=="collinear c")
    if(cabs(c)==0)
      z=w
    else
      v1par=c
      v1perp=0
      v2par=(real(w)*real(c)+imag(w)*imag(c))/|c|*c
      v2perp=w-v2par
      z=(v1par+v2par)/(1+(real(v1par)*real(v2par)+imag(v1par)*imag(v2par))*\
        recip_speed2)+v1perp+v2perp
    endif
  elseif(@vtype=="collinear z")
    if(cabs(w)==0)
      z=c
    else
      v1par=w
      v1perp=0
      v2par=(real(c)*real(w)+imag(c)*imag(w))/|w|*w
      v2perp=c-v2par
      z=(v1par+v2par)/(1+(real(v1par)*real(v2par)+imag(v1par)*imag(v2par))*\
        recip_speed2)+v1perp+v2perp
    endif
  elseif(@vtype=="collinear observer")
    if(cabs(@observer)==0)
      z=w+c
    else
      v1par=(real(w)*real(@observer)+imag(w)*imag(@observer))/|@observer|*@observer
      v1perp=w-v1par
      v2par=(real(c)*real(@observer)+imag(c)*imag(@observer))/|@observer|*@observer
      v2perp=c-v2par
      z=(v1par+v2par)/(1+(real(v1par)*real(v2par)+imag(v1par)*imag(v2par))*\
        recip_speed2)+v1perp+v2perp
    endif
  endif
bailout:
  |z|<@bailout
default:
  title="Relativistic Addition Julia"
  periodicity=0
  complex param julparam
    caption="Julia parameter"
    default=(0,1)
  endparam
  complex param power
    caption="power"
    default=2
  endparam
  float param speed
    caption="light speed"
    default=3
  endparam
  float param bailout
    caption="bailout"
    default=10
  endparam
  param vtype
    caption="reference frame"
    default=0
    enum="c frame" "z frame" "collinear c" "collinear z" "collinear observer"
  endparam
  complex param observer
    caption="observer's frame"
    default=(1,1)
    visible=(@vtype=="collinear observer")
  endparam
switch:
  type="relativistic-addition-mandelbrot"
  power=power
  speed=speed
  bailout=bailout
  vtype=vtype
  observer=observer
}

relativistic-addition-mandelbrot { ; Kerry Mitchell 03aug2007
;
; Replace the standard addition with relativistic addition
; both addends are treated like velocities
; use the full-blown equation found in wikipedia
;
; http://en.wikipedia.org/wiki/Velocity-addition_formula
;
init:
  complex c=#pixel
  complex v1=(0,0)
  complex v2=(0,0)
  complex w=(0,0)
  complex z=@manparam
  float fac1=0
  float fac2=0
  float fac3=0
  float recip_speed2=1/sqr(@speed)
loop:
  w=z^@power
  if(@vtype=="c frame")
    v1=c
    v2=w
    fac1=1+(real(v1)*real(v2)+imag(v1)*imag(v2))*recip_speed2
    fac2=recip_speed2*(1+sqrt(1-|v1|*recip_speed2))
    fac3=real(v1)*imag(v2)-real(v2)*imag(v1)
    z=(v1+v2)/fac1+fac2*(imag(v1)*fac3-flip(real(v1)*fac3))/fac1
  elseif(@vtype=="z frame")
    v2=c
    v1=w
    fac1=1+(real(v1)*real(v2)+imag(v1)*imag(v2))*recip_speed2
    fac2=recip_speed2*(1+sqrt(1-|v1|*recip_speed2))
    fac3=real(v1)*imag(v2)-real(v2)*imag(v1)
    z=(v1+v2)/fac1+fac2*(imag(v1)*fac3-flip(real(v1)*fac3))/fac1
  elseif(@vtype=="collinear c")
    if(cabs(c)==0)
      z=w
    else
      v1par=c
      v1perp=0
      v2par=(real(w)*real(c)+imag(w)*imag(c))/|c|*c
      v2perp=w-v2par
      z=(v1par+v2par)/(1+(real(v1par)*real(v2par)+imag(v1par)*imag(v2par))*\
        recip_speed2)+v1perp+v2perp
    endif
  elseif(@vtype=="collinear z")
    if(cabs(w)==0)
      z=c
    else
      v1par=w
      v1perp=0
      v2par=(real(c)*real(w)+imag(c)*imag(w))/|w|*w
      v2perp=c-v2par
      z=(v1par+v2par)/(1+(real(v1par)*real(v2par)+imag(v1par)*imag(v2par))*\
        recip_speed2)+v1perp+v2perp
    endif
  elseif(@vtype=="collinear observer")
    if(cabs(@observer)==0)
      z=w+c
    else
      v1par=(real(w)*real(@observer)+imag(w)*imag(@observer))/|@observer|*@observer
      v1perp=w-v1par
      v2par=(real(c)*real(@observer)+imag(c)*imag(@observer))/|@observer|*@observer
      v2perp=c-v2par
      z=(v1par+v2par)/(1+(real(v1par)*real(v2par)+imag(v1par)*imag(v2par))*\
        recip_speed2)+v1perp+v2perp
    endif
  endif
bailout:
  |z|<@bailout
default:
  title="Relativistic Addition Mandelbrot"
  periodicity=0
  complex param manparam
    caption="initial z"
    default=0
  endparam
  complex param power
    caption="power"
    default=2
  endparam
  float param speed
    caption="light speed"
    default=3
  endparam
  float param bailout
    caption="bailout"
    default=10
  endparam
  param vtype
    caption="reference frame"
    default=0
    enum="c frame" "z frame" "collinear c" "collinear z" "collinear observer"
  endparam
  complex param observer
    caption="observer's frame"
    default=(1,1)
    visible=(@vtype=="collinear observer")
  endparam
switch:
  type="relativistic-addition-julia"
  julparam=#pixel
  power=power
  speed=speed
  bailout=bailout
  vtype=vtype
  observer=observer
}

burning-ship-mandelbrot { ; Kerry Mitchell 17jun2008
;
; Based on the algorithm by Michael Michelitsch and Otto Rossler
; and on the UF implementation written by Pasi Piitulainen.
;
; Generalized to allow for functions, powers, and weights
; for both the x- and y-parts of z. The default parameters give the
; standard 'burning ship' fractal.
;
init:
  complex c=#pixel
  complex z=@initial_z
  float zx=0
  float zy=0
loop:
  zx=abs(real(@xfunction(real(z))))^@xpower
  zy=abs(real(@yfunction(imag(z))))^@ypower
  z=(@xweight*zx+@yweight*flip(zy))^@zpower+c
bailout:
  |z|<=@bailout
default:
  title="Burning Ship Mandelbrot"
;
; general parameters
;
  heading
    caption="General"
  endheading
  complex param initial_z
    caption="initial z"
    default=(0,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  complex param zpower
    caption="power"
    default=2
  endparam
;
; x parameters
;
  heading
    caption="X"
  endheading
  float param xpower
    caption="power"
    default=1
  endparam
  func xfunction
    caption="function"
    default=ident()
  endfunc
  complex param xweight
    caption="weight"
    default=1
  endparam
;
; y parameters
;
  heading
    caption="Y"
  endheading
  float param ypower
    caption="power"
    default=1
  endparam
  func yfunction
    caption="function"
    default=ident()
  endfunc
  complex param yweight
    caption="weight"
    default=1
  endparam
switch:
  type="burning-ship-julia"
  julparam=#pixel
  bailout=bailout
  zpower=zpower
  xpower=xpower
  xfunction=xfunction
  xweight=xweight
  ypower=ypower
  yfunction=yfunction
  yweight=yweight
}

burning-ship-julia { ; Kerry Mitchell 17jun2008
;
; Based on the algorithm by Michael Michelitsch and Otto Rossler
; and on the UF implementation written by Pasi Piitulainen.
;
; Generalized to allow for functions, powers, and weights
; for both the x- and y-parts of z. The default parameters give the
; standard 'burning ship' fractal.
;
init:
  complex c=@julparam
  complex z=#pixel
  float zx=0
  float zy=0
loop:
  zx=abs(real(@xfunction(real(z))))^@xpower
  zy=abs(real(@yfunction(imag(z))))^@ypower
  z=(@xweight*zx+@yweight*flip(zy))^@zpower+c
bailout:
  |z|<=@bailout
default:
  title="Burning Ship Julia"
;
; general parameters
;
  heading
    caption="General"
  endheading
  complex param julparam
    caption="Julia parameter"
    default=(0,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  complex param zpower
    caption="power"
    default=2
  endparam
;
; x parameters
;
  heading
    caption="X"
  endheading
  float param xpower
    caption="power"
    default=1
  endparam
  func xfunction
    caption="function"
    default=ident()
  endfunc
  complex param xweight
    caption="weight"
    default=1
  endparam
;
; y parameters
;
  heading
    caption="Y"
  endheading
  float param ypower
    caption="power"
    default=1
  endparam
  func yfunction
    caption="function"
    default=ident()
  endfunc
  complex param yweight
    caption="weight"
    default=1
  endparam
switch:
  type="burning-ship-mandelbrot"
  bailout=bailout
  zpower=zpower
  xpower=xpower
  xfunction=xfunction
  xweight=xweight
  ypower=ypower
  yfunction=yfunction
  yweight=yweight
}

relaxed-newton-halley { ; Kerry Mitchell 22dec08
;
; Newton's method, generalized to include Halley's method and two relaxation
; factors.  "Solves" z^n = c and exp(z*c) = log(z)/c
;
global:
  $define debug
  complex relaxfac1=(0,0)
  complex relaxfac2=@relaxrect2
  complex weighthalley=(0,0)
  complex weightnewton=(0,0)
;
; set up relaxation coefficient 1
;
  if(@relaxcoord=="rectangular")
    relaxfac1=@relaxrect
  elseif(@relaxcoord=="polar, degrees")
    gt=@relaxdegree*#pi/180
    relaxfac1=@relaxcenter+@relaxradius*(cos(gt)+flip(sin(gt)))
  elseif(@relaxcoord=="polar, radians")
    gt=@relaxradian
    relaxfac1=@relaxcenter+@relaxradius*(cos(gt)+flip(sin(gt)))
  elseif(@relaxcoord=="polar, turns")
    if(@pifac==0)
      gt=@relaxm/@relaxn*2*#pi
    else
      gt=@relaxm/@relaxn*2*@pifac
    endif
    relaxfac1=@relaxcenter+@relaxradius*(cos(gt)+flip(sin(gt)))
  endif
  if(@printrelax)
    print("Relaxation 1: ",relaxfac1)
    print("Relaxation 2: ",relaxfac2)
  endif
;
; weights for weighted solution
;
  weightnewton=@weightn/(@weighth+@weightn)
  weighthalley=1-weightnewton
init:
  complex c=(0,0)
  complex dz=(0,0)
  complex dzh=(0,0)
  complex dzn=(0,0)
  complex f=(0,0)
  complex fdoubleprime=(0,0)
  complex fprime=(0,0)
  complex powerm1=@power-1
  complex relaxation1=(0,0)
  complex relaxation2=(0,0)
  complex znewton=(0,0)
  int iterhalley=0
  int iternewton=0
;
; initialize #pixel, c, z
;
  if(@pixeltype=="initial z")
    znewton=#pixel
    c=@root
    relaxation1=relaxfac1
    relaxation2=relaxfac2
  elseif(@pixeltype=="c")
    znewton=@initialz
    c=#pixel
    relaxation1=relaxfac1
    relaxation2=relaxfac2
  elseif(@pixeltype=="relaxation 1")
    znewton=@initialz
    c=@root
    relaxation1=#pixel
    relaxation2=relaxfac2
  elseif(@pixeltype=="relaxation 2")
    znewton=@initialz
    c=@root
    relaxation1=relaxfac1
    relaxation2=#pixel
  elseif(@pixeltype=="relaxation 1 & 2")
    znewton=@initialz
    c=@root
    relaxation1=@relaxfn1(#pixel)
    relaxation2=@relaxfn2(#pixel)
  endif
;
; toggles for switched solution
;
  if(@solntype=="switch-Newton")
    iternewton=1
  elseif(@solntype=="switch-Halley")
    iterhalley=1
  endif
loop:
;
; toggle switched solution
;
  iternewton=1-iternewton
  iterhalley=1-iterhalley
;
; compute function and derivatives
;
  if(@eqntype=="roots of c")
;
;   optimizations for low-order power functions
;
    if(@power==(2,0))
      fdoubleprime=2
      fprime=2*znewton
      f=sqr(znewton)-c
    elseif(@power==(3,0))
      fdoubleprime=6*znewton
      fprime=sqr(znewton)
      f=znewton*fprime-c
      fprime=3*fprime
    elseif(@power==(4,0))
      fdoubleprime=sqr(znewton)
      fprime=4*znewton*fdoubleprime
      f=sqr(fdoubleprime)-c
      fdoubleprime=12*fdoubleprime
    else
      fprime=znewton^powerm1
      f=znewton*fprime-c
      fdoubleprime=fprime/znewton*@power*powerm1
      fprime=@power*fprime
    endif
  elseif(@eqntype=="exp(cz) = log(z)/c")
    fprime=exp(c*znewton)
    fdoubleprime=sqr(c)*fprime+1/sqr(znewton)/c
    f=fprime-log(znewton)/c
    fprime=c*fprime-1/(c*znewton)
  endif
;
; generate next iterate
;
  if(@solntype=="Newton")
    dz=-relaxation1*(f/fprime)^relaxation2
  elseif(@solntype=="Halley")
    dz=2*relaxation1*f*fprime
    dz=dz/(relaxation2*f*fdoubleprime-2*sqr(fprime))
  elseif(@solntype=="weighted")
    dzn=-relaxation1*f/fprime
    dzh=2*relaxation1*f*fprime
    dzh=dzh/(relaxation2*f*fdoubleprime-2*sqr(fprime))
    dz=weightnewton*dzn+weighthalley*dzh
  elseif(@solntype=="switch-Newton")
    dzn=-relaxation1*f/fprime
    dzh=2*relaxation1*f*fprime
    dzh=dzh/(relaxation2*f*fdoubleprime-2*sqr(fprime))
    dz=iternewton*dzn+iterhalley*dzh
  elseif(@solntype=="switch-Halley")
    dzn=-relaxation1*f/fprime
    dzh=2*relaxation1*f*fprime
    dzh=dzh/(relaxation2*f*fdoubleprime-2*sqr(fprime))
    dz=iternewton*dzn+iterhalley*dzh
  endif
  znewton=znewton+dz
;
; update z variable
;
  if(@ztype=="iterate")
    z=znewton
  elseif(@ztype=="change")
    z=dz
  elseif(@ztype=="function")
    z=f
  elseif(@ztype=="1st derivative")
    z=fprime
  elseif(@ztype=="2nd derivative")
    z=fdoubleprime
  endif
bailout:
  |dz|>@bailin
default:
  title="Relaxed Newton/Halley"
  periodicity=0
;
; general
;
  heading
    caption="General"
  endheading
  float param bailin
    caption="bailin"
    default=1e-6
    min=0
    hint="Make small"
  endparam
  param pixeltype
    caption="pixel type"
    default=0
    enum="initial z" "c" "relaxation 1" "relaxation 2" \
      "relaxation 1 & 2"
  endparam
    complex param initialz
      caption="initial z"
      default=(2,3)
      visible=(@pixeltype!="initial z")
    endparam
    complex param root
      caption="c"
      default=(1,0)
      visible=(@pixeltype!="c")
    endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "change" "function" "1st derivative"\
      "2nd derivative"
  endparam
;
; solution method
;
  heading
    caption="Solution"
  endheading
  param eqntype
    caption="equation to solve"
    default=0
    enum="roots of c" "exp(cz) = log(z)/c"
  endparam
  complex param power
    caption="exponent"
    default=(4,0)
    visible=(@eqntype=="roots of c")
  endparam
  param solntype
    caption="solution type"
    default=0
    enum="Newton" "Halley" "weighted" "switch-Newton" "switch-Halley"
  endparam
    complex param weightn
      caption="Newton weight"
      default=(1,0)
      visible=(@solntype=="weighted")
    endparam
    complex param weighth
      caption="Halley weight"
      default=(0,1)
      visible=(@solntype=="weighted")
    endparam
;
; relaxation parameters
;
  heading
    caption="Relaxation"
  endheading
    param relaxcoord
      caption="#1 coord. type"
      default=0
      enum="rectangular" "polar, degrees" "polar, radians"\
        "polar, turns"
      visible=((@pixeltype!="relaxation 1")&&\
        (@pixeltype!="relaxation 1 & 2"))
    endparam
;
;   relaxation 1:  rectangular
;
    complex param relaxrect
      caption="relaxation 1"
      default=(1,0)
      visible=(((@pixeltype!="relaxation 1")&&\
        (@pixeltype!="relaxation 1 & 2"))&&(@relaxcoord=="rectangular"))
    endparam
;
;     relaxation 1:  polar
;
      complex param relaxcenter
        caption="center"
        default=(1,0)
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord!="rectangular"))
      endparam
      float param relaxradius
        caption="radius"
        default=1
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord!="rectangular"))
      endparam
      float param relaxdegree
        caption="angle, degrees"
        default=60
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord=="polar, degrees"))
      endparam
      float param relaxradian
        caption="angle, radians"
        default=1
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord=="polar, radians"))
      endparam
      float param relaxm
        caption="m of m/n turns"
        default=1
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord=="polar, turns"))
      endparam
      float param relaxn
        caption="n of m/n turns"
        default=3
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord=="polar, turns"))
      endparam
      float param pifac
        caption="pi factor"
        default=0
        hint="Use 0 for machine pi"
        visible=(((@pixeltype!="relaxation 1")&&\
          (@pixeltype!="relaxation 1 & 2"))&&\
          (@relaxcoord=="polar, turns"))
      endparam
;
;   relaxation 2:  rectangular only
;
    complex param relaxrect2
      caption="relaxation 2"
      default=(1,0)
      visible=((@pixeltype!="relaxation 2")&&\
        (@pixeltype!="relaxation 1 & 2"))
    endparam
;
;   relaxation 1 & 2:  modifying functions
;
    func relaxfn1
      caption="relaxation fn 1"
      default=ident()
      visible=(@pixeltype=="relaxation 1 & 2")
    endfunc
    func relaxfn2
      caption="relaxation fn 2"
      default=ident()
      visible=(@pixeltype=="relaxation 1 & 2")
    endfunc
;
; print relaxations
;
    bool param printrelax
      caption="print relaxation"
      default=false
      visible=(@pixeltype!="relaxation 1 & 2")
    endparam
}

unit-vector-tweak-mandelbrot { ; Kerry Mitchell 30dec2011
;
; Add a tweak of the unit vector for z at each iteation.
; Either add before the z^power+c step, after, or both.
;
; Based on Asdam's Mandelbrot variation, posted to Fractal Forums 12/30/2011
; http://www.fractalforums.com/new-theories-and-research/
; is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/
; msg38948/#newinit:
;
  complex c=#pixel
  complex unit_vector=(0,0)
  complex z=@initial_z
loop:
  if((@tweak_type=="before")||(@tweak_type=="both"))
    if(cabs(z)==0)
      unit_vector=1
    else
      unit_vector=z/cabs(z)
    endif
    z=z+@fac_pre*@func_pre(unit_vector)
  endif
  z=z^@power+c
  if((@tweak_type=="after")||(@tweak_type=="both"))
    if(cabs(z)==0)
      unit_vector=1
    else
      unit_vector=z/cabs(z)
    endif
    z=z+@fac_post*@func_post(unit_vector)
  endif
bailout:
  |z|<@bailout
default:
  title="Unit Vector Tweak Mandelbrot"
  complex param initial_z
    caption="initial z"
    default=(0,0)
  endparam
  complex param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  param tweak_type
    caption="tweak type"
    default=1
    enum="before" "after" "both"
  endparam
  complex param fac_pre
    caption="before factor"
    default=(-0.7,0.1)
    visible=((@tweak_type=="before")||(@tweak_type=="both"))
  endparam
  func func_pre
    caption="before function"
    default=cos()
    visible=((@tweak_type=="before")||(@tweak_type=="both"))
  endfunc
  complex param fac_post
    caption="after factor"
    default=0.26
    visible=((@tweak_type=="after")||(@tweak_type=="both"))
  endparam
  func func_post
    caption="after function"
    default=ident()
    visible=((@tweak_type=="after")||(@tweak_type=="both"))
  endfunc
switch:
  type="unit-vector-tweak-julia"
  julparam=#pixel
  power=power
  bailout=bailout
  tweak_type=tweak_type
  fac_pre=fac_pre
  func_pre=func_pre
  fac_post=fac_post
  func_post=func_post
}

unit-vector-tweak-julia { ; Kerry Mitchell 30dec2011
;
; Add a tweak of the unit vector for z at each iteation.
; Either add before the z^power+c step, after, or both.
;
; Based on Asdam's Mandelbrot variation, posted to Fractal Forums 12/30/2011
; http://www.fractalforums.com/new-theories-and-research/
; is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/
; msg38948/#newinit:
;
  complex c=@julparam
  complex unit_vector=(0,0)
  complex z=#pixel
loop:
  if((@tweak_type=="before")||(@tweak_type=="both"))
    if(cabs(z)==0)
      unit_vector=1
    else
      unit_vector=z/cabs(z)
    endif
    z=z+@fac_pre*@func_pre(unit_vector)
  endif
  z=z^@power+c
  if((@tweak_type=="after")||(@tweak_type=="both"))
    if(cabs(z)==0)
      unit_vector=1
    else
      unit_vector=z/cabs(z)
    endif
    z=z+@fac_post*@func_post(unit_vector)
  endif
bailout:
  |z|<@bailout
default:
  title="Unit Vector Tweak Julia"
  complex param julparam
    caption="Julia seed"
    default=(0.1,1)
  endparam
  complex param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  param tweak_type
    caption="tweak type"
    default=1
    enum="before" "after" "both"
  endparam
  complex param fac_pre
    caption="before factor"
    default=(-0.7,0.1)
    visible=((@tweak_type=="before")||(@tweak_type=="both"))
  endparam
  func func_pre
    caption="before function"
    default=cos()
    visible=((@tweak_type=="before")||(@tweak_type=="both"))
  endfunc
  complex param fac_post
    caption="after factor"
    default=0.26
    visible=((@tweak_type=="after")||(@tweak_type=="both"))
  endparam
  func func_post
    caption="after function"
    default=ident()
    visible=((@tweak_type=="after")||(@tweak_type=="both"))
  endfunc
switch:
  type="unit-vector-tweak-mandelbrot"
  power=power
  bailout=bailout
  tweak_type=tweak_type
  fac_pre=fac_pre
  func_pre=func_pre
  fac_post=fac_post
  func_post=func_post
}

partial-cz-mandelbrot { ; Kerry Mitchell 29apr2012
;
; Tweaks z before and/or after iteration, as well as modifying
; how c is added in z^n + c.
;
init:
  complex c=#pixel
  complex partz=(0,0)
  complex z=(0,0)
;
; initialize z
;
  if(@manparam==(0,0))
    z=c
  else
    z=@manparam
  endif
loop:
;
; determine part of z
;
  if(@parttype=="real")
    partz=real(z)
  elseif(@parttype=="imag")
    partz=imag(z)
  elseif(@parttype=="z")
    partz=z
  elseif(@parttype=="magnitude")
    partz=cabs(z)
  elseif(@parttype=="real/mag")
    partz=real(z)/cabs(z)
  elseif(@parttype=="imag/mag")
    partz=imag(z)/cabs(z)
  elseif(@parttype=="z/mag")
    partz=z/cabs(z)
  elseif(@parttype=="angle")
    partz=atan2(z)
  elseif(@parttype=="real/imag")
    partz=real(z)/imag(z)
  endif
;
; apply modifying function
;
  partz=@partscale*@partfunc(partz)
;
; apply partz to z^n
;
  if(@ztype=="add before")
    z=z+partz
    z=z^@power
  elseif(@ztype=="add after")
    z=z^@power
    z=z+partz
  elseif(@ztype=="add both")
    z=z+partz
    z=z^@power
    z=z+partz
  elseif(@ztype=="add & subtract")
    z=z+partz
    z=z^@power
    z=z-partz
  elseif(@ztype=="multiply before")
    z=z*partz
    z=z^@power
  elseif(@ztype=="multiply after")
    z=z^@power
    z=z*partz
  elseif(@ztype=="multiply both")
    z=z*partz
    z=z^@power
    z=z*partz
  elseif(@ztype=="multiply & divide")
    z=z*partz
    z=z^@power
    z=z/partz
  else                       ; none
    z=z^@power
  endif
;
; add jacked-up c
;
  if(@ctype=="add")
    z=z+c+partz
  elseif(@ctype=="subtract")
    z=z+c-partz
  elseif(@ctype=="multiply")
    z=z+c*partz
  elseif(@ctype=="divide")
    z=z+c/partz
  elseif(@ctype=="power")
    z=z+c^partz
  elseif(@ctype=="root")
    z=z+c^(1/partz)
  else                       ; none
    z=z+c
  endif
bailout:
  |z|<@bailout
default:
  title="Partial cz Mandelbrot"
  heading
    text="Tweaks z before and/or after iteration, as well as modifying \
      how c is added in z^n + c."
  endheading
  complex param manparam
    caption="initial z"
    default=(0,0)
  endparam
  complex param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  param parttype
    caption="z part type"
    enum="z" "real" "imag" "magnitude" "angle"\
      "real/mag" "imag/mag" "z/mag" "real/imag"
    default=5
  endparam
  param ztype
    caption="z tweak type"
    enum="none" "add before" "add after" "add both" "add & subtract"\
      "multiply before" "multiply after" "multiply both" "multiply & divide"
    default=1
  endparam
  param ctype
    caption="c tweak type"
    enum="none" "add" "subtract" "multiply" "divide" "power" "root"
    default=2
  endparam
  complex param partscale
    caption="amount"
    default=0.1
  endparam
  func partfunc
    caption="modifying function"
    default=ident()
  endfunc
switch:
  type="partial-cz-julia"
  julparam=pixel
  power=power
  bailout=bailout
  parttype=parttype
  ztype=ztype
  ctype=ctype
  partscale=partscale
  partfunc=partfunc
}

partial-cz-julia { ; Kerry Mitchell 29apr2012
;
; Tweaks z before and/or after iteration, as well as modifying
; how c is added in z^n + c.
;
init:
  complex c=@julparam
  complex partz=(0,0)
  complex z=#pixel
loop:
;
; determine part of z
;
  if(@parttype=="real")
    partz=real(z)
  elseif(@parttype=="imag")
    partz=imag(z)
  elseif(@parttype=="z")
    partz=z
  elseif(@parttype=="magnitude")
    partz=cabs(z)
  elseif(@parttype=="real/mag")
    partz=real(z)/cabs(z)
  elseif(@parttype=="imag/mag")
    partz=imag(z)/cabs(z)
  elseif(@parttype=="z/mag")
    partz=z/cabs(z)
  elseif(@parttype=="angle")
    partz=atan2(z)
  elseif(@parttype=="real/imag")
    partz=real(z)/imag(z)
  endif
;
; apply modifying function
;
  partz=@partscale*@partfunc(partz)
;
; apply partz to z^n
;
  if(@ztype=="add before")
    z=z+partz
    z=z^@power
  elseif(@ztype=="add after")
    z=z^@power
    z=z+partz
  elseif(@ztype=="add both")
    z=z+partz
    z=z^@power
    z=z+partz
  elseif(@ztype=="add & subtract")
    z=z+partz
    z=z^@power
    z=z-partz
  elseif(@ztype=="multiply before")
    z=z*partz
    z=z^@power
  elseif(@ztype=="multiply after")
    z=z^@power
    z=z*partz
  elseif(@ztype=="multiply both")
    z=z*partz
    z=z^@power
    z=z*partz
  elseif(@ztype=="multiply & divide")
    z=z*partz
    z=z^@power
    z=z/partz
  else                       ; none
    z=z^@power
  endif
;
; add jacked-up c
;
  if(@ctype=="add")
    z=z+c+partz
  elseif(@ctype=="subtract")
    z=z+c-partz
  elseif(@ctype=="multiply")
    z=z+c*partz
  elseif(@ctype=="divide")
    z=z+c/partz
  elseif(@ctype=="power")
    z=z+c^partz
  elseif(@ctype=="root")
    z=z+c^(1/partz)
  else                       ; none
    z=z+c
  endif
bailout:
  |z|<@bailout
default:
  title="Partial cz Julia"
  heading
    text="Tweaks z before and/or after iteration, as well as modifying \
      how c is added in z^n + c."
  endheading
  complex param julparam
    caption="Julia parameter"
    default=(0,1)
  endparam
  complex param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  param parttype
    caption="z part type"
    enum="z" "real" "imag" "magnitude" "angle"\
      "real/mag" "imag/mag" "z/mag" "real/imag"
    default=5
  endparam
  param ztype
    caption="z tweak type"
    enum="none" "add before" "add after" "add both" "add & subtract"\
      "multiply before" "multiply after" "multiply both" "multiply & divide"
    default=1
  endparam
  param ctype
    caption="c tweak type"
    enum="none" "add" "subtract" "multiply" "divide" "power" "root"
    default=2
  endparam
  complex param partscale
    caption="amount"
    default=0.1
  endparam
  func partfunc
    caption="modifying function"
    default=ident()
  endfunc
switch:
  type="partial-cz-mandelbrot"
  power=power
  bailout=bailout
  parttype=parttype
  ztype=ztype
  ctype=ctype
  partscale=partscale
  partfunc=partfunc
}

alternating-functions { ; Kerry Mitchell 02Jul2013
init:
  complex dz=(0,0)
  complex z=(0,0)
  complex xnew=(0,0)
  complex xold=(0,0)
  complex ynew=(0,0)
  complex yold=(0,0)
  complex znew=(0,0)
  complex zold=(0,0)
  float r=0
  float sincos=0
  int funtype=0
  int iter=0
  int picker=0
  int seed=@initial_seed
;
  if(@initz_type=="pixel")
    zold=#pixel
  else
    zold=@initial_z
  endif
  iter=-1
loop:
  iter=iter+1
;
; choose function
;
; 1 function
;
  if(@nfunction=="1")
    if(@fn1=="Mandelbrot 1")
      funtype=1
    elseif(@fn1=="Julia 1")
      funtype=2
    elseif(@fn1=="Newton M 1")
      funtype=3
    elseif(@fn1=="Newton J 1")
      funtype=4
    elseif(@fn1=="Popcorn C 1")
      funtype=5
    elseif(@fn1=="Popcorn R 1")
      funtype=6
    elseif(@fn1=="Mandelbrot 2")
      funtype=7
    elseif(@fn1=="Julia 2")
      funtype=8
    elseif(@fn1=="Newton M 2")
      funtype=9
    elseif(@fn1=="Newton J 2")
      funtype=10
    elseif(@fn1=="Popcorn C 2")
      funtype=11
    elseif(@fn1=="Popcorn R 2")
      funtype=12
    endif
;
; 2 functions
;
  elseif(@nfunction=="2")
    if(@oscillator_type=="ramp")
      picker=iter%2
    elseif(@oscillator_type=="sine")
      sincos=(1+sin(iter-0.125))/2
      picker=trunc(sincos*2)
    elseif(@oscillator_type=="cosine")
      sincos=(1+cos(iter+0.125))/2
      picker=trunc(sincos*2)
    elseif(@oscillator_type=="random msb")
      seed=random(seed)
      sincos=abs(seed/#randomrange)
      picker=trunc(sincos*2)
    elseif(@oscillator_type=="random lsb")
      seed=random(seed)
      picker=abs(seed)%2
    endif
    picker=picker+1
    if(picker==1)
      if(@fn1=="Mandelbrot 1")
        funtype=1
      elseif(@fn1=="Julia 1")
        funtype=2
      elseif(@fn1=="Newton M 1")
        funtype=3
      elseif(@fn1=="Newton J 1")
        funtype=4
      elseif(@fn1=="Popcorn C 1")
        funtype=5
      elseif(@fn1=="Popcorn R 1")
        funtype=6
      elseif(@fn1=="Mandelbrot 2")
        funtype=7
      elseif(@fn1=="Julia 2")
        funtype=8
      elseif(@fn1=="Newton M 2")
        funtype=9
      elseif(@fn1=="Newton J 2")
        funtype=10
      elseif(@fn1=="Popcorn C 2")
        funtype=11
      elseif(@fn1=="Popcorn R 2")
        funtype=12
      endif
    else
      if(@fn2=="Mandelbrot 1")
        funtype=1
      elseif(@fn2=="Julia 1")
        funtype=2
      elseif(@fn2=="Newton M 1")
        funtype=3
      elseif(@fn2=="Newton J 1")
        funtype=4
      elseif(@fn2=="Popcorn C 1")
        funtype=5
      elseif(@fn2=="Popcorn R 1")
        funtype=6
      elseif(@fn2=="Mandelbrot 2")
        funtype=7
      elseif(@fn2=="Julia 2")
        funtype=8
      elseif(@fn2=="Newton M 2")
        funtype=9
      elseif(@fn2=="Newton J 2")
        funtype=10
      elseif(@fn2=="Popcorn C 2")
        funtype=11
      elseif(@fn2=="Popcorn R 2")
        funtype=12
      endif
    endif
;
; 3 functions
;
  elseif(@nfunction=="3")
    if(@oscillator_type=="ramp")
      picker=iter%3
    elseif(@oscillator_type=="sine")
      sincos=(1+sin(iter-0.125))/2
      picker=trunc(sincos*3)
    elseif(@oscillator_type=="cosine")
      sincos=(1+cos(iter+0.125))/2
      picker=trunc(sincos*3)
    elseif(@oscillator_type=="random msb")
      seed=random(seed)
      sincos=abs(seed/#randomrange)
      picker=trunc(sincos*3)
    elseif(@oscillator_type=="random lsb")
      seed=random(seed)
      picker=abs(seed)%3
    endif
    picker=picker+1
    if(picker==1)
      if(@fn1=="Mandelbrot 1")
        funtype=1
      elseif(@fn1=="Julia 1")
        funtype=2
      elseif(@fn1=="Newton M 1")
        funtype=3
      elseif(@fn1=="Newton J 1")
        funtype=4
      elseif(@fn1=="Popcorn C 1")
        funtype=5
      elseif(@fn1=="Popcorn R 1")
        funtype=6
      elseif(@fn1=="Mandelbrot 2")
        funtype=7
      elseif(@fn1=="Julia 2")
        funtype=8
      elseif(@fn1=="Newton M 2")
        funtype=9
      elseif(@fn1=="Newton J 2")
        funtype=10
      elseif(@fn1=="Popcorn C 2")
        funtype=11
      elseif(@fn1=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==2)
      if(@fn2=="Mandelbrot 1")
        funtype=1
      elseif(@fn2=="Julia 1")
        funtype=2
      elseif(@fn2=="Newton M 1")
        funtype=3
      elseif(@fn2=="Newton J 1")
        funtype=4
      elseif(@fn2=="Popcorn C 1")
        funtype=5
      elseif(@fn2=="Popcorn R 1")
        funtype=6
      elseif(@fn2=="Mandelbrot 2")
        funtype=7
      elseif(@fn2=="Julia 2")
        funtype=8
      elseif(@fn2=="Newton M 2")
        funtype=9
      elseif(@fn2=="Newton J 2")
        funtype=10
      elseif(@fn2=="Popcorn C 2")
        funtype=11
      elseif(@fn2=="Popcorn R 2")
        funtype=12
      endif
    else
      if(@fn3=="Mandelbrot 1")
        funtype=1
      elseif(@fn3=="Julia 1")
        funtype=2
      elseif(@fn3=="Newton M 1")
        funtype=3
      elseif(@fn3=="Newton J 1")
        funtype=4
      elseif(@fn3=="Popcorn C 1")
        funtype=5
      elseif(@fn3=="Popcorn R 1")
        funtype=6
      elseif(@fn3=="Mandelbrot 2")
        funtype=7
      elseif(@fn3=="Julia 2")
        funtype=8
      elseif(@fn3=="Newton M 2")
        funtype=9
      elseif(@fn3=="Newton J 2")
        funtype=10
      elseif(@fn3=="Popcorn C 2")
        funtype=11
      elseif(@fn3=="Popcorn R 2")
        funtype=12
      endif
    endif
;
; 4 functions
;
  elseif(@nfunction=="4")
    if(@oscillator_type=="ramp")
      picker=iter%4
    elseif(@oscillator_type=="sine")
      sincos=(1+sin(iter-0.125))/2
      picker=trunc(sincos*4)
    elseif(@oscillator_type=="cosine")
      sincos=(1+cos(iter+0.125))/2
      picker=trunc(sincos*4)
    elseif(@oscillator_type=="random msb")
      seed=random(seed)
      sincos=abs(seed/#randomrange)
      picker=trunc(sincos*4)
    elseif(@oscillator_type=="random lsb")
      seed=random(seed)
      picker=abs(seed)%4
    endif
    picker=picker+1
    if(picker==1)
      if(@fn1=="Mandelbrot 1")
        funtype=1
      elseif(@fn1=="Julia 1")
        funtype=2
      elseif(@fn1=="Newton M 1")
        funtype=3
      elseif(@fn1=="Newton J 1")
        funtype=4
      elseif(@fn1=="Popcorn C 1")
        funtype=5
      elseif(@fn1=="Popcorn R 1")
        funtype=6
      elseif(@fn1=="Mandelbrot 2")
        funtype=7
      elseif(@fn1=="Julia 2")
        funtype=8
      elseif(@fn1=="Newton M 2")
        funtype=9
      elseif(@fn1=="Newton J 2")
        funtype=10
      elseif(@fn1=="Popcorn C 2")
        funtype=11
      elseif(@fn1=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==2)
      if(@fn2=="Mandelbrot 1")
        funtype=1
      elseif(@fn2=="Julia 1")
        funtype=2
      elseif(@fn2=="Newton M 1")
        funtype=3
      elseif(@fn2=="Newton J 1")
        funtype=4
      elseif(@fn2=="Popcorn C 1")
        funtype=5
      elseif(@fn2=="Popcorn R 1")
        funtype=6
      elseif(@fn2=="Mandelbrot 2")
        funtype=7
      elseif(@fn2=="Julia 2")
        funtype=8
      elseif(@fn2=="Newton M 2")
        funtype=9
      elseif(@fn2=="Newton J 2")
        funtype=10
      elseif(@fn2=="Popcorn C 2")
        funtype=11
      elseif(@fn2=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==3)
      if(@fn3=="Mandelbrot 1")
        funtype=1
      elseif(@fn3=="Julia 1")
        funtype=2
      elseif(@fn3=="Newton M 1")
        funtype=3
      elseif(@fn3=="Newton J 1")
        funtype=4
      elseif(@fn3=="Popcorn C 1")
        funtype=5
      elseif(@fn3=="Popcorn R 1")
        funtype=6
      elseif(@fn3=="Mandelbrot 2")
        funtype=7
      elseif(@fn3=="Julia 2")
        funtype=8
      elseif(@fn3=="Newton M 2")
        funtype=9
      elseif(@fn3=="Newton J 2")
        funtype=10
      elseif(@fn3=="Popcorn C 2")
        funtype=11
      elseif(@fn3=="Popcorn R 2")
        funtype=12
      endif
    else
      if(@fn4=="Mandelbrot 1")
        funtype=1
      elseif(@fn4=="Julia 1")
        funtype=2
      elseif(@fn4=="Newton M 1")
        funtype=3
      elseif(@fn4=="Newton J 1")
        funtype=4
      elseif(@fn4=="Popcorn C 1")
        funtype=5
      elseif(@fn4=="Popcorn R 1")
        funtype=6
      elseif(@fn4=="Mandelbrot 2")
        funtype=7
      elseif(@fn4=="Julia 2")
        funtype=8
      elseif(@fn4=="Newton M 2")
        funtype=9
      elseif(@fn4=="Newton J 2")
        funtype=10
      elseif(@fn4=="Popcorn C 2")
        funtype=11
      elseif(@fn4=="Popcorn R 2")
        funtype=12
      endif
    endif
;
; 5 functions
;
  elseif(@nfunction=="5")
    if(@oscillator_type=="ramp")
      picker=iter%5
    elseif(@oscillator_type=="sine")
      sincos=(1+sin(iter-0.125))/2
      picker=trunc(sincos*5)
    elseif(@oscillator_type=="cosine")
      sincos=(1+cos(iter+0.125))/2
      picker=trunc(sincos*5)
    elseif(@oscillator_type=="random msb")
      seed=random(seed)
      sincos=abs(seed/#randomrange)
      picker=trunc(sincos*5)
    elseif(@oscillator_type=="random lsb")
      seed=random(seed)
      picker=abs(seed)%5
    endif
    picker=picker+1
    if(picker==1)
      if(@fn1=="Mandelbrot 1")
        funtype=1
      elseif(@fn1=="Julia 1")
        funtype=2
      elseif(@fn1=="Newton M 1")
        funtype=3
      elseif(@fn1=="Newton J 1")
        funtype=4
      elseif(@fn1=="Popcorn C 1")
        funtype=5
      elseif(@fn1=="Popcorn R 1")
        funtype=6
      elseif(@fn1=="Mandelbrot 2")
        funtype=7
      elseif(@fn1=="Julia 2")
        funtype=8
      elseif(@fn1=="Newton M 2")
        funtype=9
      elseif(@fn1=="Newton J 2")
        funtype=10
      elseif(@fn1=="Popcorn C 2")
        funtype=11
      elseif(@fn1=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==2)
      if(@fn2=="Mandelbrot 1")
        funtype=1
      elseif(@fn2=="Julia 1")
        funtype=2
      elseif(@fn2=="Newton M 1")
        funtype=3
      elseif(@fn2=="Newton J 1")
        funtype=4
      elseif(@fn2=="Popcorn C 1")
        funtype=5
      elseif(@fn2=="Popcorn R 1")
        funtype=6
      elseif(@fn2=="Mandelbrot 2")
        funtype=7
      elseif(@fn2=="Julia 2")
        funtype=8
      elseif(@fn2=="Newton M 2")
        funtype=9
      elseif(@fn2=="Newton J 2")
        funtype=10
      elseif(@fn2=="Popcorn C 2")
        funtype=11
      elseif(@fn2=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==3)
      if(@fn3=="Mandelbrot 1")
        funtype=1
      elseif(@fn3=="Julia 1")
        funtype=2
      elseif(@fn3=="Newton M 1")
        funtype=3
      elseif(@fn3=="Newton J 1")
        funtype=4
      elseif(@fn3=="Popcorn C 1")
        funtype=5
      elseif(@fn3=="Popcorn R 1")
        funtype=6
      elseif(@fn3=="Mandelbrot 2")
        funtype=7
      elseif(@fn3=="Julia 2")
        funtype=8
      elseif(@fn3=="Newton M 2")
        funtype=9
      elseif(@fn3=="Newton J 2")
        funtype=10
      elseif(@fn3=="Popcorn C 2")
        funtype=11
      elseif(@fn3=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==4)
      if(@fn4=="Mandelbrot 1")
        funtype=1
      elseif(@fn4=="Julia 1")
        funtype=2
      elseif(@fn4=="Newton M 1")
        funtype=3
      elseif(@fn4=="Newton J 1")
        funtype=4
      elseif(@fn4=="Popcorn C 1")
        funtype=5
      elseif(@fn4=="Popcorn R 1")
        funtype=6
      elseif(@fn4=="Mandelbrot 2")
        funtype=7
      elseif(@fn4=="Julia 2")
        funtype=8
      elseif(@fn4=="Newton M 2")
        funtype=9
      elseif(@fn4=="Newton J 2")
        funtype=10
      elseif(@fn4=="Popcorn C 2")
        funtype=11
      elseif(@fn4=="Popcorn R 2")
        funtype=12
      endif
    else
      if(@fn5=="Mandelbrot 1")
        funtype=1
      elseif(@fn5=="Julia 1")
        funtype=2
      elseif(@fn5=="Newton M 1")
        funtype=3
      elseif(@fn5=="Newton J 1")
        funtype=4
      elseif(@fn5=="Popcorn C 1")
        funtype=5
      elseif(@fn5=="Popcorn R 1")
        funtype=6
      elseif(@fn5=="Mandelbrot 2")
        funtype=7
      elseif(@fn5=="Julia 2")
        funtype=8
      elseif(@fn5=="Newton M 2")
        funtype=9
      elseif(@fn5=="Newton J 2")
        funtype=10
      elseif(@fn5=="Popcorn C 2")
        funtype=11
      elseif(@fn5=="Popcorn R 2")
        funtype=12
      endif
    endif
;
; 6 functions
;
  elseif(@nfunction=="6")
    if(@oscillator_type=="ramp")
      picker=iter%6
    elseif(@oscillator_type=="sine")
      sincos=(1+sin(iter-0.125))/2
      picker=trunc(sincos*6)
    elseif(@oscillator_type=="cosine")
      sincos=(1+cos(iter+0.125))/2
      picker=trunc(sincos*6)
    elseif(@oscillator_type=="random msb")
      seed=random(seed)
      sincos=abs(seed/#randomrange)
      picker=trunc(sincos*6)
    elseif(@oscillator_type=="random lsb")
      seed=random(seed)
      picker=abs(seed)%6
    endif
    picker=picker+1
    if(picker==1)
      if(@fn1=="Mandelbrot 1")
        funtype=1
      elseif(@fn1=="Julia 1")
        funtype=2
      elseif(@fn1=="Newton M 1")
        funtype=3
      elseif(@fn1=="Newton J 1")
        funtype=4
      elseif(@fn1=="Popcorn C 1")
        funtype=5
      elseif(@fn1=="Popcorn R 1")
        funtype=6
      elseif(@fn1=="Mandelbrot 2")
        funtype=7
      elseif(@fn1=="Julia 2")
        funtype=8
      elseif(@fn1=="Newton M 2")
        funtype=9
      elseif(@fn1=="Newton J 2")
        funtype=10
      elseif(@fn1=="Popcorn C 2")
        funtype=11
      elseif(@fn1=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==2)
      if(@fn2=="Mandelbrot 1")
        funtype=1
      elseif(@fn2=="Julia 1")
        funtype=2
      elseif(@fn2=="Newton M 1")
        funtype=3
      elseif(@fn2=="Newton J 1")
        funtype=4
      elseif(@fn2=="Popcorn C 1")
        funtype=5
      elseif(@fn2=="Popcorn R 1")
        funtype=6
      elseif(@fn2=="Mandelbrot 2")
        funtype=7
      elseif(@fn2=="Julia 2")
        funtype=8
      elseif(@fn2=="Newton M 2")
        funtype=9
      elseif(@fn2=="Newton J 2")
        funtype=10
      elseif(@fn2=="Popcorn C 2")
        funtype=11
      elseif(@fn2=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==3)
      if(@fn3=="Mandelbrot 1")
        funtype=1
      elseif(@fn3=="Julia 1")
        funtype=2
      elseif(@fn3=="Newton M 1")
        funtype=3
      elseif(@fn3=="Newton J 1")
        funtype=4
      elseif(@fn3=="Popcorn C 1")
        funtype=5
      elseif(@fn3=="Popcorn R 1")
        funtype=6
      elseif(@fn3=="Mandelbrot 2")
        funtype=7
      elseif(@fn3=="Julia 2")
        funtype=8
      elseif(@fn3=="Newton M 2")
        funtype=9
      elseif(@fn3=="Newton J 2")
        funtype=10
      elseif(@fn3=="Popcorn C 2")
        funtype=11
      elseif(@fn3=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==4)
      if(@fn4=="Mandelbrot 1")
        funtype=1
      elseif(@fn4=="Julia 1")
        funtype=2
      elseif(@fn4=="Newton M 1")
        funtype=3
      elseif(@fn4=="Newton J 1")
        funtype=4
      elseif(@fn4=="Popcorn C 1")
        funtype=5
      elseif(@fn4=="Popcorn R 1")
        funtype=6
      elseif(@fn4=="Mandelbrot 2")
        funtype=7
      elseif(@fn4=="Julia 2")
        funtype=8
      elseif(@fn4=="Newton M 2")
        funtype=9
      elseif(@fn4=="Newton J 2")
        funtype=10
      elseif(@fn4=="Popcorn C 2")
        funtype=11
      elseif(@fn4=="Popcorn R 2")
        funtype=12
      endif
    elseif(picker==5)
      if(@fn5=="Mandelbrot 1")
        funtype=1
      elseif(@fn5=="Julia 1")
        funtype=2
      elseif(@fn5=="Newton M 1")
        funtype=3
      elseif(@fn5=="Newton J 1")
        funtype=4
      elseif(@fn5=="Popcorn C 1")
        funtype=5
      elseif(@fn5=="Popcorn R 1")
        funtype=6
      elseif(@fn5=="Mandelbrot 2")
        funtype=7
      elseif(@fn5=="Julia 2")
        funtype=8
      elseif(@fn5=="Newton M 2")
        funtype=9
      elseif(@fn5=="Newton J 2")
        funtype=10
      elseif(@fn5=="Popcorn C 2")
        funtype=11
      elseif(@fn5=="Popcorn R 2")
        funtype=12
      endif
    else
      if(@fn6=="Mandelbrot 1")
        funtype=1
      elseif(@fn6=="Julia 1")
        funtype=2
      elseif(@fn6=="Newton M 1")
        funtype=3
      elseif(@fn6=="Newton J 1")
        funtype=4
      elseif(@fn6=="Popcorn C 1")
        funtype=5
      elseif(@fn6=="Popcorn R 1")
        funtype=6
      elseif(@fn6=="Mandelbrot 2")
        funtype=7
      elseif(@fn6=="Julia 2")
        funtype=8
      elseif(@fn6=="Newton M 2")
        funtype=9
      elseif(@fn6=="Newton J 2")
        funtype=10
      elseif(@fn6=="Popcorn C 2")
        funtype=11
      elseif(@fn6=="Popcorn R 2")
        funtype=12
      endif
    endif
  endif
;
; iterate
;
  if(funtype==1) ; Mandelbrot 1
    znew=(1-@m1weight)*zold+@m1weight*(zold^@m1power+#pixel)
    r=|znew|
  elseif(funtype==2) ; Julia 1
    znew=(1-@j1weight)*zold+@j1weight*(zold^@j1power+@j1param)
    r=|znew|
  elseif(funtype==3) ; Newton M 1, roots of pixel
    dz=(zold^@nm1power-#pixel)/(@nm1power*zold^(@nm1power-1))
    znew=zold-@nm1weight*dz
    r=1/|dz|
  elseif(funtype==4) ; Newton J 1, roots of specific value
    dz=(zold^@nj1power-@nj1param)/(@nj1power*zold^(@nj1power-1))
    znew=zold-@nj1weight*dz
    r=1/|dz|
  elseif(funtype==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(zold)
    yold=imag(zold)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    znew=xnew+flip(ynew)
    r=|znew|
  elseif(funtype==6) ; Popcorn R 1, popcorn with real variables
    xold=real(zold)
    yold=imag(zold)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    znew=xnew+flip(ynew)
    r=|znew|
  elseif(funtype==7) ; Mandelbrot 2
    znew=(1-@m2weight)*zold+@m2weight*(zold^@m2power+#pixel)
    r=|znew|
  elseif(funtype==8) ; Julia 2
    znew=(1-@j2weight)*zold+@j2weight*(zold^@j2power+@j2param)
    r=|znew|
  elseif(funtype==9) ; Newton M 2, roots of pixel
    dz=(zold^@nm2power-#pixel)/(@nm2power*zold^(@nm2power-1))
    znew=zold-@nm2weight*dz
    r=1/|dz|
  elseif(funtype==10) ; Newton J 2, roots of specific value
    dz=(zold^@nj2power-@nj2param)/(@nj2power*zold^(@nj2power-1))
    znew=zold-@nj2weight*dz
    r=1/|dz|
  elseif(funtype==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(zold)
    yold=imag(zold)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    znew=xnew+flip(ynew)
    r=|znew|
  elseif(funtype==12) ; Popcorn R 2, popcorn with real variables
    xold=real(zold)
    yold=imag(zold)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    znew=xnew+flip(ynew)
    r=|znew|
  endif
;
; establish z
;
  if(@ztype=="iterate")
    z=znew
  elseif(@ztype=="difference")
    z=znew-zold
  elseif(@ztype=="ratio")
    z=znew/zold
  elseif(@ztype=="weird")
    if(@nfunction=="1")
      z=zold^znew
    elseif(@nfunction=="2")
      z=(-zold)^znew
    elseif(@nfunction=="3")
      z=(-zold)^(-znew)
    elseif(@nfunction=="4")
      z=zold^(-znew)
    elseif(@nfunction=="5")
      z=(zold+znew)^(zold-znew)
    elseif(@nfunction=="6")
      z=(znew-zold)^(zold+znew)
    endif
  endif
  zold=znew
bailout:
  r<@bailout
default:
  title="Alternating Functions"
  heading
    text="Allows you to alternate between several functions every iteration."
  endheading
;
; general parameters
;
  heading
    caption="General parameters"
  endheading
  param initz_type
    caption="initial z type"
    default=1
    enum="manual" "pixel"
  endparam
  complex param initial_z
    caption="initial z"
    default=(0,0)
    visible=(@initz_type=="manual")
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "difference" "ratio" "weird"
  endparam
;
; alternating function defintions
;
  heading
    caption="Alternating functions"
  endheading
  param nfunction
    caption="# of functions"
    default=1
    enum="1" "2" "3" "4" "5" "6"
  endparam
  param fn1
    caption="function 1"
    default=4
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
  endparam
  param fn2
    caption="function 2"
    default=1
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>0)
  endparam
  param fn3
    caption="function 3"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>1)
  endparam
  param fn4
    caption="function 4"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>2)
  endparam
  param fn5
    caption="function 5"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>3)
  endparam
  param fn6
    caption="function 6"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>4)
  endparam
;
  complex param m1power
    caption="Mandelbrot 1 power"
    default=(2,0)
    visible=((@fn1=="Mandelbrot 1")||(@fn2=="Mandelbrot 1")||(@fn3=="Mandelbrot 1")||\
      (@fn4=="Mandelbrot 1")||(@fn5=="Mandelbrot 1")||(@fn6=="Mandelbrot 1"))
  endparam
  complex param m1weight
    caption="Mandelbrot 1 weight"
    default=1
    visible=((@fn1=="Mandelbrot 1")||(@fn2=="Mandelbrot 1")||(@fn3=="Mandelbrot 1")||\
      (@fn4=="Mandelbrot 1")||(@fn5=="Mandelbrot 1")||(@fn6=="Mandelbrot 1"))
  endparam
  complex param j1power
    caption="Julia 1 power"
    default=(2,0)
    visible=((@fn1=="Julia 1")||(@fn2=="Julia 1")||(@fn3=="Julia 1")||\
      (@fn4=="Julia 1")||(@fn5=="Julia 1")||(@fn6=="Julia 1"))
  endparam
  complex param j1param
    caption="Julia 1 parameter"
    default=(0,1)
    visible=((@fn1=="Julia 1")||(@fn2=="Julia 1")||(@fn3=="Julia 1")||\
      (@fn4=="Julia 1")||(@fn5=="Julia 1")||(@fn6=="Julia 1"))
  endparam
  complex param j1weight
    caption="Julia 1 weight"
    default=1
    visible=((@fn1=="Julia 1")||(@fn2=="Julia 1")||(@fn3=="Julia 1")||\
      (@fn4=="Julia 1")||(@fn5=="Julia 1")||(@fn6=="Julia 1"))
  endparam
  complex param nm1power
    caption="Newton M 1 power"
    default=(3,0)
    visible=((@fn1=="Newton M 1")||(@fn2=="Newton M 1")||(@fn3=="Newton M 1")||\
      (@fn4=="Newton M 1")||(@fn5=="Newton M 1")||(@fn6=="Newton M 1"))
  endparam
  complex param nm1weight
    caption="Newton M 1 nova fac"
    default=1
    visible=((@fn1=="Newton M 1")||(@fn2=="Newton M 1")||(@fn3=="Newton M 1")||\
      (@fn4=="Newton M 1")||(@fn5=="Newton M 1")||(@fn6=="Newton M 1"))
  endparam
  complex param nj1power
    caption="Newton J 1 power"
    default=(4,0)
    visible=((@fn1=="Newton J 1")||(@fn2=="Newton J 1")||(@fn3=="Newton J 1")||\
      (@fn4=="Newton J 1")||(@fn5=="Newton J 1")||(@fn6=="Newton J 1"))
  endparam
  complex param nj1param
    caption="Newton J 1 parameter"
    default=(1,0)
    visible=((@fn1=="Newton J 1")||(@fn2=="Newton J 1")||(@fn3=="Newton J 1")||\
      (@fn4=="Newton J 1")||(@fn5=="Newton J 1")||(@fn6=="Newton J 1"))
  endparam
  complex param nj1weight
    caption="Newton J 1 nova fac"
    default=1
    visible=((@fn1=="Newton J 1")||(@fn2=="Newton J 1")||(@fn3=="Newton J 1")||\
      (@fn4=="Newton J 1")||(@fn5=="Newton J 1")||(@fn6=="Newton J 1"))
  endparam
  complex param pc1h
    caption="complex step size 1"
    default=(0.1,0)
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endparam
  complex param pc1c
    caption="complex frequency 1"
    default=(3,0)
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endparam
  func pc1out
    caption="complex outer function 1"
    default=sin()
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endfunc
  func pc1in
    caption="complex inner function 1"
    default=tan()
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endfunc
  float param pr1h
    caption="real step size 1"
    default=0.1
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endparam
  float param pr1c
    caption="real frequency 1"
    default=3
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endparam
  func pr1out
    caption="real outer function 1"
    default=sin()
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endfunc
  func pr1in
    caption="real inner function 1"
    default=tan()
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endfunc
  complex param m2power
    caption="Mandelbrot 2 power"
    default=(2,0)
    visible=((@fn1=="Mandelbrot 2")||(@fn2=="Mandelbrot 2")||(@fn3=="Mandelbrot 2")||\
      (@fn4=="Mandelbrot 2")||(@fn5=="Mandelbrot 2")||(@fn6=="Mandelbrot 2"))
  endparam
  complex param m2weight
    caption="Mandelbrot 2 weight"
    default=1
    visible=((@fn1=="Mandelbrot 2")||(@fn2=="Mandelbrot 2")||(@fn3=="Mandelbrot 2")||\
      (@fn4=="Mandelbrot 2")||(@fn5=="Mandelbrot 2")||(@fn6=="Mandelbrot 2"))
  endparam
  complex param j2power
    caption="Julia 2 power"
    default=(2,0)
    visible=((@fn1=="Julia 2")||(@fn2=="Julia 2")||(@fn3=="Julia 2")||\
      (@fn4=="Julia 2")||(@fn5=="Julia 2")||(@fn6=="Julia 2"))
  endparam
  complex param j2param
    caption="Julia 2 parameter"
    default=(0,1)
    visible=((@fn1=="Julia 2")||(@fn2=="Julia 2")||(@fn3=="Julia 2")||\
      (@fn4=="Julia 2")||(@fn5=="Julia 2")||(@fn6=="Julia 2"))
  endparam
  complex param j2weight
    caption="Julia 2 weight"
    default=1
    visible=((@fn1=="Julia 2")||(@fn2=="Julia 2")||(@fn3=="Julia 2")||\
      (@fn4=="Julia 2")||(@fn5=="Julia 2")||(@fn6=="Julia 2"))
  endparam
  complex param nm2power
    caption="Newton M 2 power"
    default=(3,0)
    visible=((@fn1=="Newton M 2")||(@fn2=="Newton M 2")||(@fn3=="Newton M 2")||\
      (@fn4=="Newton M 2")||(@fn5=="Newton M 2")||(@fn6=="Newton M 2"))
  endparam
  complex param nm2weight
    caption="Newton M 2 nova fac"
    default=1
    visible=((@fn1=="Newton M 2")||(@fn2=="Newton M 2")||(@fn3=="Newton M 2")||\
      (@fn4=="Newton M 2")||(@fn5=="Newton M 2")||(@fn6=="Newton M 2"))
  endparam
  complex param nj2power
    caption="Newton J 2 power"
    default=(4,0)
    visible=((@fn1=="Newton J 2")||(@fn2=="Newton J 2")||(@fn3=="Newton J 2")||\
      (@fn4=="Newton J 2")||(@fn5=="Newton J 2")||(@fn6=="Newton J 2"))
  endparam
  complex param nj2param
    caption="Newton J 2 parameter"
    default=(1,0)
    visible=((@fn1=="Newton J 2")||(@fn2=="Newton J 2")||(@fn3=="Newton J 2")||\
      (@fn4=="Newton J 2")||(@fn5=="Newton J 2")||(@fn6=="Newton J 2"))
  endparam
  complex param nj2weight
    caption="Newton J 2 nova fac"
    default=(1,0)
    visible=((@fn1=="Newton J 2")||(@fn2=="Newton J 2")||(@fn3=="Newton J 2")||\
      (@fn4=="Newton J 2")||(@fn5=="Newton J 2")||(@fn6=="Newton J 2"))
  endparam
  complex param pc2h
    caption="complex step size 2"
    default=(0.1,0)
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endparam
  complex param pc2c
    caption="complex frequency 2"
    default=(3,0)
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endparam
  func pc2out
    caption="complex outer function 2"
    default=sin()
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endfunc
  func pc2in
    caption="complex inner function 2"
    default=tan()
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endfunc
  float param pr2h
    caption="real step size 2"
    default=0.1
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endparam
  float param pr2c
    caption="real frequency 2"
    default=3
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endparam
  func pr2out
    caption="real outer function 2"
    default=sin()
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endfunc
  func pr2in
    caption="real inner function 2"
    default=tan()
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endfunc
;
; oscillator/ramp
;
  heading
    caption="Oscillator"
  endheading
  param oscillator_type
    caption="type"
    default=0
    enum="ramp" "sine" "cosine" "random msb" "random lsb"
  endparam
  int param initial_seed
    caption="random seed"
    default=2357
    visible=((@oscillator_type=="random msb")||(@oscillator_type=="random lsb"))
  endparam
}

interpolating-functions { ; Kerry Mitchell 06 July 2014
;
; based on my Alternating Functions formula
; allows you to interpolate between up to 6 functions for each iteration
; supports multiple interpolation methods
;
init:
  complex a0=(0,0)
  complex a1=(0,0)
  complex a2=(0,0)
  complex a3=(0,0)
  complex b1=(0,0)
  complex b2=(0,0)
  complex b3=(0,0)
  complex dz=(0,0)
  complex xnew=(0,0)
  complex xold=(0,0)
  complex ynew=(0,0)
  complex yold=(0,0)
  complex z=(0,0)
  complex z1=(0,0)
  complex z2=(0,0)
  complex z3=(0,0)
  complex z4=(0,0)
  complex z5=(0,0)
  complex z6=(0,0)
  float t=0
  float weight1=0
  float weight2=0
  int fun1type=0
  int fun2type=0
  int fun3type=0
  int fun4type=0
  int fun5type=0
  int fun6type=0
  int iter=0
;
; initialize z
;
  iter=0
  if(@initz_type=="pixel")
    z=#pixel
  else
    z=@initial_z
  endif
loop:
  iter=iter+1
;
; choose functions to interpolate between
;
; 1 function
;
  if(@nfunction=="1")
    if(@fn1=="Mandelbrot 1")
      fun1type=1
    elseif(@fn1=="Julia 1")
      fun1type=2
    elseif(@fn1=="Newton M 1")
      fun1type=3
    elseif(@fn1=="Newton J 1")
      fun1type=4
    elseif(@fn1=="Popcorn C 1")
      fun1type=5
    elseif(@fn1=="Popcorn R 1")
      fun1type=6
    elseif(@fn1=="Mandelbrot 2")
      fun1type=7
    elseif(@fn1=="Julia 2")
      fun1type=8
    elseif(@fn1=="Newton M 2")
      fun1type=9
    elseif(@fn1=="Newton J 2")
      fun1type=10
    elseif(@fn1=="Popcorn C 2")
      fun1type=11
    elseif(@fn1=="Popcorn R 2")
      fun1type=12
    endif
;
; 2 functions
;
  elseif(@nfunction=="2")
    if(@fn1=="Mandelbrot 1")
      fun1type=1
    elseif(@fn1=="Julia 1")
      fun1type=2
    elseif(@fn1=="Newton M 1")
      fun1type=3
    elseif(@fn1=="Newton J 1")
      fun1type=4
    elseif(@fn1=="Popcorn C 1")
      fun1type=5
    elseif(@fn1=="Popcorn R 1")
      fun1type=6
    elseif(@fn1=="Mandelbrot 2")
      fun1type=7
    elseif(@fn1=="Julia 2")
      fun1type=8
    elseif(@fn1=="Newton M 2")
      fun1type=9
    elseif(@fn1=="Newton J 2")
      fun1type=10
    elseif(@fn1=="Popcorn C 2")
      fun1type=11
    elseif(@fn1=="Popcorn R 2")
      fun1type=12
    endif
    if(@fn2=="Mandelbrot 1")
      fun2type=1
    elseif(@fn2=="Julia 1")
      fun2type=2
    elseif(@fn2=="Newton M 1")
      fun2type=3
    elseif(@fn2=="Newton J 1")
      fun2type=4
    elseif(@fn2=="Popcorn C 1")
      fun2type=5
    elseif(@fn2=="Popcorn R 1")
      fun2type=6
    elseif(@fn2=="Mandelbrot 2")
      fun2type=7
    elseif(@fn2=="Julia 2")
      fun2type=8
    elseif(@fn2=="Newton M 2")
      fun2type=9
    elseif(@fn2=="Newton J 2")
      fun2type=10
    elseif(@fn2=="Popcorn C 2")
      fun2type=11
    elseif(@fn2=="Popcorn R 2")
      fun2type=12
    endif
;
; 3 functions
;
  elseif(@nfunction=="3")
    if(@fn1=="Mandelbrot 1")
      fun1type=1
    elseif(@fn1=="Julia 1")
      fun1type=2
    elseif(@fn1=="Newton M 1")
      fun1type=3
    elseif(@fn1=="Newton J 1")
      fun1type=4
    elseif(@fn1=="Popcorn C 1")
      fun1type=5
    elseif(@fn1=="Popcorn R 1")
      fun1type=6
    elseif(@fn1=="Mandelbrot 2")
      fun1type=7
    elseif(@fn1=="Julia 2")
      fun1type=8
    elseif(@fn1=="Newton M 2")
      fun1type=9
    elseif(@fn1=="Newton J 2")
      fun1type=10
    elseif(@fn1=="Popcorn C 2")
      fun1type=11
    elseif(@fn1=="Popcorn R 2")
      fun1type=12
    endif
    if(@fn2=="Mandelbrot 1")
      fun2type=1
    elseif(@fn2=="Julia 1")
      fun2type=2
    elseif(@fn2=="Newton M 1")
      fun2type=3
    elseif(@fn2=="Newton J 1")
      fun2type=4
    elseif(@fn2=="Popcorn C 1")
      fun2type=5
    elseif(@fn2=="Popcorn R 1")
      fun2type=6
    elseif(@fn2=="Mandelbrot 2")
      fun2type=7
    elseif(@fn2=="Julia 2")
      fun2type=8
    elseif(@fn2=="Newton M 2")
      fun2type=9
    elseif(@fn2=="Newton J 2")
      fun2type=10
    elseif(@fn2=="Popcorn C 2")
      fun2type=11
    elseif(@fn2=="Popcorn R 2")
      fun2type=12
    endif
    if(@fn3=="Mandelbrot 1")
      fun3type=1
    elseif(@fn3=="Julia 1")
      fun3type=2
    elseif(@fn3=="Newton M 1")
      fun3type=3
    elseif(@fn3=="Newton J 1")
      fun3type=4
    elseif(@fn3=="Popcorn C 1")
      fun3type=5
    elseif(@fn3=="Popcorn R 1")
      fun3type=6
    elseif(@fn3=="Mandelbrot 2")
      fun3type=7
    elseif(@fn3=="Julia 2")
      fun3type=8
    elseif(@fn3=="Newton M 2")
      fun3type=9
    elseif(@fn3=="Newton J 2")
      fun3type=10
    elseif(@fn3=="Popcorn C 2")
      fun3type=11
    elseif(@fn3=="Popcorn R 2")
      fun3type=12
    endif
;
; 4 functions
;
  elseif(@nfunction=="4")
    if(@fn1=="Mandelbrot 1")
      fun1type=1
    elseif(@fn1=="Julia 1")
      fun1type=2
    elseif(@fn1=="Newton M 1")
      fun1type=3
    elseif(@fn1=="Newton J 1")
      fun1type=4
    elseif(@fn1=="Popcorn C 1")
      fun1type=5
    elseif(@fn1=="Popcorn R 1")
      fun1type=6
    elseif(@fn1=="Mandelbrot 2")
      fun1type=7
    elseif(@fn1=="Julia 2")
      fun1type=8
    elseif(@fn1=="Newton M 2")
      fun1type=9
    elseif(@fn1=="Newton J 2")
      fun1type=10
    elseif(@fn1=="Popcorn C 2")
      fun1type=11
    elseif(@fn1=="Popcorn R 2")
      fun1type=12
    endif
    if(@fn2=="Mandelbrot 1")
      fun2type=1
    elseif(@fn2=="Julia 1")
      fun2type=2
    elseif(@fn2=="Newton M 1")
      fun2type=3
    elseif(@fn2=="Newton J 1")
      fun2type=4
    elseif(@fn2=="Popcorn C 1")
      fun2type=5
    elseif(@fn2=="Popcorn R 1")
      fun2type=6
    elseif(@fn2=="Mandelbrot 2")
      fun2type=7
    elseif(@fn2=="Julia 2")
      fun2type=8
    elseif(@fn2=="Newton M 2")
      fun2type=9
    elseif(@fn2=="Newton J 2")
      fun2type=10
    elseif(@fn2=="Popcorn C 2")
      fun2type=11
    elseif(@fn2=="Popcorn R 2")
      fun2type=12
    endif
    if(@fn3=="Mandelbrot 1")
      fun3type=1
    elseif(@fn3=="Julia 1")
      fun3type=2
    elseif(@fn3=="Newton M 1")
      fun3type=3
    elseif(@fn3=="Newton J 1")
      fun3type=4
    elseif(@fn3=="Popcorn C 1")
      fun3type=5
    elseif(@fn3=="Popcorn R 1")
      fun3type=6
    elseif(@fn3=="Mandelbrot 2")
      fun3type=7
    elseif(@fn3=="Julia 2")
      fun3type=8
    elseif(@fn3=="Newton M 2")
      fun3type=9
    elseif(@fn3=="Newton J 2")
      fun3type=10
    elseif(@fn3=="Popcorn C 2")
      fun3type=11
    elseif(@fn3=="Popcorn R 2")
      fun3type=12
    endif
    if(@fn4=="Mandelbrot 1")
      fun4type=1
    elseif(@fn4=="Julia 1")
      fun4type=2
    elseif(@fn4=="Newton M 1")
      fun4type=3
    elseif(@fn4=="Newton J 1")
      fun4type=4
    elseif(@fn4=="Popcorn C 1")
      fun4type=5
    elseif(@fn4=="Popcorn R 1")
      fun4type=6
    elseif(@fn4=="Mandelbrot 2")
      fun4type=7
    elseif(@fn4=="Julia 2")
      fun4type=8
    elseif(@fn4=="Newton M 2")
      fun4type=9
    elseif(@fn4=="Newton J 2")
      fun4type=10
    elseif(@fn4=="Popcorn C 2")
      fun4type=11
    elseif(@fn4=="Popcorn R 2")
      fun4type=12
    endif
;
; 5 functions
;
  elseif(@nfunction=="5")
    if(@fn1=="Mandelbrot 1")
      fun1type=1
    elseif(@fn1=="Julia 1")
      fun1type=2
    elseif(@fn1=="Newton M 1")
      fun1type=3
    elseif(@fn1=="Newton J 1")
      fun1type=4
    elseif(@fn1=="Popcorn C 1")
      fun1type=5
    elseif(@fn1=="Popcorn R 1")
      fun1type=6
    elseif(@fn1=="Mandelbrot 2")
      fun1type=7
    elseif(@fn1=="Julia 2")
      fun1type=8
    elseif(@fn1=="Newton M 2")
      fun1type=9
    elseif(@fn1=="Newton J 2")
      fun1type=10
    elseif(@fn1=="Popcorn C 2")
      fun1type=11
    elseif(@fn1=="Popcorn R 2")
      fun1type=12
    endif
    if(@fn2=="Mandelbrot 1")
      fun2type=1
    elseif(@fn2=="Julia 1")
      fun2type=2
    elseif(@fn2=="Newton M 1")
      fun2type=3
    elseif(@fn2=="Newton J 1")
      fun2type=4
    elseif(@fn2=="Popcorn C 1")
      fun2type=5
    elseif(@fn2=="Popcorn R 1")
      fun2type=6
    elseif(@fn2=="Mandelbrot 2")
      fun2type=7
    elseif(@fn2=="Julia 2")
      fun2type=8
    elseif(@fn2=="Newton M 2")
      fun2type=9
    elseif(@fn2=="Newton J 2")
      fun2type=10
    elseif(@fn2=="Popcorn C 2")
      fun2type=11
    elseif(@fn2=="Popcorn R 2")
      fun2type=12
    endif
    if(@fn3=="Mandelbrot 1")
      fun3type=1
    elseif(@fn3=="Julia 1")
      fun3type=2
    elseif(@fn3=="Newton M 1")
      fun3type=3
    elseif(@fn3=="Newton J 1")
      fun3type=4
    elseif(@fn3=="Popcorn C 1")
      fun3type=5
    elseif(@fn3=="Popcorn R 1")
      fun3type=6
    elseif(@fn3=="Mandelbrot 2")
      fun3type=7
    elseif(@fn3=="Julia 2")
      fun3type=8
    elseif(@fn3=="Newton M 2")
      fun3type=9
    elseif(@fn3=="Newton J 2")
      fun3type=10
    elseif(@fn3=="Popcorn C 2")
      fun3type=11
    elseif(@fn3=="Popcorn R 2")
      fun3type=12
    endif
    if(@fn4=="Mandelbrot 1")
      fun4type=1
    elseif(@fn4=="Julia 1")
      fun4type=2
    elseif(@fn4=="Newton M 1")
      fun4type=3
    elseif(@fn4=="Newton J 1")
      fun4type=4
    elseif(@fn4=="Popcorn C 1")
      fun4type=5
    elseif(@fn4=="Popcorn R 1")
      fun4type=6
    elseif(@fn4=="Mandelbrot 2")
      fun4type=7
    elseif(@fn4=="Julia 2")
      fun4type=8
    elseif(@fn4=="Newton M 2")
      fun4type=9
    elseif(@fn4=="Newton J 2")
      fun4type=10
    elseif(@fn4=="Popcorn C 2")
      fun4type=11
    elseif(@fn4=="Popcorn R 2")
      fun4type=12
    endif
    if(@fn5=="Mandelbrot 1")
      fun5type=1
    elseif(@fn5=="Julia 1")
      fun5type=2
    elseif(@fn5=="Newton M 1")
      fun5type=3
    elseif(@fn5=="Newton J 1")
      fun5type=4
    elseif(@fn5=="Popcorn C 1")
      fun5type=5
    elseif(@fn5=="Popcorn R 1")
      fun5type=6
    elseif(@fn5=="Mandelbrot 2")
      fun5type=7
    elseif(@fn5=="Julia 2")
      fun5type=8
    elseif(@fn5=="Newton M 2")
      fun5type=9
    elseif(@fn5=="Newton J 2")
      fun5type=10
    elseif(@fn5=="Popcorn C 2")
      fun5type=11
    elseif(@fn5=="Popcorn R 2")
      fun5type=12
    endif
;
; 6 functions
;
  elseif(@nfunction=="6")
    if(@fn1=="Mandelbrot 1")
      fun1type=1
    elseif(@fn1=="Julia 1")
      fun1type=2
    elseif(@fn1=="Newton M 1")
      fun1type=3
    elseif(@fn1=="Newton J 1")
      fun1type=4
    elseif(@fn1=="Popcorn C 1")
      fun1type=5
    elseif(@fn1=="Popcorn R 1")
      fun1type=6
    elseif(@fn1=="Mandelbrot 2")
      fun1type=7
    elseif(@fn1=="Julia 2")
      fun1type=8
    elseif(@fn1=="Newton M 2")
      fun1type=9
    elseif(@fn1=="Newton J 2")
      fun1type=10
    elseif(@fn1=="Popcorn C 2")
      fun1type=11
    elseif(@fn1=="Popcorn R 2")
      fun1type=12
    endif
    if(@fn2=="Mandelbrot 1")
      fun2type=1
    elseif(@fn2=="Julia 1")
      fun2type=2
    elseif(@fn2=="Newton M 1")
      fun2type=3
    elseif(@fn2=="Newton J 1")
      fun2type=4
    elseif(@fn2=="Popcorn C 1")
      fun2type=5
    elseif(@fn2=="Popcorn R 1")
      fun2type=6
    elseif(@fn2=="Mandelbrot 2")
      fun2type=7
    elseif(@fn2=="Julia 2")
      fun2type=8
    elseif(@fn2=="Newton M 2")
      fun2type=9
    elseif(@fn2=="Newton J 2")
      fun2type=10
    elseif(@fn2=="Popcorn C 2")
      fun2type=11
    elseif(@fn2=="Popcorn R 2")
      fun2type=12
    endif
    if(@fn3=="Mandelbrot 1")
      fun3type=1
    elseif(@fn3=="Julia 1")
      fun3type=2
    elseif(@fn3=="Newton M 1")
      fun3type=3
    elseif(@fn3=="Newton J 1")
      fun3type=4
    elseif(@fn3=="Popcorn C 1")
      fun3type=5
    elseif(@fn3=="Popcorn R 1")
      fun3type=6
    elseif(@fn3=="Mandelbrot 2")
      fun3type=7
    elseif(@fn3=="Julia 2")
      fun3type=8
    elseif(@fn3=="Newton M 2")
      fun3type=9
    elseif(@fn3=="Newton J 2")
      fun3type=10
    elseif(@fn3=="Popcorn C 2")
      fun3type=11
    elseif(@fn3=="Popcorn R 2")
      fun3type=12
    endif
    if(@fn4=="Mandelbrot 1")
      fun4type=1
    elseif(@fn4=="Julia 1")
      fun4type=2
    elseif(@fn4=="Newton M 1")
      fun4type=3
    elseif(@fn4=="Newton J 1")
      fun4type=4
    elseif(@fn4=="Popcorn C 1")
      fun4type=5
    elseif(@fn4=="Popcorn R 1")
      fun4type=6
    elseif(@fn4=="Mandelbrot 2")
      fun4type=7
    elseif(@fn4=="Julia 2")
      fun4type=8
    elseif(@fn4=="Newton M 2")
      fun4type=9
    elseif(@fn4=="Newton J 2")
      fun4type=10
    elseif(@fn4=="Popcorn C 2")
      fun4type=11
    elseif(@fn4=="Popcorn R 2")
      fun4type=12
    endif
    if(@fn5=="Mandelbrot 1")
      fun5type=1
    elseif(@fn5=="Julia 1")
      fun5type=2
    elseif(@fn5=="Newton M 1")
      fun5type=3
    elseif(@fn5=="Newton J 1")
      fun5type=4
    elseif(@fn5=="Popcorn C 1")
      fun5type=5
    elseif(@fn5=="Popcorn R 1")
      fun5type=6
    elseif(@fn5=="Mandelbrot 2")
      fun5type=7
    elseif(@fn5=="Julia 2")
      fun5type=8
    elseif(@fn5=="Newton M 2")
      fun5type=9
    elseif(@fn5=="Newton J 2")
      fun5type=10
    elseif(@fn5=="Popcorn C 2")
      fun5type=11
    elseif(@fn5=="Popcorn R 2")
      fun5type=12
    endif
    if(@fn6=="Mandelbrot 1")
      fun6type=1
    elseif(@fn6=="Julia 1")
      fun6type=2
    elseif(@fn6=="Newton M 1")
      fun6type=3
    elseif(@fn6=="Newton J 1")
      fun6type=4
    elseif(@fn6=="Popcorn C 1")
      fun6type=5
    elseif(@fn6=="Popcorn R 1")
      fun6type=6
    elseif(@fn6=="Mandelbrot 2")
      fun6type=7
    elseif(@fn6=="Julia 2")
      fun6type=8
    elseif(@fn6=="Newton M 2")
      fun6type=9
    elseif(@fn6=="Newton J 2")
      fun6type=10
    elseif(@fn6=="Popcorn C 2")
      fun6type=11
    elseif(@fn6=="Popcorn R 2")
      fun6type=12
    endif
  endif
;
; iterate each interpolating function
;
  if(fun1type==1) ; Mandelbrot 1
    z1=z^@m1power+#pixel
  elseif(fun1type==2) ; Julia 1
    z1=z^@j1power+@j1param
  elseif(fun1type==3) ; Newton M 1, roots of pixel
    dz=(z^@nm1power-#pixel)/(@nm1power*z^(@nm1power-1))
    z1=z-@nm1weight*dz
  elseif(fun1type==4) ; Newton J 1, roots of specific value
    dz=(z^@nj1power-@nj1param)/(@nj1power*z^(@nj1power-1))
    z1=z-@nj1weight*dz
  elseif(fun1type==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    z1=xnew+flip(ynew)
  elseif(fun1type==6) ; Popcorn R 1, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    z1=xnew+flip(ynew)
  elseif(fun1type==7) ; Mandelbrot 2
    z1=z^@m2power+#pixel
  elseif(fun1type==8) ; Julia 2
    z1=z^@j2power+@j2param
  elseif(fun1type==9) ; Newton M 2, roots of pixel
    dz=(z^@nm2power-#pixel)/(@nm2power*z^(@nm2power-1))
    z1=z-@nm2weight*dz
  elseif(fun1type==10) ; Newton J 2, roots of specific value
    dz=(z^@nj2power-@nj2param)/(@nj2power*z^(@nj2power-1))
    z1=z-@nj2weight*dz
  elseif(fun1type==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    z1=xnew+flip(ynew)
  elseif(fun1type==12) ; Popcorn R 2, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    z1=xnew+flip(ynew)
  endif
;
  if(fun2type==1) ; Mandelbrot 1
    z2=z^@m1power+#pixel
  elseif(fun2type==2) ; Julia 1
    z2=z^@j1power+@j1param
  elseif(fun2type==3) ; Newton M 1, roots of pixel
    dz=(z^@nm1power-#pixel)/(@nm1power*z^(@nm1power-1))
    z2=z-@nm1weight*dz
  elseif(fun2type==4) ; Newton J 1, roots of specific value
    dz=(z^@nj1power-@nj1param)/(@nj1power*z^(@nj1power-1))
    z2=z-@nj1weight*dz
  elseif(fun2type==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    z2=xnew+flip(ynew)
  elseif(fun2type==6) ; Popcorn R 1, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    z2=xnew+flip(ynew)
  elseif(fun2type==7) ; Mandelbrot 2
    z2=z^@m2power+#pixel
  elseif(fun2type==8) ; Julia 2
    z2=z^@j2power+@j2param
  elseif(fun2type==9) ; Newton M 2, roots of pixel
    dz=(z^@nm2power-#pixel)/(@nm2power*z^(@nm2power-1))
    z2=z-@nm2weight*dz
  elseif(fun2type==10) ; Newton J 2, roots of specific value
    dz=(z^@nj2power-@nj2param)/(@nj2power*z^(@nj2power-1))
    z2=z-@nj2weight*dz
  elseif(fun2type==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    z2=xnew+flip(ynew)
  elseif(fun2type==12) ; Popcorn R 2, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    z2=xnew+flip(ynew)
  endif
;
  if(fun3type==1) ; Mandelbrot 1
    z3=z^@m1power+#pixel
  elseif(fun3type==2) ; Julia 1
    z3=z^@j1power+@j1param
  elseif(fun3type==3) ; Newton M 1, roots of pixel
    dz=(z^@nm1power-#pixel)/(@nm1power*z^(@nm1power-1))
    z3=z-@nm1weight*dz
  elseif(fun3type==4) ; Newton J 1, roots of specific value
    dz=(z^@nj1power-@nj1param)/(@nj1power*z^(@nj1power-1))
    z3=z-@nj1weight*dz
  elseif(fun3type==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    z3=xnew+flip(ynew)
  elseif(fun3type==6) ; Popcorn R 1, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    z3=xnew+flip(ynew)
  elseif(fun3type==7) ; Mandelbrot 2
    z3=z^@m2power+#pixel
  elseif(fun3type==8) ; Julia 2
    z3=z^@j2power+@j2param
  elseif(fun3type==9) ; Newton M 2, roots of pixel
    dz=(z^@nm2power-#pixel)/(@nm2power*z^(@nm2power-1))
    z3=z-@nm2weight*dz
  elseif(fun3type==10) ; Newton J 2, roots of specific value
    dz=(z^@nj2power-@nj2param)/(@nj2power*z^(@nj2power-1))
    z3=z-@nj2weight*dz
  elseif(fun3type==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    z3=xnew+flip(ynew)
  elseif(fun3type==12) ; Popcorn R 2, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    z3=xnew+flip(ynew)
  endif
;
  if(fun4type==1) ; Mandelbrot 1
    z4=z^@m1power+#pixel
  elseif(fun4type==2) ; Julia 1
    z4=z^@j1power+@j1param
  elseif(fun4type==3) ; Newton M 1, roots of pixel
    dz=(z^@nm1power-#pixel)/(@nm1power*z^(@nm1power-1))
    z4=z-@nm1weight*dz
  elseif(fun4type==4) ; Newton J 1, roots of specific value
    dz=(z^@nj1power-@nj1param)/(@nj1power*z^(@nj1power-1))
    z4=z-@nj1weight*dz
  elseif(fun4type==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    z4=xnew+flip(ynew)
  elseif(fun4type==6) ; Popcorn R 1, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    z4=xnew+flip(ynew)
  elseif(fun4type==7) ; Mandelbrot 2
    z4=z^@m2power+#pixel
  elseif(fun4type==8) ; Julia 2
    z4=z^@j2power+@j2param
  elseif(fun4type==9) ; Newton M 2, roots of pixel
    dz=(z^@nm2power-#pixel)/(@nm2power*z^(@nm2power-1))
    z4=z-@nm2weight*dz
  elseif(fun4type==10) ; Newton J 2, roots of specific value
    dz=(z^@nj2power-@nj2param)/(@nj2power*z^(@nj2power-1))
    z4=z-@nj2weight*dz
  elseif(fun4type==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    z4=xnew+flip(ynew)
  elseif(fun4type==12) ; Popcorn R 2, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    z4=xnew+flip(ynew)
  endif
;
  if(fun5type==1) ; Mandelbrot 1
    z5=z^@m1power+#pixel
  elseif(fun5type==2) ; Julia 1
    z5=z^@j1power+@j1param
  elseif(fun5type==3) ; Newton M 1, roots of pixel
    dz=(z^@nm1power-#pixel)/(@nm1power*z^(@nm1power-1))
    z5=z-@nm1weight*dz
  elseif(fun5type==4) ; Newton J 1, roots of specific value
    dz=(z^@nj1power-@nj1param)/(@nj1power*z^(@nj1power-1))
    z5=z-@nj1weight*dz
  elseif(fun5type==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    z5=xnew+flip(ynew)
  elseif(fun5type==6) ; Popcorn R 1, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    z5=xnew+flip(ynew)
  elseif(fun5type==7) ; Mandelbrot 2
    z5=z^@m2power+#pixel
  elseif(fun5type==8) ; Julia 2
    z5=z^@j2power+@j2param
  elseif(fun5type==9) ; Newton M 2, roots of pixel
    dz=(z^@nm2power-#pixel)/(@nm2power*z^(@nm2power-1))
    z5=z-@nm2weight*dz
  elseif(fun5type==10) ; Newton J 2, roots of specific value
    dz=(z^@nj2power-@nj2param)/(@nj2power*z^(@nj2power-1))
    z5=z-@nj2weight*dz
  elseif(fun5type==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    z5=xnew+flip(ynew)
  elseif(fun5type==12) ; Popcorn R 2, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    z5=xnew+flip(ynew)
  endif
;
  if(fun6type==1) ; Mandelbrot 1
    z6=z^@m1power+#pixel
  elseif(fun6type==2) ; Julia 1
    z6=z^@j1power+@j1param
  elseif(fun6type==3) ; Newton M 1, roots of pixel
    dz=(z^@nm1power-#pixel)/(@nm1power*z^(@nm1power-1))
    z6=z-@nm1weight*dz
  elseif(fun6type==4) ; Newton J 1, roots of specific value
    dz=(z^@nj1power-@nj1param)/(@nj1power*z^(@nj1power-1))
    z6=z-@nj1weight*dz
  elseif(fun6type==5) ; Popcorn C 1, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc1h*@pc1out(yold+@pc1in(@pc1c*yold))
    ynew=yold-@pc1h*@pc1out(xold+@pc1in(@pc1c*xold))
    z6=xnew+flip(ynew)
  elseif(fun6type==6) ; Popcorn R 1, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr1h*real(@pr1out(yold+real(@pr1in(@pr1c*yold))))
    ynew=yold-@pr1h*real(@pr1out(xold+real(@pr1in(@pr1c*xold))))
    z6=xnew+flip(ynew)
  elseif(fun6type==7) ; Mandelbrot 2
    z6=z^@m2power+#pixel
  elseif(fun6type==8) ; Julia 2
    z6=z^@j2power+@j2param
  elseif(fun6type==9) ; Newton M 2, roots of pixel
    dz=(z^@nm2power-#pixel)/(@nm2power*z^(@nm2power-1))
    z6=z-@nm2weight*dz
  elseif(fun6type==10) ; Newton J 2, roots of specific value
    dz=(z^@nj2power-@nj2param)/(@nj2power*z^(@nj2power-1))
    z6=z-@nj2weight*dz
  elseif(fun6type==11) ; Popcorn C 2, popcorn with complex variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pc2h*@pc2out(yold+@pc2in(@pc2c*yold))
    ynew=yold-@pc2h*@pc2out(xold+@pc2in(@pc2c*xold))
    z6=xnew+flip(ynew)
  elseif(fun6type==12) ; Popcorn R 2, popcorn with real variables
    xold=real(z)
    yold=imag(z)
    xnew=xold-@pr2h*real(@pr2out(yold+real(@pr2in(@pr2c*yold))))
    ynew=yold-@pr2h*real(@pr2out(xold+real(@pr2in(@pr2c*xold))))
    z6=xnew+flip(ynew)
  endif
;
; interpolate
;
;   1 function--no interpolation
;
  if(@nfunction=="1")
    z=z1
;
;   2 functions
;
  elseif(@nfunction=="2")
    if(@slider2_type=="constant")
      weight2=@slider2-1
      weight1=1-weight2
    elseif(@slider2_type=="cos(iter)")
      weight2=(1-cos(@freq2*iter))/2
      weight1=1-weight2
    elseif(@slider2_type=="sin(iter)")
      weight2=(1+sin(@freq2*iter))/2
      weight1=1-weight2
    elseif(@slider2_type=="tanh(iter)")
      weight2=tanh(@freq2*iter)
      weight1=1-weight2
    elseif(@slider2_type=="cos(real)")
      weight2=(1-cos(@freq2*real(z)))/2
      weight1=1-weight2
    elseif(@slider2_type=="sin(imag)")
      weight2=(1+sin(@freq2*imag(z)))/2
      weight1=1-weight2
    elseif(@slider2_type=="tanh(mag)")
      weight2=tanh(@freq2*cabs(z))
      weight1=1-weight2
    elseif(@slider2_type=="angle")
      weight2=(atan2(z)/#pi+1)/2
      weight2=(weight2*@freq2)%1
      weight1=1-weight2
    endif
    z=weight1*z1+weight2*z2
;
;   3 functions
;
  elseif(@nfunction=="3")
    if(@slider3_type=="constant")
      t=(@slider3-1)/2
    elseif(@slider3_type=="cos(iter)")
      t=(1-cos(@freq3*iter))/2
    elseif(@slider3_type=="sin(iter)")
      t=(1+sin(@freq3*iter))/2
    elseif(@slider3_type=="tanh(iter)")
      t=tanh(@freq3*iter)
    elseif(@slider3_type=="cos(real)")
      t=(1-cos(@freq3*real(z)))/2
    elseif(@slider3_type=="sin(imag)")
      t=(1+sin(@freq3*imag(z)))/2
    elseif(@slider3_type=="tanh(mag)")
      t=tanh(@freq3*cabs(z))
    elseif(@slider3_type=="angle")
      t=(atan2(z)/#pi+1)/2
      t=(t*@freq3)%1
    endif
    if(@interp3_type=="Lagrange")
      t=t*2+1
      z=z1*(t-2)*(t-3)/((1-2)*(1-3))\
       +z2*(t-1)*(t-3)/((2-1)*(2-3))\
       +z3*(t-1)*(t-2)/((3-1)*(3-2))
    elseif(@interp3_type=="Bezier")
      z=sqr(1-t)*z1+2*(1-t)*t*z2+sqr(t)*z3
    elseif(@interp3_type=="Fourier")
      a0=2/3*(z1+z2+z3)
      a1=2/3*(z1-z2/2-z3/2)
      b1=sqrt(3)/3*(z2-z3)
      t=t*4*#pi/3
      z=a0/2+a1*cos(t)+b1*sin(t)
    endif
;
;   4 functions
;
  elseif(@nfunction=="4")
    if(@slider4_type=="constant")
      t=(@slider4-1)/3
    elseif(@slider4_type=="cos(iter)")
      t=(1-cos(@freq4*iter))/2
    elseif(@slider4_type=="sin(iter)")
      t=(1+sin(@freq4*iter))/2
    elseif(@slider4_type=="tanh(iter)")
      t=tanh(@freq4*iter)
    elseif(@slider4_type=="cos(real)")
      t=(1-cos(@freq4*real(z)))/2
    elseif(@slider4_type=="sin(imag)")
      t=(1+sin(@freq4*imag(z)))/2
    elseif(@slider4_type=="tanh(mag)")
      t=tanh(@freq4*cabs(z))
    elseif(@slider4_type=="angle")
      t=(atan2(z)/#pi+1)/2
      t=(t*@freq4)%1
    endif
    if(@interp4_type=="Lagrange")
      t=t*3+1
      z=z1*(t-2)*(t-3)*(t-4)/((1-2)*(1-3)*(1-4))\
       +z2*(t-1)*(t-3)*(t-4)/((2-1)*(2-3)*(2-4))\
       +z3*(t-1)*(t-2)*(t-4)/((3-1)*(3-2)*(3-4))\
       +z4*(t-1)*(t-2)*(t-3)/((4-1)*(4-2)*(4-3))
    elseif(@interp4_type=="Bezier")
      z=(1-t)^3*z1+3*sqr(1-t)*t*z2+3*(1-t)*sqr(t)*z3+t^3*z4
    elseif(@interp4_type=="Fourier")
      a0=(z1+z2+z3+z4)/2
      a1=(z1-z3)/2
      a2=(z1-z2+z3-z4)/2
      b1=(z2-z4)/2
      b2=0
      t=t*1.5*#pi
      z=a0/2+a1*cos(t)+a2/2*cos(2*t)+b1*sin(t)+b2*sin(2*t)
    endif
;
;   5 functions
;
  elseif(@nfunction=="5")
    if(@slider5_type=="constant")
      t=(@slider5-1)/4
    elseif(@slider5_type=="cos(iter)")
      t=(1-cos(@freq5*iter))/2
    elseif(@slider5_type=="sin(iter)")
      t=(1+sin(@freq5*iter))/2
    elseif(@slider5_type=="tanh(iter)")
      t=tanh(@freq5*iter)
    elseif(@slider5_type=="cos(real)")
      t=(1-cos(@freq5*real(z)))/2
    elseif(@slider5_type=="sin(imag)")
      t=(1+sin(@freq5*imag(z)))/2
    elseif(@slider5_type=="tanh(mag)")
      t=tanh(@freq5*cabs(z))
    elseif(@slider5_type=="angle")
      t=(atan2(z)/#pi+1)/2
      t=(t*@freq5)%1
    endif
    if(@interp5_type=="Lagrange")
      t=t*4+1
      z=z1*(t-2)*(t-3)*(t-4)*(t-5)/((1-2)*(1-3)*(1-4)*(1-5))\
       +z2*(t-1)*(t-3)*(t-4)*(t-5)/((2-1)*(2-3)*(2-4)*(2-5))\
       +z3*(t-1)*(t-2)*(t-4)*(t-5)/((3-1)*(3-2)*(3-4)*(3-5))\
       +z4*(t-1)*(t-2)*(t-3)*(t-5)/((4-1)*(4-2)*(4-3)*(4-5))\
       +z5*(t-1)*(t-2)*(t-3)*(t-4)/((5-1)*(5-2)*(5-3)*(5-4))
    elseif(@interp5_type=="Bezier")
      z=(1-t)^4*z1+4*(1-t)^3*t*z2+6*sqr(1-t)*sqr(t)*z3+4*(1-t)*t^3*z4\
       +t^4*z5
    elseif(@interp5_type=="Fourier")
      a0=0.4*(z1+z2+z3+z4+z5)
      a1=0.4*(z1+z2*cos(0.4*#pi)+z3*cos(0.8*#pi)+z4*cos(1.2*#pi)+z5*cos(1.6*#pi))
      a2=0.4*(z1+z2*cos(0.8*#pi)+z3*cos(1.6*#pi)+z4*cos(2.4*#pi)+z5*cos(3.2*#pi))
      b1=0.4*(z2*sin(0.4*#pi)+z3*sin(0.8*#pi)+z4*sin(1.2*#pi)+z5*sin(1.6*#pi))
      b2=0.4*(z2*sin(0.8*#pi)+z3*sin(1.6*#pi)+z4*sin(2.4*#pi)+z5*sin(3.2*#pi))
      t=t*1.6*#pi
      z=a0/2+a1*cos(t)+a2*cos(2*t)+b1*sin(t)+b2*sin(2*t)
    endif
;
;   6 functions
;
  elseif(@nfunction=="6")
    if(@slider6_type=="constant")
      t=(@slider6-1)/5
    elseif(@slider6_type=="cos(iter)")
      t=(1-cos(@freq6*iter))/2
    elseif(@slider6_type=="sin(iter)")
      t=(1+sin(@freq6*iter))/2
    elseif(@slider6_type=="tanh(iter)")
      t=tanh(@freq6*iter)
    elseif(@slider6_type=="cos(real)")
      t=(1-cos(@freq6*real(z)))/2
    elseif(@slider6_type=="sin(imag)")
      t=(1+sin(@freq6*imag(z)))/2
    elseif(@slider6_type=="tanh(mag)")
      t=tanh(@freq6*cabs(z))
    elseif(@slider6_type=="angle")
      t=(atan2(z)/#pi+1)/2
      t=(t*@freq6)%1
    endif
    if(@interp6_type=="Lagrange")
      t=t*5+1
      z=z1*(t-2)*(t-3)*(t-4)*(t-5)*(t-6)/((1-2)*(1-3)*(1-4)*(1-5)*(1-6))\
       +z2*(t-1)*(t-3)*(t-4)*(t-5)*(t-6)/((2-1)*(2-3)*(2-4)*(2-5)*(2-6))\
       +z3*(t-1)*(t-2)*(t-4)*(t-5)*(t-6)/((3-1)*(3-2)*(3-4)*(3-5)*(3-6))\
       +z4*(t-1)*(t-2)*(t-3)*(t-5)*(t-6)/((4-1)*(4-2)*(4-3)*(4-5)*(4-6))\
       +z5*(t-1)*(t-2)*(t-3)*(t-4)*(t-6)/((5-1)*(5-2)*(5-3)*(5-4)*(5-6))\
       +z6*(t-1)*(t-2)*(t-3)*(t-4)*(t-5)/((6-1)*(6-2)*(6-3)*(6-4)*(6-5))
    elseif(@interp6_type=="Bezier")
      z=(1-t)^5*z1+5*(1-t)^4*t*z2+10*(1-t)^3*sqr(t)*z3+10*sqr(1-t)*t^3*z4\
       +5*(1-t)*t^4*z5+t^5*z6
    elseif(@interp6_type=="Fourier")
      a0=(z1+z2+z3+z4+z5+z6)/3
      a1=(z1+z2/2-z3/2-z4-z5/2+z6/2)/3
      a2=(z1-z2/2-z3/2+z4-z5/2-z6/2)/3
      a3=(z1-z2+z3-z4+z5-z6)/3
      b1=sqrt(3)/6*(z2+z3-z5-z6)
      b2=sqrt(3)/6*(z2-z3+z5-z6)
      b3=0
      t=t*5*#pi/3
      z=a0/2+a1*cos(t)+a2*cos(2*t)+a3/2*cos(3*t)+b1*sin(t)+b2*sin(2*t)+b3*sin(3*t)
    endif
  endif
bailout:
  |z|<@bailout
default:
  title="Interpolating Functions"
  heading
    text="Allows you to interpolate between several functions every iteration."
  endheading
;
; general parameters
;
  heading
    caption="General parameters"
  endheading
  param initz_type
    caption="initial z type"
    default=1
    enum="manual" "pixel"
  endparam
  complex param initial_z
    caption="initial z"
    default=(0,0)
    visible=(@initz_type=="manual")
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; interpolating function defintions
;
  heading
    caption="Interpolating functions"
  endheading
  param nfunction
    caption="# of functions"
    default=1
    enum="1" "2" "3" "4" "5" "6"
  endparam
  param fn1
    caption="function 1"
    default=1
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
  endparam
  param fn2
    caption="function 2"
    default=7
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>0)
  endparam
  param fn3
    caption="function 3"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>1)
  endparam
  param fn4
    caption="function 4"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>2)
  endparam
  param fn5
    caption="function 5"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>3)
  endparam
  param fn6
    caption="function 6"
    default=0
    enum=" " "Mandelbrot 1" "Julia 1" "Newton M 1" "Newton J 1" "Popcorn C 1" "Popcorn R 1"\
      "Mandelbrot 2" "Julia 2" "Newton M 2" "Newton J 2" "Popcorn C 2" "Popcorn R 2"
    visible=(@nfunction>4)
  endparam
;
  complex param m1power
    caption="Mandelbrot 1 power"
    default=(2,0)
    visible=((@fn1=="Mandelbrot 1")||(@fn2=="Mandelbrot 1")||(@fn3=="Mandelbrot 1")||\
      (@fn4=="Mandelbrot 1")||(@fn5=="Mandelbrot 1")||(@fn6=="Mandelbrot 1"))
  endparam
  complex param j1power
    caption="Julia 1 power"
    default=(2,0)
    visible=((@fn1=="Julia 1")||(@fn2=="Julia 1")||(@fn3=="Julia 1")||\
      (@fn4=="Julia 1")||(@fn5=="Julia 1")||(@fn6=="Julia 1"))
  endparam
  complex param j1param
    caption="Julia 1 parameter"
    default=(0,1)
    visible=((@fn1=="Julia 1")||(@fn2=="Julia 1")||(@fn3=="Julia 1")||\
      (@fn4=="Julia 1")||(@fn5=="Julia 1")||(@fn6=="Julia 1"))
  endparam
  complex param nm1power
    caption="Newton M 1 power"
    default=(3,0)
    visible=((@fn1=="Newton M 1")||(@fn2=="Newton M 1")||(@fn3=="Newton M 1")||\
      (@fn4=="Newton M 1")||(@fn5=="Newton M 1")||(@fn6=="Newton M 1"))
  endparam
  complex param nm1weight
    caption="Newton M 1 nova fac"
    default=1
    visible=((@fn1=="Newton M 1")||(@fn2=="Newton M 1")||(@fn3=="Newton M 1")||\
      (@fn4=="Newton M 1")||(@fn5=="Newton M 1")||(@fn6=="Newton M 1"))
  endparam
  complex param nj1power
    caption="Newton J 1 power"
    default=(4,0)
    visible=((@fn1=="Newton J 1")||(@fn2=="Newton J 1")||(@fn3=="Newton J 1")||\
      (@fn4=="Newton J 1")||(@fn5=="Newton J 1")||(@fn6=="Newton J 1"))
  endparam
  complex param nj1param
    caption="Newton J 1 parameter"
    default=(1,0)
    visible=((@fn1=="Newton J 1")||(@fn2=="Newton J 1")||(@fn3=="Newton J 1")||\
      (@fn4=="Newton J 1")||(@fn5=="Newton J 1")||(@fn6=="Newton J 1"))
  endparam
  complex param nj1weight
    caption="Newton J 1 nova fac"
    default=1
    visible=((@fn1=="Newton J 1")||(@fn2=="Newton J 1")||(@fn3=="Newton J 1")||\
      (@fn4=="Newton J 1")||(@fn5=="Newton J 1")||(@fn6=="Newton J 1"))
  endparam
  complex param pc1h
    caption="complex step size 1"
    default=(0.1,0)
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endparam
  complex param pc1c
    caption="complex frequency 1"
    default=(3,0)
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endparam
  func pc1out
    caption="complex outer function 1"
    default=sin()
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endfunc
  func pc1in
    caption="complex inner function 1"
    default=tan()
    visible=((@fn1=="Popcorn C 1")||(@fn2=="Popcorn C 1")||(@fn3=="Popcorn C 1")||\
      (@fn4=="Popcorn C 1")||(@fn5=="Popcorn C 1")||(@fn6=="Popcorn C 1"))
  endfunc
  float param pr1h
    caption="real step size 1"
    default=0.1
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endparam
  float param pr1c
    caption="real frequency 1"
    default=3
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endparam
  func pr1out
    caption="real outer function 1"
    default=sin()
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endfunc
  func pr1in
    caption="real inner function 1"
    default=tan()
    visible=((@fn1=="Popcorn R 1")||(@fn2=="Popcorn R 1")||(@fn3=="Popcorn R 1")||\
      (@fn4=="Popcorn R 1")||(@fn5=="Popcorn R 1")||(@fn6=="Popcorn R 1"))
  endfunc
  complex param m2power
    caption="Mandelbrot 2 power"
    default=(4,0)
    visible=((@fn1=="Mandelbrot 2")||(@fn2=="Mandelbrot 2")||(@fn3=="Mandelbrot 2")||\
      (@fn4=="Mandelbrot 2")||(@fn5=="Mandelbrot 2")||(@fn6=="Mandelbrot 2"))
  endparam
  complex param j2power
    caption="Julia 2 power"
    default=(4,0)
    visible=((@fn1=="Julia 2")||(@fn2=="Julia 2")||(@fn3=="Julia 2")||\
      (@fn4=="Julia 2")||(@fn5=="Julia 2")||(@fn6=="Julia 2"))
  endparam
  complex param j2param
    caption="Julia 2 parameter"
    default=(-1.26,0)
    visible=((@fn1=="Julia 2")||(@fn2=="Julia 2")||(@fn3=="Julia 2")||\
      (@fn4=="Julia 2")||(@fn5=="Julia 2")||(@fn6=="Julia 2"))
  endparam
  complex param nm2power
    caption="Newton M 2 power"
    default=(3,0)
    visible=((@fn1=="Newton M 2")||(@fn2=="Newton M 2")||(@fn3=="Newton M 2")||\
      (@fn4=="Newton M 2")||(@fn5=="Newton M 2")||(@fn6=="Newton M 2"))
  endparam
  complex param nm2weight
    caption="Newton M 2 nova fac"
    default=1
    visible=((@fn1=="Newton M 2")||(@fn2=="Newton M 2")||(@fn3=="Newton M 2")||\
      (@fn4=="Newton M 2")||(@fn5=="Newton M 2")||(@fn6=="Newton M 2"))
  endparam
  complex param nj2power
    caption="Newton J 2 power"
    default=(4,0)
    visible=((@fn1=="Newton J 2")||(@fn2=="Newton J 2")||(@fn3=="Newton J 2")||\
      (@fn4=="Newton J 2")||(@fn5=="Newton J 2")||(@fn6=="Newton J 2"))
  endparam
  complex param nj2param
    caption="Newton J 2 parameter"
    default=(1,0)
    visible=((@fn1=="Newton J 2")||(@fn2=="Newton J 2")||(@fn3=="Newton J 2")||\
      (@fn4=="Newton J 2")||(@fn5=="Newton J 2")||(@fn6=="Newton J 2"))
  endparam
  complex param nj2weight
    caption="Newton J 2 nova fac"
    default=(1,0)
    visible=((@fn1=="Newton J 2")||(@fn2=="Newton J 2")||(@fn3=="Newton J 2")||\
      (@fn4=="Newton J 2")||(@fn5=="Newton J 2")||(@fn6=="Newton J 2"))
  endparam
  complex param pc2h
    caption="complex step size 2"
    default=(0.1,0)
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endparam
  complex param pc2c
    caption="complex frequency 2"
    default=(3,0)
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endparam
  func pc2out
    caption="complex outer function 2"
    default=sin()
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endfunc
  func pc2in
    caption="complex inner function 2"
    default=tan()
    visible=((@fn1=="Popcorn C 2")||(@fn2=="Popcorn C 2")||(@fn3=="Popcorn C 2")||\
      (@fn4=="Popcorn C 2")||(@fn5=="Popcorn C 2")||(@fn6=="Popcorn C 2"))
  endfunc
  float param pr2h
    caption="real step size 2"
    default=0.1
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endparam
  float param pr2c
    caption="real frequency 2"
    default=3
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endparam
  func pr2out
    caption="real outer function 2"
    default=sin()
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endfunc
  func pr2in
    caption="real inner function 2"
    default=tan()
    visible=((@fn1=="Popcorn R 2")||(@fn2=="Popcorn R 2")||(@fn3=="Popcorn R 2")||\
      (@fn4=="Popcorn R 2")||(@fn5=="Popcorn R 2")||(@fn6=="Popcorn R 2"))
  endfunc
;
; interpolation
;
  heading
    caption="Interpolation"
  endheading
;
;   2 functions
;
  param slider2_type
    caption="slider type"
    default=0
    enum="constant" "cos(iter)" "sin(iter)" "tanh(iter)" \
      "cos(real)" "sin(imag)" "tanh(mag)" "angle"
    visible=(@nfunction=="2")
  endparam
  float param slider2
    caption="slider value"
    default=1.5
    min=1
    max=2
    hint="1 for z1, 2 for z2"
    visible=((@nfunction=="2")&&(@slider2_type=="constant"))
  endparam
  float param freq2
    caption="frequency"
    default=1
    min=0
    visible=((@nfunction=="2")&&(@slider2_type!="constant"))
  endparam
;
;  3 functions
;
  param interp3_type
    caption="interpolation type"
    default=0
    enum="Lagrange" "Bezier" "Fourier"
    visible=(@nfunction=="3")
  endparam
  param slider3_type
    caption="slider type"
    default=0
    enum="constant" "cos(iter)" "sin(iter)" "tanh(iter)" \
      "cos(real)" "sin(imag)" "tanh(mag)" "angle"
    visible=(@nfunction=="3")
  endparam
  float param slider3
    caption="slider value"
    default=2
    min=1
    max=3
    hint="1 for z1, 2 for z2, 3 for z3"
    visible=((@nfunction=="3")&&(@slider3_type=="constant"))
  endparam
  float param freq3
    caption="frequency"
    default=1
    min=0
    visible=((@nfunction=="3")&&(@slider3_type!="constant"))
  endparam
;
;  4 functions
;
  param interp4_type
    caption="interpolation type"
    default=0
    enum="Lagrange" "Bezier" "Fourier"
    visible=(@nfunction=="4")
  endparam
  param slider4_type
    caption="slider type"
    default=0
    enum="constant" "cos(iter)" "sin(iter)" "tanh(iter)" \
      "cos(real)" "sin(imag)" "tanh(mag)" "angle"
    visible=(@nfunction=="4")
  endparam
  float param slider4
    caption="slider value"
    default=2.5
    min=1
    max=4
    hint="1 for z1 to 4 for z4"
    visible=((@nfunction=="4")&&(@slider4_type=="constant"))
  endparam
  float param freq4
    caption="frequency"
    default=1
    min=0
    visible=((@nfunction=="4")&&(@slider4_type!="constant"))
  endparam
;
;  5 functions
;
  param interp5_type
    caption="interpolation type"
    default=0
    enum="Lagrange" "Bezier" "Fourier"
    visible=(@nfunction=="5")
  endparam
  param slider5_type
    caption="slider type"
    default=0
    enum="constant" "cos(iter)" "sin(iter)" "tanh(iter)" \
      "cos(real)" "sin(imag)" "tanh(mag)" "angle"
    visible=(@nfunction=="5")
  endparam
  float param slider5
    caption="slider value"
    default=3
    min=1
    max=5
    hint="1 for z1 to 5 for z5"
    visible=((@nfunction=="5")&&(@slider5_type=="constant"))
  endparam
  float param freq5
    caption="frequency"
    default=1
    min=0
    visible=((@nfunction=="5")&&(@slider5_type!="constant"))
  endparam
;
;  6 functions
;
  param interp6_type
    caption="interpolation type"
    default=0
    enum="Lagrange" "Bezier" "Fourier"
    visible=(@nfunction=="6")
  endparam
  param slider6_type
    caption="slider type"
    default=0
    enum="constant" "cos(iter)" "sin(iter)" "tanh(iter)" \
      "cos(real)" "sin(imag)" "tanh(mag)" "angle"
    visible=(@nfunction=="6")
  endparam
  float param slider6
    caption="slider value"
    default=3.5
    min=1
    max=6
    hint="1 for z1 to 6 for z6"
    visible=((@nfunction=="6")&&(@slider6_type=="constant"))
  endparam
  float param freq6
    caption="frequency"
    default=1
    min=0
    visible=((@nfunction=="6")&&(@slider6_type!="constant"))
  endparam
}

fourier-popcorn { ; Kerry Mitchell 17May2015
;
; This is a complex-valued implementation and generalization
; of Cliff Pickover's "popcorn" algorithm.  His method iterates
; 2 real variables like this:
;
; xnew = xold - h * sin(a*y + tan(b*y))
; ynew = yold - h * sin(a*x + tan(b*x))
;
; where a = 1. The current method generalizes the above by:
;   using truncated Fourier series instead of just single sin or tan function
;   allowing for complex values of a (slope)
;   having the same Fourier series parameters (basic controls) or
;   having different parameters for the 4 series (x inner, x outer,
;     y inner, and y outer)
;
init:
  complex temp=(0,0)
  complex tempx=(0,0)
  complex tempy=(0,0)
  complex temp2=(0,0)
  complex temp2x=(0,0)
  complex temp2y=(0,0)
  complex tempin=(0,0)
  complex tempinx=(0,0)
  complex tempiny=(0,0)
  complex tempout=(0,0)
  complex tempoutx=(0,0)
  complex tempouty=(0,0)
  complex x=real(#pixel)
  complex y=imag(#pixel)
  complex xnew=(0,0)
  complex ynew=(0,0)
  complex z=#pixel
  int iterm=0
  int jterm=0
  int parity=1
loop:
;
; basic controls
;
  if(@control_type=="basic")
;
; choose what inner variable to use
;
  tempinx=y
  tempiny=x
;
; compute the inner Fourier series
;
  tempx=0
  tempy=0
  iterm=0
  while(iterm<@ntermall)
    iterm=iterm+1
    if(@functypeall=="sin odd pos")
      jterm=2*iterm-1
      tempx=tempx+sin(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+sin(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos odd pos")
      jterm=2*iterm-1
      tempx=tempx+cos(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+cos(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin all pos")
      jterm=iterm
      tempx=tempx+sin(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+sin(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos all pos")
      jterm=iterm
      tempx=tempx+cos(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+cos(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      tempx=tempx+parity*sin(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+parity*sin(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      tempx=tempx+parity*cos(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+parity*cos(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      tempx=tempx+parity*sin(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+parity*sin(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      tempx=tempx+parity*cos(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+parity*cos(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin general")
      jterm=iterm
      tempx=tempx+@funcall(jterm)*sin(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+@funcall(jterm)*sin(tempiny*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos general")
      jterm=iterm
      tempx=tempx+@funcall(jterm)*cos(tempinx*jterm)/(jterm^@powerall)
      tempy=tempy+@funcall(jterm)*cos(tempiny*jterm)/(jterm^@powerall)
    endif
  endwhile
;
; choose what outer variable to use
;
  tempoutx=y
  tempouty=x
;
; compute the outer Fourier series
;
  temp2x=0
  temp2y=0
  iterm=0
  while(iterm<@ntermall)
    iterm=iterm+1
    if(@functypeall=="sin odd pos")
      jterm=2*iterm-1
      temp2x=temp2x+sin((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+sin((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos odd pos")
      jterm=2*iterm-1
      temp2x=temp2x+cos((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+cos((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin all pos")
      jterm=iterm
      temp2x=temp2x+sin((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+sin((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos all pos")
      jterm=iterm
      temp2x=temp2x+cos((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+cos((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp2x=temp2x+parity*sin((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+parity*sin((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp2x=temp2x+parity*cos((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+parity*cos((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp2x=temp2x+parity*sin((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+parity*sin((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp2x=temp2x+parity*cos((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+parity*cos((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="sin general")
      jterm=iterm
      temp2x=temp2x+@funcall(jterm)*sin((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+@funcall(jterm)*sin((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    elseif(@functypeall=="cos general")
      jterm=iterm
      temp2x=temp2x+@funcall(jterm)*cos((@slopeall*tempoutx+tempx)*jterm)/(jterm^@powerall)
      temp2y=temp2y+@funcall(jterm)*cos((@slopeall*tempouty+tempy)*jterm)/(jterm^@powerall)
    endif
  endwhile
  xnew=x+@hall*temp2x
  ynew=y+@hall*temp2y
  x=xnew
  y=ynew
  z=x+flip(y)
;
; advanced controls
;
  else
;
; x equation
;
;   choose what inner variable to use
;
  if(@vartypexin=="x")
    tempin=x
  elseif(@vartypexin=="y")
    tempin=y
  elseif(@vartypexin=="x+y")
    tempin=x+y
  elseif(@vartypexin=="y-x")
    tempin=y-x
  elseif(@vartypexin=="x*y")
    tempin=x*y
  elseif(@vartypexin=="y/x")
    tempin=y/x
  endif
;
;   compute the inner Fourier series
;
  temp=0
  iterm=0
  while(iterm<@ntermxin)
    iterm=iterm+1
    if(@functypexin=="sin odd pos")
      jterm=2*iterm-1
      temp=temp+sin(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="cos odd pos")
      jterm=2*iterm-1
      temp=temp+cos(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="sin all pos")
      jterm=iterm
      temp=temp+sin(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="cos all pos")
      jterm=iterm
      temp=temp+cos(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="sin odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp=temp+parity*sin(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="cos odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp=temp+parity*cos(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="sin all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp=temp+parity*sin(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="cos all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp=temp+parity*cos(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="sin general")
      jterm=iterm
      temp=temp+@funcxin(jterm)*sin(tempin*jterm)/(jterm^@powerxin)
    elseif(@functypexin=="cos general")
      jterm=iterm
      temp=temp+@funcxin(jterm)*cos(tempin*jterm)/(jterm^@powerxin)
    endif
  endwhile
;
;   choose what outer variable to use
;
  if(@vartypexout=="x")
    tempout=x
  elseif(@vartypexout=="y")
    tempout=y
  elseif(@vartypexout=="x+y")
    tempout=x+y
  elseif(@vartypexout=="y-x")
    tempout=y-x
  elseif(@vartypexout=="x*y")
    tempout=x*y
  elseif(@vartypexout=="y/x")
    tempout=y/x
  endif
;
;   compute the outer Fourier series
;
  temp2=0
  iterm=0
  while(iterm<@ntermxout)
    iterm=iterm+1
    if(@functypexout=="sin odd pos")
      jterm=2*iterm-1
      temp2=temp2+sin((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="cos odd pos")
      jterm=2*iterm-1
      temp2=temp2+cos((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="sin all pos")
      jterm=iterm
      temp2=temp2+sin((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="cos all pos")
      jterm=iterm
      temp2=temp2+cos((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="sin odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp2=temp2+parity*sin((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="cos odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp2=temp2+parity*cos((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="sin all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp2=temp2+parity*sin((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="cos all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp2=temp2+parity*cos((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="sin general")
      jterm=iterm
      temp2=temp2+@funcxout(jterm)*sin((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    elseif(@functypexout=="cos general")
      jterm=iterm
      temp2=temp2+@funcxout(jterm)*cos((@slopexout*tempout+temp)*jterm)/(jterm^@powerxout)
    endif
  endwhile
  xnew=x+@hx*temp2
;
; y equation
;
;   choose what inner variable to use
;
  if(@vartypeyin=="x")
    tempin=x
  elseif(@vartypeyin=="y")
    tempin=y
  elseif(@vartypeyin=="x+y")
    tempin=x+y
  elseif(@vartypeyin=="y-x")
    tempin=y-x
  elseif(@vartypeyin=="x*y")
    tempin=x*y
  elseif(@vartypeyin=="y/x")
    tempin=y/x
  endif
;
;   compute the inner Fourier series
;
  temp=0
  iterm=0
  while(iterm<@ntermyin)
    iterm=iterm+1
    if(@functypeyin=="sin odd pos")
      jterm=2*iterm-1
      temp=temp+sin(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="cos odd pos")
      jterm=2*iterm-1
      temp=temp+cos(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="sin all pos")
      jterm=iterm
      temp=temp+sin(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="cos all pos")
      jterm=iterm
      temp=temp+cos(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="sin odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp=temp+parity*sin(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="cos odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp=temp+parity*cos(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="sin all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp=temp+parity*sin(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="cos all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp=temp+parity*cos(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="sin general")
      jterm=iterm
      temp=temp+@funcyin(jterm)*sin(tempin*jterm)/(jterm^@poweryin)
    elseif(@functypeyin=="cos general")
      jterm=iterm
      temp=temp+@funcyin(jterm)*cos(tempin*jterm)/(jterm^@poweryin)
    endif
  endwhile
;
;   choose what outer variable to use
;
  if(@vartypeyout=="x")
    tempout=x
  elseif(@vartypeyout=="y")
    tempout=y
  elseif(@vartypeyout=="x+y")
    tempout=x+y
  elseif(@vartypeyout=="y-x")
    tempout=y-x
  elseif(@vartypeyout=="x*y")
    tempout=x*y
  elseif(@vartypeyout=="y/x")
    tempout=y/x
  endif
;
;   compute the outer Fourier series
;
  temp2=0
  iterm=0
  while(iterm<@ntermyout)
    iterm=iterm+1
    if(@functypeyout=="sin odd pos")
      jterm=2*iterm-1
      temp2=temp2+sin((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="cos odd pos")
      jterm=2*iterm-1
      temp2=temp2+cos((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="sin all pos")
      jterm=iterm
      temp2=temp2+sin((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="cos all pos")
      jterm=iterm
      temp2=temp2+cos((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="sin odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp2=temp2+parity*sin((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="cos odd alt")
      jterm=2*iterm-1
      parity=(iterm%2)*2-1
      temp2=temp2+parity*cos((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="sin all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp2=temp2+parity*sin((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="cos all alt")
      jterm=iterm
      parity=(iterm%2)*2-1
      temp2=temp2+parity*cos((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="sin general")
      jterm=iterm
      temp2=temp2+@funcyout(jterm)*sin((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    elseif(@functypeyout=="cos general")
      jterm=iterm
      temp2=temp2+@funcyout(jterm)*cos((@slopeyout*tempout+temp)*jterm)/(jterm^@poweryout)
    endif
  endwhile
  ynew=y+@hy*temp2
  x=xnew
  y=ynew
  z=x+flip(y)
  endif
bailout:
  |z|<@bailout
default:
  title="Fourier Popcorn"
  heading
    text="Uses Cliff Pickover's Popcorn routine, substituting Fourier \
      series for the sine and tangent functions."
  endheading
;
; general parameters
;
  heading
    caption="General parameters"
  endheading
  param control_type
    caption="control type"
    default=0
    enum="basic" "advanced"
  endparam
  float param bailout
    caption="bailout value"
    default=1000
  endparam
;
; basic parameters
;
  heading
    caption="Fourier series parameters"
    visible=(@control_type=="basic")
  endheading
  complex param hall
    caption="step size"
    default=0.1
    visible=(@control_type=="basic")
  endparam
  param functypeall
    caption="function type"
    default=0
    enum="sin odd pos" "cos odd pos" "sin all pos" "cos all pos" \
      "sin odd alt" "cos odd alt" "sin all alt" "cos all alt" \
      "sin general" "cos general"
    visible=(@control_type=="basic")
  endparam
  complex param powerall
    caption="power"
    default=1
    visible=(@control_type=="basic")
  endparam
  int param ntermall
    caption="# terms"
    default=10
    min=0
    visible=(@control_type=="basic")
  endparam
  complex param slopeall
    caption="slope"
    default=1
    visible=(@control_type=="basic")
  endparam
  func funcall
    caption="function"
    default=tan()
    visible=((@control_type=="basic")&&((@functypeall=="sin general")\
      ||(@functypeall=="cos general")))
  endfunc
;
; advanced parameters
;
; x inner parameters
;
  heading
    caption="x inner parameters"
    visible=(@control_type=="advanced")
  endheading
  complex param hx
    caption="step size"
    default=0.1
    visible=(@control_type=="advanced")
  endparam
  param vartypexin
    caption="inner variable type"
    default=1
    enum="x" "y" "x+y" "y-x" "x*y" "y/x"
    visible=(@control_type=="advanced")
  endparam
  param functypexin
    caption="function type"
    default=1
    enum="sin odd pos" "cos odd pos" "sin all pos" "cos all pos" \
      "sin odd alt" "cos odd alt" "sin all alt" "cos all alt" \
      "sin general" "cos general"
    visible=(@control_type=="advanced")
  endparam
  complex param powerxin
    caption="power"
    default=1
    visible=(@control_type=="advanced")
  endparam
  int param ntermxin
    caption="# terms"
    default=10
    min=0
    visible=(@control_type=="advanced")
  endparam
  func funcxin
    caption="function"
    default=cos()
    visible=((@control_type=="advanced")&&((@functypexin=="sin general")\
      ||(@functypexin=="cos general")))
  endfunc
;
; x outer parameters
;
  heading
    caption="x outer parameters"
    visible=(@control_type=="advanced")
  endheading
  param vartypexout
    caption="outer variable type"
    default=1
    enum="x" "y" "x+y" "y-x" "x*y" "y/x"
    visible=(@control_type=="advanced")
  endparam
  param functypexout
    caption="function type"
    default=1
    enum="sin odd pos" "cos odd pos" "sin all pos" "cos all pos" \
      "sin odd alt" "cos odd alt" "sin all alt" "cos all alt" \
      "sin general" "cos general"
    visible=(@control_type=="advanced")
  endparam
  complex param powerxout
    caption="power"
    default=1
    visible=(@control_type=="advanced")
  endparam
  int param ntermxout
    caption="# terms"
    default=10
    min=0
    visible=(@control_type=="advanced")
  endparam
  complex param slopexout
    caption="slope"
    default=1
    visible=(@control_type=="advanced")
  endparam
  func funcxout
    caption="function"
    default=cos()
    visible=((@control_type=="advanced")&&((@functypexout=="sin general")\
      ||(@functypexout=="cos general")))
  endfunc
;
; y inner parameters
;
  heading
    caption="y inner parameters"
    visible=(@control_type=="advanced")
  endheading
  complex param hy
    caption="step size"
    default=0.1
    visible=(@control_type=="advanced")
  endparam
  param vartypeyin
    caption="inner variable type"
    default=0
    enum="x" "y" "x+y" "y-x" "x*y" "y/x"
    visible=(@control_type=="advanced")
  endparam
  param functypeyin
    caption="function type"
    default=0
    enum="sin odd pos" "cos odd pos" "sin all pos" "cos all pos" \
      "sin odd alt" "cos odd alt" "sin all alt" "cos all alt" \
      "sin general" "cos general"
    visible=(@control_type=="advanced")
  endparam
  complex param poweryin
    caption="power"
    default=1
    visible=(@control_type=="advanced")
  endparam
  int param ntermyin
    caption="# terms"
    default=10
    min=0
    visible=(@control_type=="advanced")
  endparam
  func funcyin
    caption="function"
    default=sin()
    visible=((@control_type=="advanced")&&((@functypeyin=="sin general")\
      ||(@functypeyin=="cos general")))
  endfunc
;
; y outer parameters
;
  heading
    caption="y outer parameters"
    visible=(@control_type=="advanced")
  endheading
  param vartypeyout
    caption="outer variable type"
    default=0
    enum="x" "y" "x+y" "y-x" "x*y" "y/x"
    visible=(@control_type=="advanced")
  endparam
  param functypeyout
    caption="function type"
    default=0
    enum="sin odd pos" "cos odd pos" "sin all pos" "cos all pos" \
      "sin odd alt" "cos odd alt" "sin all alt" "cos all alt" \
      "sin general" "cos general"
    visible=(@control_type=="advanced")
  endparam
  complex param poweryout
    caption="power"
    default=1
    visible=(@control_type=="advanced")
  endparam
  int param ntermyout
    caption="# terms"
    default=10
    min=0
    visible=(@control_type=="advanced")
  endparam
  complex param slopeyout
    caption="slope"
    default=1
    visible=(@control_type=="advanced")
  endparam
  func funcyout
    caption="function"
    default=sin()
    visible=((@control_type=="advanced")&&((@functypeyout=="sin general")\
      ||(@functypeyout=="cos general")))
  endfunc
}

breaking-newton-real { ; Kerry Mitchell 28Jul2018
;
; force chaos into newton's method
; by changing c and/or z and/or Newton factor each iteration
;
init:
  complex c=(0,0)
  complex dz=(0,0)
  complex g=(0,0)
  complex gp=(0,0)
  complex incrementc=(0,0)
  complex incrementfac=(0,0)
  complex incrementz=(0,0)
  complex newtfac=(0,0)
  complex z=(0,0)
  complex zc=(0,0)
  float cx=0
  float cy=0
  float dx=0
  float dy=0
  float facx=0
  float facy=0
  float gx=0
  float gxp=0
  float gy=0
  float gyp=0
  float x=0
  float y=0
;
; initializations
;
  if(@pixel_type=="initial c")
    c=#pixel
    incrementc=@cinc
    zc=@initialz
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="c change")
    c=@initialc
    incrementc=#pixel
    zc=@initialz
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="initial z")
    c=@initialc
    incrementc=@cinc
    zc=#pixel
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="z change")
    c=@initialc
    incrementc=@cinc
    zc=@initialz
    incrementz=#pixel
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="initial Newton factor")
    c=@initialc
    incrementc=@cinc
    zc=@initialz
    incrementz=@zinc
    newtfac=#pixel
    incrementfac=@facinc
  elseif(@pixel_type=="Newton factor change")
    c=@initialc
    incrementc=@cinc
    zc=@initialz
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=#pixel
  endif
loop:
;
; iterate
;
;  x equation
;
  cx=real(c)
  facx=real(newtfac)
  x=real(zc)
  if(@x_type=="x^2 = c")
    gx=x^2-cx
    gxp=2*x
  elseif(@x_type=="x^4 = c")
    gx=x^4-cx
    gxp=4*x^3
  elseif(@x_type=="x^6 = c")
    gx=x^6-cx
    gxp=6*x^5
  elseif(@x_type=="sin(x) = c")
    gx=sin(@freqx*x)-cx
    gxp=@freqx*cos(@freqx*x)
  elseif(@x_type=="cos(x) = c")
    gx=cos(@freqx*x)-cx
    gxp=-@freqx*sin(@freqx*x)
  elseif(@x_type=="tanh(x) = c")
    gx=tanh(@freqx*x)-cx
    gxp=@freqx/sqr(cosh(@freqx*x))
  elseif(@x_type=="exp(x) = c")
    gxp=exp(x)
    gx=gxp-cx
  endif
  dx=gx/gxp
  x=x-facx*dx
;
;  y equation
;
  cy=imag(c)
  facy=imag(newtfac)
  y=imag(zc)
  if(@y_type=="y^2 = c")
    gy=y^2-cy
    gyp=2*y
  elseif(@y_type=="y^4 = c")
    gy=y^4-cy
    gyp=4*y^3
  elseif(@y_type=="y^6 = c")
    gy=y^6-cy
    gyp=6*y^5
  elseif(@y_type=="sin(y) = c")
    gy=sin(@freqy*y)-cy
    gyp=@freqy*cos(@freqy*y)
  elseif(@y_type=="cos(y) = c")
    gy=cos(@freqy*y)-cy
    gyp=-@freqy*sin(@freqy*y)
  elseif(@y_type=="tanh(y) = c")
    gy=tanh(@freqy*y)-cy
    gyp=@freqy/sqr(cosh(@freqy*y))
  elseif(@y_type=="exp(y) = c")
    gyp=exp(y)
    gy=gyp-cy
  endif
  dy=gy/gyp
  y=y-facy*dy
;
;   combine both equations & find new zc
;
  g=gx+flip(gy)
  gp=gxp+flip(gyp)
  dz=dx+flip(dy)
  zc=x+flip(y)
;
; change c
;
  if(@changec=="add")
    c=c+incrementc
  elseif(@changec=="multiply")
    c=c*incrementc
  elseif(@changec=="power")
    c=c^incrementc
  endif
;
; change z
;
  if(@changez=="add")
    zc=zc+incrementz
  elseif(@changez=="multiply")
    zc=zc*incrementz
  elseif(@changez=="power")
    zc=zc^incrementz
  endif
;
; change factor
;
  if(@changefac=="add")
    newtfac=newtfac+incrementfac
  elseif(@changefac=="multiply")
    newtfac=newtfac*incrementfac
  elseif(@changefac=="power")
    newtfac=newtfac^incrementfac
  endif
;
; set z for coloring
;
  if(@ztype=="iterate")
    z=zc
  elseif(@ztype=="function")
    z=g
  elseif(@ztype=="derivative")
    z=gp
  elseif(@ztype=="difference")
    z=1/dz
  endif
bailout:
  |1/dz|<@bailout
default:
  title="Breaking Newton - Real"
  heading
    text="Attempts to solve two real-valued equations,\
      one for x and one for y, using Newton's method.\
      The equations generally don't have solutions. \
      Chaos may be introduced by changing c, z, and/\
      or the Newton factor at each iteration."
  endheading
;
; general parameters
;
  heading
    caption="general parameters"
  endheading
  param pixel_type
    caption="pixel type"
    default=4
    enum="initial c" "c change" "initial z" "z change" "initial Newton factor" "Newton factor change"
  endparam
  param x_type
    caption="x type"
    default=0
    enum="x^2 = c" "x^4 = c" "x^6 = c" "sin(x) = c" "cos(x) = c" "tanh(x) = c" "exp(x) = c"
  endparam
  float param freqx
    caption="frequency"
    default=2
    visible=((@x_type=="sin(x) = c")||(@x_type=="cos(x) = c")||(@x_type=="tanh(x) = c"))
  endparam
  param y_type
    caption="y type"
    default=0
    enum="y^2 = c" "y^4 = c" "y^6 = c" "sin(y) = c" "cos(y) = c" "tanh(y) = c" "exp(y) = c"
  endparam
  float param freqy
    caption="frequency"
    default=3
    visible=((@y_type=="sin(y) = c")||(@y_type=="cos(y) = c")||(@y_type=="tanh(y) = c"))
  endparam
  float param bailout
    caption="bailout"
    default=1000000
  endparam
;
; c parameters
;
  heading
    caption="c parameters"
  endheading
  complex param initialc
    caption="initial c"
    default=(2,3)
    visible=(@pixel_type!="initial c")
  endparam
  param changec
    caption="change c"
    default=0
    enum="none" "add" "multiply" "power"
  endparam
  complex param cinc
    caption="change amount"
    default=1
    visible=((@changec!="none")&&(@pixel_type!="c change"))
  endparam
;
; z parameters
;
  heading
    caption="z parameters"
  endheading
  complex param initialz
    caption="initial z"
    default=(1,2)
    visible=(@pixel_type!="initial z")
  endparam
  param changez
    caption="change z"
    default=2
    enum="none" "add" "multiply" "power"
  endparam
  complex param zinc
    caption="change amount"
    default=(2,-3)
    visible=((@changez!="none")&&(@pixel_type!="z change"))
  endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "function" "derivative" "difference"
  endparam
;
; Newton factor parameters
;
  heading
    caption="Newton factor parameters"
  endheading
  complex param initialfac
    caption="initial factor"
    default=(1,1)
    visible=(@pixel_type!="initial Newton factor")
  endparam
  param changefac
    caption="change factor"
    default=0
    enum="none" "add" "multiply" "power"
  endparam
  complex param facinc
    caption="change amount"
    default=1
    visible=((@changefac!="none")&&(@pixel_type!="Newton factor change"))
  endparam
}

breaking-newton-complex { ; Kerry Mitchell 28Jul2018
;
; force chaos into newton's method
; by changing c and/or z and/or Newton factor each iteration
;
init:
  complex c=(0,0)
  complex dz=(0,0)
  complex g=(0,0)
  complex gp=(0,0)
  complex incrementc=(0,0)
  complex incrementfac=(0,0)
  complex incrementz=(0,0)
  complex lhs=(0,0)
  complex lhsp=(0,0)
  complex newtfac=(0,0)
  complex rhs=(0,0)
  complex rhsp=(0,0)
  complex z=(0,0)
  complex zc=(0,0)
;
; initializations
;
  if(@pixel_type=="initial c")
    c=#pixel
    incrementc=@cinc
    zc=@initialz
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="c change")
    c=@initialc
    incrementc=#pixel
    zc=@initialz
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="initial z")
    c=@initialc
    incrementc=@cinc
    zc=#pixel
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="z change")
    c=@initialc
    incrementc=@cinc
    zc=@initialz
    incrementz=#pixel
    newtfac=@initialfac
    incrementfac=@facinc
  elseif(@pixel_type=="initial Newton factor")
    c=@initialc
    incrementc=@cinc
    zc=@initialz
    incrementz=@zinc
    newtfac=#pixel
    incrementfac=@facinc
  elseif(@pixel_type=="Newton factor change")
    c=@initialc
    incrementc=@cinc
    zc=@initialz
    incrementz=@zinc
    newtfac=@initialfac
    incrementfac=#pixel
  endif
loop:
;
; iterate
;
;  lefthand side of equation
;
  if(@lhs_type=="power")
    lhs=zc^@lpower
    lhsp=@lpower*zc^(@lpower-1)
  elseif(@lhs_type=="exp")
    lhs=exp(zc)
    lhsp=lhs
  elseif(@lhs_type=="log")
    lhs=log(zc)
    lhsp=1/zc
  elseif(@lhs_type=="sin")
    lhs=sin(@lfreq*zc)
    lhsp=@lfreq*cos(@lfreq*zc)
  elseif(@lhs_type=="cos")
    lhs=cos(@lfreq*zc)
    lhsp=-@lfreq*sin(@lfreq*zc)
  elseif(@lhs_type=="tan")
    lhs=tan(@lfreq*zc)
    lhsp=@lfreq/sqr(cos(@lfreq*zc))
  elseif(@lhs_type=="sinh")
    lhs=sinh(@lfreq*zc)
    lhsp=@lfreq*cosh(@lfreq*zc)
  elseif(@lhs_type=="cosh")
    lhs=cosh(@lfreq*zc)
    lhsp=@lfreq*sinh(@lfreq*zc)
  elseif(@lhs_type=="tanh")
    lhs=tanh(@lfreq*zc)
    lhsp=@lfreq/sqr(cosh(@lfreq*zc))
  else
    lhs=c
    lhsp=0
  endif
;
;  righthand side of equation
;
  if(@rhs_type=="power")
    rhs=zc^@rpower
    rhsp=@rpower*zc^(@rpower-1)
  elseif(@rhs_type=="exp")
    rhs=exp(zc)
    rhsp=rhs
  elseif(@rhs_type=="log")
    rhs=log(zc)
    rhsp=1/zc
  elseif(@rhs_type=="sin")
    rhs=sin(@rfreq*zc)
    rhsp=@rfreq*cos(@rfreq*zc)
  elseif(@rhs_type=="cos")
    rhs=cos(@rfreq*zc)
    rhsp=-@rfreq*sin(@rfreq*zc)
  elseif(@rhs_type=="tan")
    rhs=tan(@rfreq*zc)
    rhsp=@rfreq/sqr(cos(@rfreq*zc))
  elseif(@rhs_type=="sinh")
    rhs=sinh(@rfreq*zc)
    rhsp=@rfreq*cosh(@rfreq*zc)
  elseif(@rhs_type=="cosh")
    rhs=cosh(@rfreq*zc)
    rhsp=@rfreq*sinh(@rfreq*zc)
  elseif(@rhs_type=="tanh")
    rhs=tanh(@rfreq*zc)
    rhsp=@rfreq/sqr(cosh(@rfreq*zc))
  else
    rhs=c
    rhsp=0
  endif
;
;   combine both sides & find new zc
;
  g=lhs-rhs
  gp=lhsp-rhsp
  dz=g/gp
  zc=zc-newtfac*dz
;
; change c
;
  if(@changec=="add")
    c=c+incrementc
  elseif(@changec=="multiply")
    c=c*incrementc
  elseif(@changec=="power")
    c=c^incrementc
  endif
;
; change z
;
  if(@changez=="add")
    zc=zc+incrementz
  elseif(@changez=="multiply")
    zc=zc*incrementz
  elseif(@changez=="power")
    zc=zc^incrementz
  endif
;
; change factor
;
  if(@changefac=="add")
    newtfac=newtfac+incrementfac
  elseif(@changefac=="multiply")
    newtfac=newtfac*incrementfac
  elseif(@changefac=="power")
    newtfac=newtfac^incrementfac
  endif
;
; set z for coloring
;
  if(@ztype=="iterate")
    z=zc
  elseif(@ztype=="function")
    z=g
  elseif(@ztype=="derivative")
    z=gp
  elseif(@ztype=="difference")
    z=1/dz
  endif
bailout:
  |1/dz|<@bailout
default:
  title="Breaking Newton - Complex"
  heading
    text="Attempts to solve equations of the form: lefthand = \
      righthand, using Newton's method. Chaos may be \
      introduced by changing c, z, and/or the Newton \
      factor at each iteration."
  endheading
;
; general parameters
;
  heading
    caption="general parameters"
  endheading
  param pixel_type
    caption="pixel type"
    default=4
    enum="initial c" "c change" "initial z" "z change" "initial Newton factor" "Newton factor change"
  endparam
  param lhs_type
    caption="lefthand type"
    default=1
    enum="c" "power" "exp" "log" "sin" "cos" "tan" "sinh" "cosh" "tanh"
  endparam
  complex param lpower
    caption="power"
    default=4
    visible=(@lhs_type=="power")
  endparam
  complex param lfreq
    caption="frequency"
    default=2
    visible=(@lhs_type>3)
  endparam
  param rhs_type
    caption="righthand type"
    default=0
    enum="c" "power" "exp" "log" "sin" "cos" "tan" "sinh" "cosh" "tanh"
  endparam
  complex param rpower
    caption="power"
    default=3
    visible=(@rhs_type=="power")
  endparam
  complex param rfreq
    caption="frequency"
    default=3
    visible=(@rhs_type>3)
  endparam
  float param bailout
    caption="bailout"
    default=1000000
  endparam
;
; c parameters
;
  heading
    caption="c parameters"
    visible=((@lhs_type=="c")||(@rhs_type=="c"))
  endheading
  complex param initialc
    caption="initial c"
    default=1
    visible=((@lhs_type=="c")||(@rhs_type=="c")&&(@pixel_type!="initial c"))
  endparam
  param changec
    caption="change c"
    default=0
    enum="none" "add" "multiply" "power"
    visible=((@lhs_type=="c")||(@rhs_type=="c"))
  endparam
  complex param cinc
    caption="change amount"
    default=1
    visible=(((@lhs_type=="c")||(@rhs_type=="c"))&&(@changec!="none")&&(@pixel_type!="c change"))
  endparam
;
; z parameters
;
  heading
    caption="z parameters"
  endheading
  complex param initialz
    caption="initial z"
    default=0.01
    visible=(@pixel_type!="initial z")
  endparam
  param changez
    caption="change z"
    default=0
    enum="none" "add" "multiply" "power"
  endparam
  complex param zinc
    caption="change amount"
    default=1
    visible=((@changez!="none")&&(@pixel_type!="z change"))
  endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "function" "derivative" "difference"
  endparam
;
; Newton factor parameters
;
  heading
    caption="Newton factor parameters"
  endheading
  complex param initialfac
    caption="initial factor"
    default=1
    visible=(@pixel_type!="initial Newton factor")
  endparam
  param changefac
    caption="change factor"
    default=0
    enum="none" "add" "multiply" "power"
  endparam
  complex param facinc
    caption="change amount"
    default=1
    visible=((@changefac!="none")&&(@pixel_type!="Newton factor change"))
  endparam
}

compound-mandelbrot { ; Kerry Mitchell 26Nov2019
;
; f(z) = ((z^n+c)/(z^m+c))^p+c
;
init:
  complex c=#pixel
  complex z=(0,0)
  if(@initialz_type=="pixel")
    z=c
  else
    z=@initialz
  endif
loop:
  z=((z^@npower+c)/(z^@mpower+c))^@ppower+c
bailout:
  |z|<@bailout
default:
  title="Compound Mandelbrot"
  heading
    text="Uses the formula z = ((z^m + c)/(z^n + c))^p + c; 3 exponents."
  endheading
  param initialz_type
    caption="initial z"
    default=0
    enum="pixel" "constant"
  endparam
  complex param initialz
    caption="initial z value"
    default=(0,0)
    visible=(@initialz_type=="constant")
  endparam
  complex param npower
    caption="numerator power"
    default=(2,0)
  endparam
  complex param mpower
    caption="denominator power"
    default=(1,0)
  endparam
  complex param ppower
    caption="overall power"
    default=(2,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
switch:
  type="compound-julia"
  julparam=#pixel
  npower=npower
  mpower=mpower
  ppower=ppower
  bailout=bailout
}

compound-julia { ; Kerry Mitchell 26Nov2019
;
; f(z) = ((z^n+c)/(z^m+c))^p+c
;
init:
  complex c=@julparam
  complex z=#pixel
loop:
  z=((z^@npower+c)/(z^@mpower+c))^@ppower+c
bailout:
  |z|<@bailout
default:
  title="Compound Julia"
  heading
    text="Uses the formula z = ((z^m + c)/(z^n + c))^p + c; 3 exponents."
  endheading
  complex param julparam
    caption="Julia parameter"
    default=(0,1)
  endparam
  complex param npower
    caption="numerator power"
    default=(2,0)
  endparam
  complex param mpower
    caption="denominator power"
    default=(1,0)
  endparam
  complex param ppower
    caption="overall power"
    default=(2,0)
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
switch:
  type="compound-mandelbrot"
  npower=npower
  mpower=mpower
  ppower=ppower
  bailout=bailout
}

mixed-up-multiplication-mandelbrot { ; Kerry Mitchell 27Nov2019
;
; Instead of using regular complex multiplication, this method uses a base
; shape (= unit circle normally). Use the polar angle of z to reach back
; to the base shape and find its coordinates. Then, use the actual and base
; magnitudes to determine a scale factor, and the arc length to determine
; the angle (angle = arc length / base magnitude). From there, the magnitude
; of z^power = scale factor ^ power, and angle of z^power = power * angle.
;
$define debug
global:
  complex corner[20]
  float pitch_s=0
  float pitch_trad=0
  float smax=0
  float tdegmax[20]
  float twopi=2*#pi
  int gi=0
;
; cruciform settings: central square side & four other squares, one
; on each side of central square
;
  if(@shape_type=="cruciform")
    corner[0]=(3,0)
    corner[1]=(3,1)
    corner[2]=(1,1)
    corner[3]=(1,3)
    corner[4]=(-1,3)
    corner[5]=(-1,1)
    corner[6]=(-3,1)
    corner[7]=(-3,-1)
    corner[8]=(-1,-1)
    corner[9]=(-1,-3)
    corner[10]=(1,-3)
    corner[11]=(1,-1)
    corner[12]=(3,-1)
    corner[13]=(3,0)
;
;   rescale so that corner[0] = (1,0)
;   find degree measures of corners
;
    gi=-1
    while(gi<12)
      gi=gi+1
      corner[gi]=corner[gi]/3
      tdegmax[gi]=(atan2(corner[gi])/#pi*180+720)%360
    endwhile
  elseif(@shape_type=="triangle")
    corner[0]=(1,0)
    corner[1]=(-1+flip(sqrt(3)))/2
    corner[2]=(-1-flip(sqrt(3)))/2
    tdegmax[0]=0
    tdegmax[1]=120
    tdegmax[2]=240
    tdegmax[3]=360
  elseif(@shape_type=="star")
    corner[0]=(1,0)
    corner[1]=@inrad*(1,1)/sqrt(2)
    corner[2]=(0,1)
    corner[3]=@inrad*(-1,1)/sqrt(2)
    corner[4]=(-1,0)
    corner[5]=@inrad*(-1,-1)/sqrt(2)
    corner[6]=(0,-1)
    corner[7]=@inrad*(1,-1)/sqrt(2)
    corner[8]=(1,0)
    tdegmax[0]=0
    tdegmax[1]=45
    tdegmax[2]=90
    tdegmax[3]=135
    tdegmax[4]=180
    tdegmax[5]=225
    tdegmax[6]=270
    tdegmax[7]=315
    tdegmax[8]=360
  endif
;
; gear settings
;
  if(@shape_type=="gear")
    pitch_trad=twopi/@nteeth
    smax=#pi*(1+@rsmall)
    pitch_s=smax/@nteeth
  endif
init:
  complex arccenter=(0,0)
  complex c=#pixel
  complex compk=(0,0)
  complex w=(0,0)
  complex z=#pixel
  float afac=0
  float bfac=0
  float cfac=0
  float dcrit=1/(1+@rsmall)
  float dtooth=0
  float h1=0
  float k1=0
  float k=0
  float phi=0
  float phimax=0
  float r=0
  float s=0
  float slope=0
  float tdeg=0
  float third=1/3
  float trad=0
  float x=0
  float xb=0
  float x3=0
  float y=0
  float yb=0
  float y3=0
  int itooth=0
loop:
;
; decompose z
;
  x=real(z), y=imag(z)
;
; square
;
  if(@shape_type=="square")
;
;   determine arc length from basis shape
;
    phimax=8
;
;   find scale factor k
;
    k=abs(x)
    if(abs(y)>k)
      k=abs(y)
    endif
;
;   find standard polar angle
;
    trad=atan2(z)
    if(trad<0)
      trad=trad+2*#pi
    endif
    tdeg=trad/#pi*180
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<45) ; upper right side
      xb=1, yb=y/k, s=yb
    elseif(tdeg<135) ; top
      xb=x/k, yb=1, s=2-xb
    elseif(tdeg<225) ; left side
      xb=-1, yb=y/k, s=4-yb
    elseif(tdeg<315) ; bottom
      xb=x/k, yb=-1, s=6+xb
    else ; lower right side
      xb=1, yb=y/k, s=8+yb
    endif
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<1) ; upper right side
      xb=1, yb=phi
    elseif(phi<3) ; top
      xb=2-phi, yb=1
    elseif(phi<5) ; left side
      xb=-1, yb=4-phi
    elseif(phi<7) ; bottom
      xb=phi-6, yb=-1
    else ; lower right side
      xb=1, yb=phi-8
    endif
    x=xb*k, y=yb*k
    z=x+flip(y)+c
;
; triangle
;
  elseif(@shape_type=="triangle")
;
;   determine arc length from basis shape
;
    third=sqrt(3)
    phimax=3*third
;
;   find standard magnitude and polar angle
;
    r=cabs(z)
    trad=atan2(z)
    tdeg=(trad/#pi*180+720)%360
    trad=tdeg/180*#pi
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<tdegmax[1]) ; side 1 top
      xb=third/(third+3*tan(trad)), yb=third/3*(1-xb), s=cabs(xb+flip(yb)-corner[0])
    elseif(tdeg<tdegmax[2]) ; side 2 left
      xb=-0.5, yb=xb*tan(trad), s=third+cabs(xb+flip(yb)-corner[1])
    else ; side 3 bottom
      xb=third/(third-3*tan(trad)), yb=third/3*(xb-1), s=2*third+cabs(xb+flip(yb)-corner[2])
    endif
    k=r/cabs(xb+flip(yb))
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<third) ; side 1 top
      w=phi/third*(corner[1]-corner[0])+corner[0]
    elseif(phi<(2*third)) ; side 2 left
      w=(phi/third-1)*(corner[2]-corner[1])+corner[1]
    else ; side 2 bottom
      w=(phi/third-2)*(corner[0]-corner[2])+corner[2]
    endif
    w=w*k, z=w+c
;
; quarter-circle arcs in quadrants
;
  elseif(@shape_type=="quarter-circle")
    phimax=2*#pi
;
;   z: find standard magnitude and polar angle
;
    r=cabs(z)
    trad=(atan2(z)+2*phimax)%phimax
    tdeg=trad/#pi*180
;
;   z: find coordinates of point on base shape and base shape arc length
;     equate angle of base point with angle of field point
;
    if(x==0)
      xb=0, yb=1
      s=#pi/2
      if(y<0)
        yb=-1
        s=3*#pi/2
      endif
    else
      if(tdeg<90) ; quadrant 1
        if(@q1_type=="in")
          arccenter=(1,1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac-sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=#pi/2-atan2((1-xb)+flip(1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      elseif(tdeg<180) ; quadrant 2
        if(@q2_type=="in")
          arccenter=(-1,1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac+sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=3*#pi/2-atan2((-1-xb)+flip(1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      elseif(tdeg<270) ; quadrant 3
        if(@q3_type=="in")
          arccenter=(-1,-1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac+sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=#pi/2-atan2((-1-xb)+flip(-1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      else ; quadrant 4
        if(@q4_type=="in")
          arccenter=(1,-1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac-sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=3*#pi/2-atan2((1-xb)+flip(-1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      endif
    endif
    phi=s
    k=r/cabs(xb+flip(yb))
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    tdeg=phi/#pi*180
    if(tdeg<90) ; quadrant 1
      if(@q1_type=="in")
        xb=1-sin(phi), yb=1-cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    elseif(tdeg<180) ; quadrant 2
      if(@q2_type=="in")
        xb=-1+sin(phi), yb=1+cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    elseif(tdeg<270) ; quadrant 3
      if(@q3_type=="in")
        xb=-1-sin(phi), yb=-1-cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    else ; quadrant 4
      if(@q4_type=="in")
        xb=1+sin(phi), yb=-1+cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    endif
    z=k*w+c
;
; cruciform
;
  elseif(@shape_type=="cruciform")
;
;   determine arc length from basis shape
;
    phimax=8
;
;   find standard magnitude and polar angle
;
    r=cabs(z)
    trad=atan2(z)
    tdeg=(trad/#pi*180+720)%360
    trad=tdeg/180*#pi
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<tdegmax[1]) ; side 1 (far right) upper
      xb=1, yb=xb*tan(trad), s=yb
    elseif(tdeg<tdegmax[2]) ; side 2
      yb=third, xb=yb/tan(trad), s=(1-xb)+third
    elseif(tdeg<tdegmax[3]) ; side 3
      xb=third, yb=xb*tan(trad), s=1+(yb-third)
    elseif(tdeg<tdegmax[4]) ; side 4
      yb=1, xb=yb/tan(trad), s=2-xb
    elseif(tdeg<tdegmax[5]) ; side 5
      xb=-third, yb=xb*tan(trad), s=7*third+(1-yb)
    elseif(tdeg<tdegmax[6]) ; side 6
      yb=third, xb=yb/tan(trad), s=3-(xb+third)
    elseif(tdeg<tdegmax[7]) ; side 7
      xb=-1, yb=xb*tan(trad), s=4-yb
    elseif(tdeg<tdegmax[8]) ; side 8
      yb=-third, xb=yb/tan(trad), s=16*third+xb
    elseif(tdeg<tdegmax[9]) ; side 9
      xb=-third, yb=xb*tan(trad), s=14*third-yb
    elseif(tdeg<tdegmax[10]) ; side 10
      yb=-1, xb=yb/tan(trad), s=6+xb
    elseif(tdeg<tdegmax[11]) ; side 11
      xb=third, yb=xb*tan(trad), s=22*third+yb
    elseif(tdeg<tdegmax[12]) ; side 12
      yb=-third, xb=yb/tan(trad), s=7+(xb-third)
    else ; side 1 lower
      xb=1, yb=xb*tan(trad), s=8+yb
    endif
    k=r/cabs(xb+flip(yb))
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<(1/3)) ; side 1 (far right) upper
      xb=1, yb=phi
    elseif(phi<1) ; side 2
      xb=third+1-phi, yb=third
    elseif(phi<(5/3)) ; side 3
      xb=third, yb=phi-2*third
    elseif(phi<(7/3)) ; side 4
      xb=2-phi, yb=1
    elseif(phi<3) ; side 5
      xb=-third, yb=10*third-phi
    elseif(phi<(11/3)) ; side 6
      xb=8/3-phi, yb=third
    elseif(phi<(13/3)) ; side 7
      xb=-1, yb=4-phi
    elseif(phi<5) ; side 8
      xb=-16/3+phi, yb=-third
    elseif(phi<(17/3)) ; side 9
      xb=-third, yb=14/3-phi
    elseif(phi<(19/3)) ; side 10
      xb=phi-6, yb=-1
    elseif(phi<7) ; side 11
      xb=third, yb=phi-22/3
    elseif(phi<(23/3)) ; side 12
      xb=phi-20/3, yb=-third
    else ; side 1 lower
      xb=1, yb=phi-8
    endif
    x=xb*k, y=yb*k
    z=x+flip(y)+c
;
; rose curve
;
  elseif(@shape_type=="rose curve")
    trad=atan2(z)
    r=1+@amp*sin(@freq*trad)
    xb=r*cos(trad), yb=r*sin(trad), w=xb+flip(yb)
    phi=trad, compk=z/w
    compk=compk^@power, phi=@power*phi
    r=1+@amp*sin(@freq*phi)
    xb=r*cos(phi), yb=r*sin(phi), w=xb+flip(yb)
    z=compk*w+c
;
; gear
;
  elseif(@shape_type=="gear")
;
;   z: find theta, which tooth z projects to, and radius (1 or rsmall)
;     itooth from 0 to nteeth-1, dtooth = fractional tooth, 0 to 1
;
    trad=(atan2(z)+2*twopi)%twopi
    dtooth=trad/pitch_trad
    itooth=trunc(dtooth), dtooth=dtooth-itooth
    if(dtooth<0.5) ; radius = 1
      k=cabs(z)
      s=itooth*pitch_s+dtooth*pitch_trad
    else ; radius = rsmall
      k=cabs(z)/@rsmall
      s=itooth*pitch_s+0.5*pitch_trad+(dtooth-0.5)*pitch_trad*@rsmall
    endif
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%smax
;
;   use arc length to find which tooth, radius, and project to new z and add c
;
    dtooth=phi/pitch_s
    itooth=trunc(dtooth), dtooth=dtooth-itooth
    if(dtooth<dcrit) ; radius = 1
      r=k
      trad=itooth*pitch_trad+dtooth/dcrit*pitch_trad/2
    else ; radius = rsmall
      r=k*@rsmall
      trad=itooth*pitch_trad+(1+(dtooth-dcrit)/(1-dcrit))*pitch_trad/2
    endif
    z=r*(cos(trad)+flip(sin(trad)))+c
;
; star
;
  elseif(@shape_type=="star")
;
;   determine arc length from basis shape
;
    phimax=cabs(corner[1]-corner[0])
;
;   find standard magnitude and polar angle
;
    r=cabs(z)
    trad=(atan2(z)+2*twopi)%twopi
    tdeg=trad/#pi*180
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<tdegmax[1]) ; side 1
      k1=imag(corner[1]+corner[0])/2, h1=real(corner[1]+corner[0])/2
      slope=imag(corner[1]-corner[0])/real(corner[1]-corner[0])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[0])
    elseif(tdeg<tdegmax[2]) ; side 2
      k1=imag(corner[2]+corner[1])/2, h1=real(corner[2]+corner[1])/2
      slope=imag(corner[2]-corner[1])/real(corner[2]-corner[1])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[1])+phimax
    elseif(tdeg<tdegmax[3]) ; side 3
      k1=imag(corner[3]+corner[2])/2, h1=real(corner[3]+corner[2])/2
      slope=imag(corner[3]-corner[2])/real(corner[3]-corner[2])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[2])+2*phimax
    elseif(tdeg<tdegmax[4]) ; side 4
      k1=imag(corner[4]+corner[3])/2, h1=real(corner[4]+corner[3])/2
      slope=imag(corner[4]-corner[3])/real(corner[4]-corner[3])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[3])+3*phimax
    elseif(tdeg<tdegmax[5]) ; side 5
      k1=imag(corner[5]+corner[4])/2, h1=real(corner[5]+corner[4])/2
      slope=imag(corner[5]-corner[4])/real(corner[5]-corner[4])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[4])+4*phimax
    elseif(tdeg<tdegmax[6]) ; side 6
      k1=imag(corner[6]+corner[5])/2, h1=real(corner[6]+corner[5])/2
      slope=imag(corner[6]-corner[5])/real(corner[6]-corner[5])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[5])+5*phimax
    elseif(tdeg<tdegmax[7]) ; side 7
      k1=imag(corner[7]+corner[6])/2, h1=real(corner[7]+corner[6])/2
      slope=imag(corner[7]-corner[6])/real(corner[7]-corner[6])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[6])+6*phimax
    else ; side 8
      k1=imag(corner[8]+corner[7])/2, h1=real(corner[8]+corner[7])/2
      slope=imag(corner[8]-corner[7])/real(corner[8]-corner[7])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[7])+7*phimax
    endif
    k=r/cabs(w)
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%(8*phimax)
    phi=phi/phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<1) ; side 1
      w=phi*(corner[1]-corner[0])+corner[0]
    elseif(phi<2) ; side 2
      w=(phi-1)*(corner[2]-corner[1])+corner[1]
    elseif(phi<3) ; side 3
      w=(phi-2)*(corner[3]-corner[2])+corner[2]
    elseif(phi<4) ; side 4
      w=(phi-3)*(corner[4]-corner[3])+corner[3]
    elseif(phi<5) ; side 5
      w=(phi-4)*(corner[5]-corner[4])+corner[4]
    elseif(phi<6) ; side 6
      w=(phi-5)*(corner[6]-corner[5])+corner[5]
    elseif(phi<7) ; side 7
      w=(phi-6)*(corner[7]-corner[6])+corner[6]
    else ; side 8
      w=(phi-7)*(corner[8]-corner[7])+corner[7]
    endif
    z=k*w+c
;
; unit circle (standard)
;
  elseif(@shape_type=="unit circle")
    z=z^@power+c
  endif
bailout:
  (|z|<@bailout)
default:
  title="Mixed-Up Multiplication Mandelbrot"
;
; iteration parameters
;
  heading
    caption="Iteration Parameters"
  endheading
  float param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; base shape parameters
;
  heading
    caption="Base Shape Parameters"
  endheading
  param shape_type
    caption="shape type"
    default=1
    enum="unit circle" "square" "triangle" "quarter-circle" "cruciform" \
      "rose curve" "gear" "star"
  endparam
  param q1_type
    caption="quadrant 1"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  param q2_type
    caption="quadrant 2"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  param q3_type
    caption="quadrant 3"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  param q4_type
    caption="quadrant 4"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  int param freq
    caption="frequency"
    default=2
    visible=(@shape_type=="rose curve")
  endparam
  float param amp
    caption="amplitude"
    default=0.5
    max=1
    visible=(@shape_type=="rose curve")
  endparam
  int param nteeth
    caption="# teeth"
    default=2
    min=1
    visible=(@shape_type=="gear")
  endparam
  float param rsmall
    caption="small radius"
    default=0.5
    min=0
    visible=(@shape_type=="gear")
  endparam
  float param inrad
    caption="inner radius"
    default=0.25
    min=0
    visible=(@shape_type=="star")
  endparam
switch:
  type="mixed-up-multiplication-julia"
  julparam=pixel
  power=power
  bailout=bailout
  shape_type=shape_type
  q1_type=q1_type
  q2_type=q2_type
  q3_type=q3_type
  q4_type=q4_type
  freq=freq
  amp=amp
  nteeth=nteeth
  rsmall=rsmall
  inrad=inrad
}

mixed-up-multiplication-julia { ; Kerry Mitchell 27Nov2019
;
; Instead of using regular complex multiplication, this method uses a base
; shape (= unit circle normally). Use the polar angle of z to reach back
; to the base shape and find its coordinates. Then, use the actual and base
; magnitudes to determine a scale factor, and the arc length to determine
; the angle (angle = arc length / base magnitude). From there, the magnitude
; of z^power = scale factor ^ power, and angle of z^power = power * angle.
;
$define debug
global:
  complex corner[20]
  float pitch_s=0
  float pitch_trad=0
  float smax=0
  float tdegmax[20]
  float twopi=2*#pi
  int gi=0
;
; cruciform settings: central square side & four other squares, one
; on each side of central square
;
  if(@shape_type=="cruciform")
    corner[0]=(3,0)
    corner[1]=(3,1)
    corner[2]=(1,1)
    corner[3]=(1,3)
    corner[4]=(-1,3)
    corner[5]=(-1,1)
    corner[6]=(-3,1)
    corner[7]=(-3,-1)
    corner[8]=(-1,-1)
    corner[9]=(-1,-3)
    corner[10]=(1,-3)
    corner[11]=(1,-1)
    corner[12]=(3,-1)
    corner[13]=(3,0)
;
;   rescale so that corner[0] = (1,0)
;   find degree measures of corners
;
    gi=-1
    while(gi<12)
      gi=gi+1
      corner[gi]=corner[gi]/3
      tdegmax[gi]=(atan2(corner[gi])/#pi*180+720)%360
    endwhile
  elseif(@shape_type=="triangle")
    corner[0]=(1,0)
    corner[1]=(-1+flip(sqrt(3)))/2
    corner[2]=(-1-flip(sqrt(3)))/2
    tdegmax[0]=0
    tdegmax[1]=120
    tdegmax[2]=240
    tdegmax[3]=360
  elseif(@shape_type=="star")
    corner[0]=(1,0)
    corner[1]=@inrad*(1,1)/sqrt(2)
    corner[2]=(0,1)
    corner[3]=@inrad*(-1,1)/sqrt(2)
    corner[4]=(-1,0)
    corner[5]=@inrad*(-1,-1)/sqrt(2)
    corner[6]=(0,-1)
    corner[7]=@inrad*(1,-1)/sqrt(2)
    corner[8]=(1,0)
    tdegmax[0]=0
    tdegmax[1]=45
    tdegmax[2]=90
    tdegmax[3]=135
    tdegmax[4]=180
    tdegmax[5]=225
    tdegmax[6]=270
    tdegmax[7]=315
    tdegmax[8]=360
  endif
;
; gear settings
;
  if(@shape_type=="gear")
    pitch_trad=twopi/@nteeth
    smax=#pi*(1+@rsmall)
    pitch_s=smax/@nteeth
  endif
init:
  complex arccenter=(0,0)
  complex c=@julparam
  complex compk=(0,0)
  complex w=(0,0)
  complex z=#pixel
  float afac=0
  float bfac=0
  float cfac=0
  float dcrit=1/(1+@rsmall)
  float dtooth=0
  float h1=0
  float k1=0
  float k=0
  float phi=0
  float phimax=0
  float r=0
  float s=0
  float slope=0
  float tdeg=0
  float third=1/3
  float trad=0
  float x=0
  float xb=0
  float x3=0
  float y=0
  float yb=0
  float y3=0
  int itooth=0
loop:
;
; decompose z
;
  x=real(z), y=imag(z)
;
; square
;
  if(@shape_type=="square")
;
;   determine arc length from basis shape
;
    phimax=8
;
;   find scale factor k
;
    k=abs(x)
    if(abs(y)>k)
      k=abs(y)
    endif
;
;   find standard polar angle
;
    trad=atan2(z)
    if(trad<0)
      trad=trad+2*#pi
    endif
    tdeg=trad/#pi*180
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<45) ; upper right side
      xb=1, yb=y/k, s=yb
    elseif(tdeg<135) ; top
      xb=x/k, yb=1, s=2-xb
    elseif(tdeg<225) ; left side
      xb=-1, yb=y/k, s=4-yb
    elseif(tdeg<315) ; bottom
      xb=x/k, yb=-1, s=6+xb
    else ; lower right side
      xb=1, yb=y/k, s=8+yb
    endif
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<1) ; upper right side
      xb=1, yb=phi
    elseif(phi<3) ; top
      xb=2-phi, yb=1
    elseif(phi<5) ; left side
      xb=-1, yb=4-phi
    elseif(phi<7) ; bottom
      xb=phi-6, yb=-1
    else ; lower right side
      xb=1, yb=phi-8
    endif
    x=xb*k, y=yb*k
    z=x+flip(y)+c
;
; triangle
;
  elseif(@shape_type=="triangle")
;
;   determine arc length from basis shape
;
    third=sqrt(3)
    phimax=3*third
;
;   find standard magnitude and polar angle
;
    r=cabs(z)
    trad=atan2(z)
    tdeg=(trad/#pi*180+720)%360
    trad=tdeg/180*#pi
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<tdegmax[1]) ; side 1 top
      xb=third/(third+3*tan(trad)), yb=third/3*(1-xb), s=cabs(xb+flip(yb)-corner[0])
    elseif(tdeg<tdegmax[2]) ; side 2 left
      xb=-0.5, yb=xb*tan(trad), s=third+cabs(xb+flip(yb)-corner[1])
    else ; side 3 bottom
      xb=third/(third-3*tan(trad)), yb=third/3*(xb-1), s=2*third+cabs(xb+flip(yb)-corner[2])
    endif
    k=r/cabs(xb+flip(yb))
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<third) ; side 1 top
      w=phi/third*(corner[1]-corner[0])+corner[0]
    elseif(phi<(2*third)) ; side 2 left
      w=(phi/third-1)*(corner[2]-corner[1])+corner[1]
    else ; side 2 bottom
      w=(phi/third-2)*(corner[0]-corner[2])+corner[2]
    endif
    w=w*k, z=w+c
;
; quarter-circle arcs in quadrants
;
  elseif(@shape_type=="quarter-circle")
    phimax=2*#pi
;
;   z: find standard magnitude and polar angle
;
    r=cabs(z)
    trad=(atan2(z)+2*phimax)%phimax
    tdeg=trad/#pi*180
;
;   z: find coordinates of point on base shape and base shape arc length
;     equate angle of base point with angle of field point
;
    if(x==0)
      xb=0, yb=1
      s=#pi/2
      if(y<0)
        yb=-1
        s=3*#pi/2
      endif
    else
      if(tdeg<90) ; quadrant 1
        if(@q1_type=="in")
          arccenter=(1,1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac-sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=#pi/2-atan2((1-xb)+flip(1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      elseif(tdeg<180) ; quadrant 2
        if(@q2_type=="in")
          arccenter=(-1,1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac+sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=3*#pi/2-atan2((-1-xb)+flip(1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      elseif(tdeg<270) ; quadrant 3
        if(@q3_type=="in")
          arccenter=(-1,-1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac+sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=#pi/2-atan2((-1-xb)+flip(-1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      else ; quadrant 4
        if(@q4_type=="in")
          arccenter=(1,-1), x3=real(arccenter), y3=imag(arccenter)
          afac=r*r, bfac=-2*x*(x*x3+y*y3), cfac=x*x*(x3*x3+y3*y3-1)
          yb=bfac*bfac-4*afac*cfac
          xb=(-bfac-sqrt(yb))/(2*afac)
          yb=xb*y/x
          s=3*#pi/2-atan2((1-xb)+flip(-1-yb))
        else
          xb=cos(trad), yb=sin(trad), s=trad
        endif
      endif
    endif
    phi=s
    k=r/cabs(xb+flip(yb))
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    tdeg=phi/#pi*180
    if(tdeg<90) ; quadrant 1
      if(@q1_type=="in")
        xb=1-sin(phi), yb=1-cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    elseif(tdeg<180) ; quadrant 2
      if(@q2_type=="in")
        xb=-1+sin(phi), yb=1+cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    elseif(tdeg<270) ; quadrant 3
      if(@q3_type=="in")
        xb=-1-sin(phi), yb=-1-cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    else ; quadrant 4
      if(@q4_type=="in")
        xb=1+sin(phi), yb=-1+cos(phi), w=xb+flip(yb)
      else
        xb=cos(phi), yb=sin(phi), w=xb+flip(yb)
      endif
    endif
    z=k*w+c
;
; cruciform
;
  elseif(@shape_type=="cruciform")
;
;   determine arc length from basis shape
;
    phimax=8
;
;   find standard magnitude and polar angle
;
    r=cabs(z)
    trad=atan2(z)
    tdeg=(trad/#pi*180+720)%360
    trad=tdeg/180*#pi
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<tdegmax[1]) ; side 1 (far right) upper
      xb=1, yb=xb*tan(trad), s=yb
    elseif(tdeg<tdegmax[2]) ; side 2
      yb=third, xb=yb/tan(trad), s=(1-xb)+third
    elseif(tdeg<tdegmax[3]) ; side 3
      xb=third, yb=xb*tan(trad), s=1+(yb-third)
    elseif(tdeg<tdegmax[4]) ; side 4
      yb=1, xb=yb/tan(trad), s=2-xb
    elseif(tdeg<tdegmax[5]) ; side 5
      xb=-third, yb=xb*tan(trad), s=7*third+(1-yb)
    elseif(tdeg<tdegmax[6]) ; side 6
      yb=third, xb=yb/tan(trad), s=3-(xb+third)
    elseif(tdeg<tdegmax[7]) ; side 7
      xb=-1, yb=xb*tan(trad), s=4-yb
    elseif(tdeg<tdegmax[8]) ; side 8
      yb=-third, xb=yb/tan(trad), s=16*third+xb
    elseif(tdeg<tdegmax[9]) ; side 9
      xb=-third, yb=xb*tan(trad), s=14*third-yb
    elseif(tdeg<tdegmax[10]) ; side 10
      yb=-1, xb=yb/tan(trad), s=6+xb
    elseif(tdeg<tdegmax[11]) ; side 11
      xb=third, yb=xb*tan(trad), s=22*third+yb
    elseif(tdeg<tdegmax[12]) ; side 12
      yb=-third, xb=yb/tan(trad), s=7+(xb-third)
    else ; side 1 lower
      xb=1, yb=xb*tan(trad), s=8+yb
    endif
    k=r/cabs(xb+flip(yb))
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<(1/3)) ; side 1 (far right) upper
      xb=1, yb=phi
    elseif(phi<1) ; side 2
      xb=third+1-phi, yb=third
    elseif(phi<(5/3)) ; side 3
      xb=third, yb=phi-2*third
    elseif(phi<(7/3)) ; side 4
      xb=2-phi, yb=1
    elseif(phi<3) ; side 5
      xb=-third, yb=10*third-phi
    elseif(phi<(11/3)) ; side 6
      xb=8/3-phi, yb=third
    elseif(phi<(13/3)) ; side 7
      xb=-1, yb=4-phi
    elseif(phi<5) ; side 8
      xb=-16/3+phi, yb=-third
    elseif(phi<(17/3)) ; side 9
      xb=-third, yb=14/3-phi
    elseif(phi<(19/3)) ; side 10
      xb=phi-6, yb=-1
    elseif(phi<7) ; side 11
      xb=third, yb=phi-22/3
    elseif(phi<(23/3)) ; side 12
      xb=phi-20/3, yb=-third
    else ; side 1 lower
      xb=1, yb=phi-8
    endif
    x=xb*k, y=yb*k
    z=x+flip(y)+c
;
; rose curve
;
  elseif(@shape_type=="rose curve")
    trad=atan2(z)
    r=1+@amp*sin(@freq*trad)
    xb=r*cos(trad), yb=r*sin(trad), w=xb+flip(yb)
    phi=trad, compk=z/w
    compk=compk^@power, phi=@power*phi
    r=1+@amp*sin(@freq*phi)
    xb=r*cos(phi), yb=r*sin(phi), w=xb+flip(yb)
    z=compk*w+c
;
; gear
;
  elseif(@shape_type=="gear")
;
;   z: find theta, which tooth z projects to, and radius (1 or rsmall)
;     itooth from 0 to nteeth-1, dtooth = fractional tooth, 0 to 1
;
    trad=(atan2(z)+2*twopi)%twopi
    dtooth=trad/pitch_trad
    itooth=trunc(dtooth), dtooth=dtooth-itooth
    if(dtooth<0.5) ; radius = 1
      k=cabs(z)
      s=itooth*pitch_s+dtooth*pitch_trad
    else ; radius = rsmall
      k=cabs(z)/@rsmall
      s=itooth*pitch_s+0.5*pitch_trad+(dtooth-0.5)*pitch_trad*@rsmall
    endif
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%smax
;
;   use arc length to find which tooth, radius, and project to new z and add c
;
    dtooth=phi/pitch_s
    itooth=trunc(dtooth), dtooth=dtooth-itooth
    if(dtooth<dcrit) ; radius = 1
      r=k
      trad=itooth*pitch_trad+dtooth/dcrit*pitch_trad/2
    else ; radius = rsmall
      r=k*@rsmall
      trad=itooth*pitch_trad+(1+(dtooth-dcrit)/(1-dcrit))*pitch_trad/2
    endif
    z=r*(cos(trad)+flip(sin(trad)))+c
;
; star
;
  elseif(@shape_type=="star")
;
;   determine arc length from basis shape
;
    phimax=cabs(corner[1]-corner[0])
;
;   find standard magnitude and polar angle
;
    r=cabs(z)
    trad=(atan2(z)+2*twopi)%twopi
    tdeg=trad/#pi*180
;
;   use polar angle to find coordinates of point on base shape
;   and base shape arc length
;
    if(tdeg<tdegmax[1]) ; side 1
      k1=imag(corner[1]+corner[0])/2, h1=real(corner[1]+corner[0])/2
      slope=imag(corner[1]-corner[0])/real(corner[1]-corner[0])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[0])
    elseif(tdeg<tdegmax[2]) ; side 2
      k1=imag(corner[2]+corner[1])/2, h1=real(corner[2]+corner[1])/2
      slope=imag(corner[2]-corner[1])/real(corner[2]-corner[1])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[1])+phimax
    elseif(tdeg<tdegmax[3]) ; side 3
      k1=imag(corner[3]+corner[2])/2, h1=real(corner[3]+corner[2])/2
      slope=imag(corner[3]-corner[2])/real(corner[3]-corner[2])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[2])+2*phimax
    elseif(tdeg<tdegmax[4]) ; side 4
      k1=imag(corner[4]+corner[3])/2, h1=real(corner[4]+corner[3])/2
      slope=imag(corner[4]-corner[3])/real(corner[4]-corner[3])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[3])+3*phimax
    elseif(tdeg<tdegmax[5]) ; side 5
      k1=imag(corner[5]+corner[4])/2, h1=real(corner[5]+corner[4])/2
      slope=imag(corner[5]-corner[4])/real(corner[5]-corner[4])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[4])+4*phimax
    elseif(tdeg<tdegmax[6]) ; side 6
      k1=imag(corner[6]+corner[5])/2, h1=real(corner[6]+corner[5])/2
      slope=imag(corner[6]-corner[5])/real(corner[6]-corner[5])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[5])+5*phimax
    elseif(tdeg<tdegmax[7]) ; side 7
      k1=imag(corner[7]+corner[6])/2, h1=real(corner[7]+corner[6])/2
      slope=imag(corner[7]-corner[6])/real(corner[7]-corner[6])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[6])+6*phimax
    else ; side 8
      k1=imag(corner[8]+corner[7])/2, h1=real(corner[8]+corner[7])/2
      slope=imag(corner[8]-corner[7])/real(corner[8]-corner[7])
      xb=(k1-slope*h1)/(tan(trad)-slope), yb=xb*tan(trad)
      w=xb+flip(yb), s=cabs(w-corner[7])+7*phimax
    endif
    k=r/cabs(w)
    phi=s
;
;   raise z to power
;
    k=k^@power, phi=(@power*phi)%(8*phimax)
    phi=phi/phimax
;
;   use phi and k to return new x & y and add c
;
    if(phi<1) ; side 1
      w=phi*(corner[1]-corner[0])+corner[0]
    elseif(phi<2) ; side 2
      w=(phi-1)*(corner[2]-corner[1])+corner[1]
    elseif(phi<3) ; side 3
      w=(phi-2)*(corner[3]-corner[2])+corner[2]
    elseif(phi<4) ; side 4
      w=(phi-3)*(corner[4]-corner[3])+corner[3]
    elseif(phi<5) ; side 5
      w=(phi-4)*(corner[5]-corner[4])+corner[4]
    elseif(phi<6) ; side 6
      w=(phi-5)*(corner[6]-corner[5])+corner[5]
    elseif(phi<7) ; side 7
      w=(phi-6)*(corner[7]-corner[6])+corner[6]
    else ; side 8
      w=(phi-7)*(corner[8]-corner[7])+corner[7]
    endif
    z=k*w+c
;
; unit circle (standard)
;
  elseif(@shape_type=="unit circle")
    z=z^@power+c
  endif
bailout:
  (|z|<@bailout)
default:
  title="Mixed-Up Multiplication Julia"
;
; iteration parameters
;
  heading
    caption="Iteration Parameters"
  endheading
  complex param julparam
    caption="Julia parameter"
    default=(0,1)
  endparam
  float param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; base shape parameters
;
  heading
    caption="Base Shape Parameters"
  endheading
  param shape_type
    caption="shape type"
    default=1
    enum="unit circle" "square" "triangle" "quarter-circle" "cruciform" \
      "rose curve" "gear" "star"
  endparam
  param q1_type
    caption="quadrant 1"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  param q2_type
    caption="quadrant 2"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  param q3_type
    caption="quadrant 3"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  param q4_type
    caption="quadrant 4"
    default=1
    enum="in" "out"
    visible=(@shape_type=="quarter-circle")
  endparam
  int param freq
    caption="frequency"
    default=2
    visible=(@shape_type=="rose curve")
  endparam
  float param amp
    caption="amplitude"
    default=0.5
    max=1
    visible=(@shape_type=="rose curve")
  endparam
  int param nteeth
    caption="# teeth"
    default=2
    min=1
    visible=(@shape_type=="gear")
  endparam
  float param rsmall
    caption="small radius"
    default=0.5
    min=0
    visible=(@shape_type=="gear")
  endparam
  float param inrad
    caption="inner radius"
    default=0.25
    min=0
    visible=(@shape_type=="star")
  endparam
switch:
  type="mixed-up-multiplication-mandelbrot"
  power=power
  bailout=bailout
  shape_type=shape_type
  q1_type=q1_type
  q2_type=q2_type
  q3_type=q3_type
  q4_type=q4_type
  freq=freq
  amp=amp
  nteeth=nteeth
  rsmall=rsmall
  inrad=inrad
}

mod-mandelbrot { ; Kerry Mitchell 17May2020
;
; Mandelbrot-type fractal for z = (z^n) modulo z, or some other base
;
init:
  complex c=#pixel
  complex fofz=(0,0)
  complex mod_base=(0,0)
  complex multiple=(0,0)
  complex quotient=(0,0)
  complex remainder=(0,0)
  complex z=(0,0)
  float r=0
  float rmin=0
  int ix=0
  int ixmin=0
  int ixmax=0
  int jy=0
  int jymin=0
  int jymax=0
  int multx=0
  int multy=0
;
  if(@base_type=="z")
    z=#pixel
  endif
loop:
;
; basic fractal iteration
;
  fofz=z^@power+c
;
; base for the mod function
;
  if(@base_type=="constant")
    mod_base=@mod_func(@modbasie)
  elseif(@base_type=="pixel")
    mod_base=@mod_func(c)
  elseif(@base_type=="z")
    mod_base=@mod_func(z)
  endif
;
; mod function: f(z) mod mod_base = remainder = new z
;
  if(@mod_type=="smaller multiple")
;
;   f(z) = multiple + remainder, multiple = Gaussian integer * mod_base
;   Gaussian integer chosen to minimize |remainder|
;   WITH the constraint that |multiple| <= |fofz|
;   e.g, 7 mod 4 = 3, Gaussian integer = 1, |multiple| = 4
;
    quotient=fofz/mod_base
    ix=round(real(quotient))
    ixmin=ix-2, ixmax=ix+2
    jy=round(imag(quotient))
    jymin=jy-2, jymax=jy+2
    rmin=1e20
    ix=ixmin-1
    while(ix<ixmax)
      ix=ix+1
      jy=jymin-1
      while(jy<jymax)
        jy=jy+1
        multiple=(ix+flip(jy))*mod_base
        remainder=fofz-multiple
        r=cabs(remainder)
        if((r<rmin)&&(cabs(multiple)<=cabs(fofz)))
          rmin=r
          multx=ix, multy=jy
        endif
      endwhile
    endwhile
    multiple=(multx+flip(multy))*mod_base
    remainder=fofz-multiple
    z=remainder
  elseif(@mod_type=="smallest remainder")
;
;   f(z) = multiple + remainder, multiple = Gaussian integer * mod_base
;   Gaussian integer chosen to minimize |remainder|
;   WITHOUT any constraint on |multiple|
;   e.g., 7 mod 4 = -1, Gaussian integer = 2, |multiple| = 8
;
    quotient=fofz/mod_base
    ix=round(real(quotient))
    ixmin=ix-2, ixmax=ix+2
    jy=round(imag(quotient))
    jymin=jy-2, jymax=jy+2
    rmin=1e20
    ix=ixmin-1
    while(ix<ixmax)
      ix=ix+1
      jy=jymin-1
      while(jy<jymax)
        jy=jy+1
        multiple=(ix+flip(jy))*mod_base
        remainder=fofz-multiple
        r=cabs(remainder)
        if(r<rmin)
          rmin=r
          multx=ix, multy=jy
        endif
      endwhile
    endwhile
    multiple=(multx+flip(multy))*mod_base
    remainder=fofz-multiple
    z=remainder
  elseif(@mod_type=="Wolfram floor")
;
; f(z) mod mod_base = f(z) - mod_base * floor(f(z)/mod_base)
; floor(q) = floor(real(q)) + i*floor(imag(q))
;
    z=fofz-mod_base*floor(fofz/mod_base)
  endif
bailout:
  |z|<@bailout
default:
  title="Mod Mandelbrot"
  param mod_type
    caption="mod type"
    default=0
    enum="smaller multiple" "smallest remainder" "Wolfram floor"
  endparam
  param base_type
    caption="mod base type"
    default=0
    enum="constant" "pixel" "z"
  endparam
  complex param modbasie
    caption="mod base"
    default=(1,1)
    visible=(@base_type=="constant")
  endparam
  func mod_func
    caption="mod base function"
    default=ident()
  endfunc
  complex param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
switch:
  type="mod-julia"
  julparam=pixel
  mod_type=mod_type
  base_type=base_type
  modbasie=modbasie
  mod_func=mod_func
  power=power
  bailout=bailout
}

mod-julia { ; Kerry Mitchell 17May2020
;
; Julia-type fractal for z = (z^n) modulo z, or some other base
;
init:
  complex c=@julparam
  complex fofz=(0,0)
  complex mod_base=(0,0)
  complex multiple=(0,0)
  complex quotient=(0,0)
  complex remainder=(0,0)
  complex z=#pixel
  float r=0
  float rmin=0
  int ix=0
  int ixmin=0
  int ixmax=0
  int jy=0
  int jymin=0
  int jymax=0
  int multx=0
  int multy=0
loop:
;
; basic fractal iteration
;
  fofz=z^@power+c
;
; base for the mod function
;
  if(@base_type=="constant")
    mod_base=@mod_func(@modbasie)
  elseif(@base_type=="pixel")
    mod_base=@mod_func(c)
  elseif(@base_type=="z")
    mod_base=@mod_func(z)
  endif
;
; mod function: f(z) mod mod_base = remainder = new z
;
  if(@mod_type=="smaller multiple")
;
;   f(z) = multiple + remainder, multiple = Gaussian integer * mod_base
;   Gaussian integer chosen to minimize |remainder|
;   WITH the constraint that |multiple| <= |fofz|
;   e.g, 7 mod 4 = 3, Gaussian integer = 1, |multiple| = 4
;
    quotient=fofz/mod_base
    ix=round(real(quotient))
    ixmin=ix-2, ixmax=ix+2
    jy=round(imag(quotient))
    jymin=jy-2, jymax=jy+2
    rmin=1e20
    ix=ixmin-1
    while(ix<ixmax)
      ix=ix+1
      jy=jymin-1
      while(jy<jymax)
        jy=jy+1
        multiple=(ix+flip(jy))*mod_base
        remainder=fofz-multiple
        r=cabs(remainder)
        if((r<rmin)&&(cabs(multiple)<=cabs(fofz)))
          rmin=r
          multx=ix, multy=jy
        endif
      endwhile
    endwhile
    multiple=(multx+flip(multy))*mod_base
    remainder=fofz-multiple
    z=remainder
  elseif(@mod_type=="smallest remainder")
;
;   f(z) = multiple + remainder, multiple = Gaussian integer * mod_base
;   Gaussian integer chosen to minimize |remainder|
;   WITHOUT any constraint on |multiple|
;   e.g., 7 mod 4 = -1, Gaussian integer = 2, |multiple| = 8
;
    quotient=fofz/mod_base
    ix=round(real(quotient))
    ixmin=ix-2, ixmax=ix+2
    jy=round(imag(quotient))
    jymin=jy-2, jymax=jy+2
    rmin=1e20
    ix=ixmin-1
    while(ix<ixmax)
      ix=ix+1
      jy=jymin-1
      while(jy<jymax)
        jy=jy+1
        multiple=(ix+flip(jy))*mod_base
        remainder=fofz-multiple
        r=cabs(remainder)
        if(r<rmin)
          rmin=r
          multx=ix, multy=jy
        endif
      endwhile
    endwhile
    multiple=(multx+flip(multy))*mod_base
    remainder=fofz-multiple
    z=remainder
  elseif(@mod_type=="Wolfram floor")
;
; f(z) mod mod_base = f(z) - mod_base * floor(f(z)/mod_base)
; floor(q) = floor(real(q)) + i*floor(imag(q))
;
    z=fofz-mod_base*floor(fofz/mod_base)
  endif
bailout:
  |z|<@bailout
default:
  title="Mod Julia"
  complex param julparam
    caption="Julia parameter"
    default=(0,1)
  endparam
  param mod_type
    caption="mod type"
    default=0
    enum="smaller multiple" "smallest remainder" "Wolfram floor"
  endparam
  param base_type
    caption="mod base type"
    default=0
    enum="constant" "pixel" "z"
  endparam
  complex param modbasie
    caption="mod base"
    default=(1,1)
    visible=(@base_type=="constant")
  endparam
  func mod_func
    caption="mod base function"
    default=ident()
  endfunc
  complex param power
    caption="power"
    default=2
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
switch:
  type="mod-mandelbrot"
  mod_type=mod_type
  base_type=base_type
  modbasie=modbasie
  mod_func=mod_func
  power=power
  bailout=bailout
}

relaxed-householder { ; Kerry Mitchell 25May2020
;
; The Householder methods are generalizations of Newton's method, using higher
; derivatives. This formula allows the user to choose from 11 different methods
; (six standard Householder + five variations), with up to six different levels
; of relaxation. The chosen methoed can be applied to 11 different formula types.
;
global:
init:
  complex c=(0,0)
  complex dz=(0,0)
  complex dz1=(0,0)
  complex dz2=(0,0)
  complex dz3=(0,0)
  complex dz4=(0,0)
  complex dz5=(0,0)
  complex dz6=(0,0)
  complex f=(0,0)
  complex fp=(0,0)
  complex fp2=(0,0)
  complex fp3=(0,0)
  complex fp4=(0,0)
  complex fp5=(0,0)
  complex fp6=(0,0)
  complex relaxation1=(0,0)
  complex relaxation2=(0,0)
  complex relaxation3=(0,0)
  complex relaxation4=(0,0)
  complex relaxation5=(0,0)
  complex relaxation6=(0,0)
  complex temp1=(0,0)
  complex temp2=(0,0)
  complex temp3=(0,0)
  complex temp4=(0,0)
  complex temp5=(0,0)
  complex temp6=(0,0)
  complex temp7=(0,0)
  complex temp8=(0,0)
  complex z=(0,0)
  complex ziterate=(0,0)
;
; initialize c, z, relaxation
;
  if(@pixeltype=="initial z")
    c=@root
    ziterate=#pixel
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="c")
    c=#pixel
    ziterate=@initialz
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="relaxation 1")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(#pixel)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="relaxation 2")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(#pixel)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="relaxation 3")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(#pixel)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="relaxation 4")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(#pixel)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="relaxation 5")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(#pixel)
    relaxation6=@relax6_func(@relax6)
  elseif(@pixeltype=="relaxation 6")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(@relax1)
    relaxation2=@relax2_func(@relax2)
    relaxation3=@relax3_func(@relax3)
    relaxation4=@relax4_func(@relax4)
    relaxation5=@relax5_func(@relax5)
    relaxation6=@relax6_func(#pixel)
  elseif(@pixeltype=="all relaxations")
    c=@root
    ziterate=@initialz
    relaxation1=@relax1_func(#pixel)
    relaxation2=@relax2_func(#pixel)
    relaxation3=@relax3_func(#pixel)
    relaxation4=@relax4_func(#pixel)
    relaxation5=@relax5_func(#pixel)
    relaxation6=@relax6_func(#pixel)
  endif
loop:
;
; compute function and derivatives
;
  if(@eqntype=="roots of c")
;
;   optimizations for low-order power functions
;
    if(@power==(2,0))
      f=sqr(ziterate)-c
      fp=2*ziterate
      fp2=2
      fp3=0
      fp4=0
      fp5=0
      fp6=0
    elseif(@power==(3,0))
      temp1=sqr(ziterate)
      f=ziterate*temp1-c
      fp=3*temp1
      fp2=6*ziterate
      fp3=6
      fp4=0
      fp5=0
      fp6=0
    elseif(@power==(4,0))
      temp1=sqr(ziterate)
      f=sqr(temp1)-c
      fp=4*ziterate*temp1
      fp2=12*temp1
      fp3=24*ziterate
      fp4=24
      fp5=0
      fp6=0
    else
      f=ziterate^@power-c
      fp=@power*(ziterate^(@power-1))
      if(@solntype>0)
        fp2=(@power-1)*@power*(ziterate^(@power-2))
        if(@solntype>2)
          fp3=(@power-2)*(@power-1)*@power*(ziterate^(@power-3))
          if(@solntype>4)
            fp4=(@power-3)*(@power-2)*(@power-1)*@power*(ziterate^(@power-4))
            if(@solntype>6)
              fp5=(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*(ziterate^(@power-5))
              if(@solntype>8)
                fp6=(@power-5)*(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*(ziterate^(@power-6))
              endif
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="sum of 4 powers = c")
    f=\
      @coeff1*ziterate^@power1+\
      @coeff2*ziterate^@power2+\
      @coeff3*ziterate^@power3+\
      @coeff4*ziterate^@power4-c
    fp=\
      @coeff1*@power1*ziterate^(@power1-1)+\
      @coeff2*@power2*ziterate^(@power2-1)+\
      @coeff3*@power3*ziterate^(@power3-1)+\
      @coeff4*@power4*ziterate^(@power4-1)
    if(@solntype>0)
      fp2=\
        @coeff1*(@power1-1)*@power1*ziterate^(@power1-2)+\
        @coeff2*(@power2-1)*@power2*ziterate^(@power2-2)+\
        @coeff3*(@power3-1)*@power3*ziterate^(@power3-2)+\
        @coeff4*(@power4-1)*@power4*ziterate^(@power4-2)
      if(@solntype>2)
        fp3=\
          @coeff1*(@power1-2)*(@power1-1)*@power1*ziterate^(@power1-3)+\
          @coeff2*(@power2-2)*(@power2-1)*@power2*ziterate^(@power2-3)+\
          @coeff3*(@power3-2)*(@power3-1)*@power3*ziterate^(@power3-3)+\
          @coeff4*(@power4-2)*(@power4-1)*@power4*ziterate^(@power4-3)
        if(@solntype>4)
          fp4=\
            @coeff1*(@power1-3)*(@power1-2)*(@power1-1)*@power1*ziterate^(@power1-4)+\
            @coeff2*(@power2-3)*(@power2-2)*(@power2-1)*@power2*ziterate^(@power2-4)+\
            @coeff3*(@power3-3)*(@power3-2)*(@power3-1)*@power3*ziterate^(@power3-4)+\
            @coeff4*(@power4-3)*(@power4-2)*(@power4-1)*@power4*ziterate^(@power4-4)
          if(@solntype>6)
            fp5=\
              @coeff1*(@power1-4)*(@power1-3)*(@power1-2)*(@power1-1)*@power1*ziterate^(@power1-5)+\
              @coeff2*(@power2-4)*(@power2-3)*(@power2-2)*(@power2-1)*@power2*ziterate^(@power2-5)+\
              @coeff3*(@power3-4)*(@power3-3)*(@power3-2)*(@power3-1)*@power3*ziterate^(@power3-5)+\
              @coeff4*(@power4-4)*(@power4-3)*(@power4-2)*(@power4-1)*@power4*ziterate^(@power4-5)
            if(@solntype>8)
              fp6=\
                @coeff1*(@power1-5)*(@power1-4)*(@power1-3)*(@power1-2)*(@power1-1)*@power1*ziterate^(@power1-6)+\
                @coeff2*(@power2-5)*(@power2-4)*(@power2-3)*(@power2-2)*(@power2-1)*@power2*ziterate^(@power2-6)+\
                @coeff3*(@power3-5)*(@power3-4)*(@power3-3)*(@power3-2)*(@power3-1)*@power3*ziterate^(@power3-6)+\
                @coeff4*(@power4-5)*(@power4-4)*(@power4-3)*(@power4-2)*(@power4-1)*@power4*ziterate^(@power4-6)
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="z^n + c = sin(z)")
    temp1=sin(ziterate), temp2=cos(ziterate)
    f=ziterate^@power+c-temp1
    fp=@power*ziterate^(@power-1)-temp2
    if(@solntype>0)
      fp2=(@power-1)*@power*ziterate^(@power-2)+temp1
      if(@solntype>2)
        fp3=(@power-2)*(@power-1)*@power*ziterate^(@power-3)+temp2
        if(@solntype>4)
          fp4=(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-4)-temp1
          if(@solntype>6)
            fp5=(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-5)-temp2
            if(@solntype>8)
              fp6=(@power-5)*(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-6)+temp1
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="z^n = c*cos(z)")
    temp1=c*cos(ziterate), temp2=c*sin(ziterate)
    f=ziterate^@power-temp1
    fp=@power*ziterate^(@power-1)+temp2
    if(@solntype>0)
      fp2=(@power-1)*@power*ziterate^(@power-2)+temp1
      if(@solntype>2)
        fp3=(@power-2)*(@power-1)*@power*ziterate^(@power-3)-temp2
        if(@solntype>4)
          fp4=(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-4)-temp1
          if(@solntype>6)
            fp5=(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-5)+temp2
            if(@solntype>8)
              fp6=(@power-5)*(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-6)+temp1
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="z^n + c = tan(z)")
    temp1=cos(ziterate), temp2=sin(ziterate), temp3=sqr(temp2)
    f=ziterate^@power+c-tan(ziterate)
    fp=@power*ziterate^(@power-1)-temp1^(-2)
    if(@solntype>0)
      fp2=(@power-1)*@power*ziterate^(@power-2)-2*temp1^(-3)*temp2
      if(@solntype>2)
        fp3=(@power-2)*(@power-1)*@power*ziterate^(@power-3)-2*temp1^(-4)*(1+2*temp3)
        if(@solntype>4)
          fp4=(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-4)-\
            8*temp2*temp1^(-5)*(2+temp3)
          if(@solntype>6)
            fp5=(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-5)-\
              8*temp1^(-6)*(temp3*(2*temp3+11)+2)
            if(@solntype>8)
              fp6=(@power-5)*(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-6)-\
                16*temp2*temp1^(-7)*(2*temp3*(temp3+13)+17)
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="z^n = c*tan(z)")
    temp1=cos(ziterate), temp2=sin(ziterate), temp3=sqr(temp2)
    f=ziterate^@power-c*tan(ziterate)
    fp=@power*ziterate^(@power-1)-c*temp1^(-2)
    if(@solntype>0)
      fp2=(@power-1)*@power*ziterate^(@power-2)-2*c*temp1^(-3)*temp2
      if(@solntype>2)
        fp3=(@power-2)*(@power-1)*@power*ziterate^(@power-3)-2*c*temp1^(-4)*(1+2*temp3)
        if(@solntype>4)
          fp4=(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-4)-\
            8*c*temp2*temp1^(-5)*(2+temp3)
          if(@solntype>6)
            fp5=(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-5)-\
              8*c*temp1^(-6)*(temp3*(2*temp3+11)+2)
            if(@solntype>8)
              fp6=(@power-5)*(@power-4)*(@power-3)*(@power-2)*(@power-1)*@power*ziterate^(@power-6)-\
                16*c*temp2*temp1^(-7)*(2*temp3*(temp3+13)+17)
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="sin(z+c) = cos(c*z)")
    temp1=sin(ziterate+c), temp2=cos(c*ziterate)
    temp3=cos(ziterate+c), temp4=sin(c*ziterate)
    f=temp1-temp2
    fp=temp3+c*temp4
    if(@solntype>0)
      fp2=-temp1+sqr(c)*temp2
      if(@solntype>2)
        fp3=-temp3-(c^3)*temp4
        if(@solntype>4)
          fp4=temp1-(c^4)*temp2
          if(@solntype>6)
            fp5=temp3+(c^5)*temp4
            if(@solntype>8)
              fp6=-temp1+(c^6)*temp2
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="sum of 4 sines = c")
    temp1=@amp1*sin(@freq1*ziterate), temp2=@amp2*sinh(@freq2*ziterate)
    temp3=@amp3*sinh(@freq3*ziterate), temp4=@amp4*sin(@freq4*ziterate)
    temp5=@amp1*cos(@freq1*ziterate), temp6=@amp2*cosh(@freq2*ziterate)
    temp7=@amp3*cosh(@freq3*ziterate), temp8=@amp4*cos(@freq4*ziterate)
    f=temp1+temp2+temp3+temp4-c
    fp=@freq1*temp5+@freq2*temp6+@freq3*temp7+@freq4*temp8
    if(@solntype>0)
      fp2=-(@freq1^2)*temp1+(@freq2^2)*temp2+(@freq3^2)*temp3-(@freq4^2)*temp4
      if(@solntype>2)
        fp3=-(@freq1^3)*temp5+(@freq2^3)*temp6+(@freq3^3)*temp7-(@freq4^3)*temp8
        if(@solntype>4)
          fp4=(@freq1^4)*temp1+(@freq2^4)*temp2+(@freq3^4)*temp3+(@freq4^4)*temp4
          if(@solntype>6)
            fp5=(@freq1^5)*temp5+(@freq2^5)*temp6+(@freq3^5)*temp7+(@freq4^5)*temp8
            if(@solntype>8)
              fp6=-(@freq1^6)*temp1+(@freq2^6)*temp2+(@freq3^6)*temp3-(@freq4^6)*temp4
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="sum of 4 cosines = c")
    temp1=@amp1*cos(@freq1*ziterate+c), temp2=@amp2*cos(@freq2*ziterate+c)
    temp3=@amp3*cosh(@freq3*ziterate+c), temp4=@amp4*cosh(@freq4*ziterate+c)
    temp5=@amp1*sin(@freq1*ziterate+c), temp6=@amp2*sin(@freq2*ziterate+c)
    temp7=@amp3*sinh(@freq3*ziterate+c), temp8=@amp4*sinh(@freq4*ziterate+c)
    f=temp1+temp2+temp3+temp4
    fp=-@freq1*temp5-@freq2*temp6+@freq3*temp7+@freq4*temp8
    if(@solntype>0)
      fp2=-(@freq1^2)*temp1-(@freq2^2)*temp2+(@freq3^2)*temp3+(@freq4^2)*temp4
      if(@solntype>2)
        fp3=+(@freq1^3)*temp5+(@freq2^3)*temp6+(@freq3^3)*temp7+(@freq4^3)*temp8
        if(@solntype>4)
          fp4=+(@freq1^4)*temp1+(@freq2^4)*temp2+(@freq3^4)*temp3+(@freq4^4)*temp4
          if(@solntype>6)
            fp5=-(@freq1^5)*temp5-(@freq2^5)*temp6+(@freq3^5)*temp7+(@freq4^5)*temp8
            if(@solntype>8)
              fp6=-(@freq1^6)*temp1-(@freq2^6)*temp2+(@freq3^6)*temp3+(@freq4^6)*temp4
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="exp(cz) = log(z)/c")
    temp1=exp(c*ziterate), temp2=log(ziterate)
    f=temp1-temp2/c
    fp=c*temp1-1/(c*ziterate)
    if(@solntype>0)
      fp2=sqr(c)*temp1+1/(c*sqr(ziterate))
      if(@solntype>2)
        fp3=(c^3)*temp1-2/(c*ziterate^3)
        if(@solntype>4)
          fp4=(c^4)*temp1+6/(c*ziterate^4)
          if(@solntype>6)
            fp5=(c^5)*temp1-24/(c*ziterate^5)
            if(@solntype>8)
              fp6=(c^6)*temp1+120/(c*ziterate^6)
            endif
          endif
        endif
      endif
    endif
  elseif(@eqntype=="z^c = c^z")
    temp1=c^ziterate, temp2=log(c)
    f=ziterate^c-temp1
    fp=c*ziterate^(c-1)-temp2*temp1
    if(@solntype>0)
      fp2=(c-1)*c*ziterate^(c-2)-(temp2^2)*temp1
      if(@solntype>2)
        fp3=(c-2)*(c-1)*c*ziterate^(c-3)-(temp2^3)*temp1
        if(@solntype>4)
          fp4=(c-3)*(c-2)*(c-1)*c*ziterate^(c-4)-(temp2^4)*temp1
          if(@solntype>6)
            fp5=(c-4)*(c-3)*(c-2)*(c-1)*c*ziterate^(c-5)-(temp2^5)*temp1
            if(@solntype>8)
              fp6=(c-5)*(c-4)*(c-3)*(c-2)*(c-1)*c*ziterate^(c-6)-(temp2^6)*temp1
            endif
          endif
        endif
      endif
    endif
  endif
;
; generate next iterate
;
  if(@solntype=="Newton (order 1)")
    dz1=-relaxation1*f/fp
    dz=dz1
  elseif(@solntype=="Halley (order 2)")
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz=dz2
  elseif(@solntype=="order 2b")
;
;   uses quadratic equation to solve for dz
;
    dz3=sqrt(sqr(fp)-2*f*fp2)
    dz1=(-f+dz3)/fp2, dz2=(-f+dz3)/fp2
    dz4=dz2
    if(cabs(f+dz1*fp)<cabs(f+dz2*fp))
      dz4=dz1
    endif
    dz=relaxation2*dz4
  elseif(@solntype=="order 3")
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz2/2*(fp2+dz1/3*fp3))
    dz=dz3
  elseif(@solntype=="order 3b")
;
; basically same as order 3, but reverse orders of dz's in dz3
;
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz1/2*(fp2+dz2/3*fp3))
    dz=dz3
  elseif(@solntype=="order 4")
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz2/2*(fp2+dz1/3*fp3))
    dz4=-relaxation4*f/(fp+dz3/2*(fp2+dz2/3*(fp3+dz1/4*fp4)))
    dz=dz4
  elseif(@solntype=="order 4b")
;
; basically same as order 4, but reverse orders of dz's in dz3 & dz4
;
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz1/2*(fp2+dz2/3*fp3))
    dz4=-relaxation4*f/(fp+dz1/2*(fp2+dz2/3*(fp3+dz3/4*fp4)))
    dz=dz4
  elseif(@solntype=="order 5")
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz2/2*(fp2+dz1/3*fp3))
    dz4=-relaxation4*f/(fp+dz3/2*(fp2+dz2/3*(fp3+dz1/4*fp4)))
    dz5=-relaxation5*f/(fp+dz4/2*(fp2+dz3/3*(fp3+dz2/4*(fp4+dz1/5*fp5))))
    dz=dz5
  elseif(@solntype=="order 5b")
;
; basically same as order 5, but reverse orders of dz's in dz3, dz4, & dz5
;
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz1/2*(fp2+dz2/3*fp3))
    dz4=-relaxation4*f/(fp+dz1/2*(fp2+dz2/3*(fp3+dz3/4*fp4)))
    dz5=-relaxation5*f/(fp+dz1/2*(fp2+dz2/3*(fp3+dz3/4*(fp4+dz4/5*fp5))))
    dz=dz5
  elseif(@solntype=="order 6")
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz2/2*(fp2+dz1/3*fp3))
    dz4=-relaxation4*f/(fp+dz3/2*(fp2+dz2/3*(fp3+dz1/4*fp4)))
    dz5=-relaxation5*f/(fp+dz4/2*(fp2+dz3/3*(fp3+dz2/4*(fp4+dz1/5*fp5))))
    dz6=-relaxation6*f/(fp+dz5/2*(fp2+dz4/3*(fp3+dz3/4*(fp4+dz2/5*(fp5+dz1/6*fp6)))))
    dz=dz6
  elseif(@solntype=="order 6b")
;
; basically same as order 6, but reverse orders of dz's in dz3, dz4, dz5 & dz6
;
    dz1=-relaxation1*f/fp
    dz2=-relaxation2*f/(fp+dz1/2*fp2)
    dz3=-relaxation3*f/(fp+dz1/2*(fp2+dz2/3*fp3))
    dz4=-relaxation4*f/(fp+dz1/2*(fp2+dz2/3*(fp3+dz3/4*fp4)))
    dz5=-relaxation5*f/(fp+dz1/2*(fp2+dz2/3*(fp3+dz3/4*(fp4+dz4/5*fp5))))
    dz6=-relaxation6*f/(fp+dz1/2*(fp2+dz2/3*(fp3+dz3/4*(fp4+dz4/5*(fp5+dz5/6*fp6)))))
    dz=dz6
  endif
  ziterate=ziterate+dz
;
; update z variable
;
  if(@ztype=="iterate")
    z=ziterate
  elseif(@ztype=="change")
    z=dz
  elseif(@ztype=="function")
    z=f
  elseif(@ztype=="1st derivative")
    z=fp
  elseif(@ztype=="2nd derivative")
    z=fp2
  elseif(@ztype=="3rd derivative")
    z=fp3
  elseif(@ztype=="4th derivative")
    z=fp4
  elseif(@ztype=="5th derivative")
    z=fp5
  elseif(@ztype=="6th derivative")
    z=fp6
  endif
bailout:
  |1/dz|<@bailout
default:
  title="Relaxed Householder's Method"
  heading
    text="Generalized Newton's method, with relaxations included to induce chaos."
  endheading
;
; general
;
  heading
    caption="General"
  endheading
  float param bailout
    caption="bailout"
    default=1000
  endparam
  param pixeltype
    caption="pixel type"
    default=0
    enum="initial z" "c" "relaxation 1" "relaxation 2" "relaxation 3"\
      "relaxation 4" "relaxation 5" "relaxation 6" "all relaxations"
  endparam
    complex param initialz
      caption="initial z"
      default=(1,1)
      visible=(@pixeltype!="initial z")
    endparam
    complex param root
      caption="c"
      default=(1,0)
      visible=(@pixeltype!="c")
    endparam
  param ztype
    caption="z type"
    default=0
    enum="iterate" "change" "function" "1st derivative" "2nd derivative"\
      "3rd derivative" "4th derivative" "5th derivative" "6th derivative"
  endparam
;
; solution method
;
  heading
    caption="Solution Method"
  endheading
  param eqntype
    caption="equation to solve"
    default=0
    enum="roots of c" "sum of 4 powers = c" "sin(z+c) = cos(c*z)"\
      "z^n + c = sin(z)" "z^n = c*cos(z)" "z^n + c = tan(z)" "z^n = c*tan(z)"\
      "sum of 4 sines = c" "sum of 4 cosines = c" "exp(cz) = log(z)/c" "z^c = c^z"
  endparam
;
  complex param power
    caption="exponent"
    default=(4,0)
    visible=((@eqntype=="roots of c")||(@eqntype=="z^n + c = sin(z)")||\
      (@eqntype=="z^n = c*cos(z)")||(@eqntype=="z^n + c = tan(z)")||\
      (@eqntype=="z^n = c*tan(z)"))
  endparam
;
  complex param coeff1
    caption="coefficient 1"
    default=(0,-4)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param power1
    caption="power 1"
    default=(4,0)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param coeff2
    caption="coefficient 2"
    default=(-4,0)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param power2
    caption="power 2"
    default=(0,4)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param coeff3
    caption="coefficient 3"
    default=(0,4)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param power3
    caption="power 3"
    default=(-4,0)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param coeff4
    caption="coefficient 4"
    default=(4,0)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
  complex param power4
    caption="power 4"
    default=(0,-4)
    visible=(@eqntype=="sum of 4 powers = c")
  endparam
;
  complex param amp1
    caption="amplitude 1"
    default=(0,-4)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param freq1
    caption="frequency 1"
    default=(1,0)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param amp2
    caption="amplitude 2"
    default=(-3,0)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param freq2
    caption="frequency 2"
    default=(0,2)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param amp3
    caption="amplitude 3"
    default=(0,2)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param freq3
    caption="frequency 3"
    default=(-3,0)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param amp4
    caption="amplitude 4"
    default=(1,0)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
  complex param freq4
    caption="frequency 4"
    default=(0,-4)
    visible=((@eqntype=="sum of 4 sines = c")||(@eqntype=="sum of 4 cosines = c"))
  endparam
;
  param solntype
    caption="solution type"
    default=0
    enum="Newton (order 1)" "Halley (order 2)" "order 2b" "order 3" "order 3b"\
      "order 4" "order 4b" "order 5" "order 5b" "order 6" "order 6b"
  endparam
;
; relaxation parameters
;
  heading
    caption="Relaxation"
  endheading
  complex param relax1
    caption="relaxation 1"
    default=(1,0)
    visible=((@pixeltype!="relaxation 1")&&(@pixeltype!="all relaxations"))
  endparam
  func relax1_func
    caption="relaxation 1 function"
    default=ident()
  endfunc
  complex param relax2
    caption="relaxation 2"
    default=(1,0)
    visible=((@pixeltype!="relaxation 2")&&(@pixeltype!="all relaxations")&&(@solntype>0))
  endparam
  func relax2_func
    caption="relaxation 2 function"
    default=ident()
    visible=(@solntype>0)
  endfunc
  complex param relax3
    caption="relaxation 3"
    default=(1,0)
    visible=((@pixeltype!="relaxation 3")&&(@pixeltype!="all relaxations")&&(@solntype>2))
  endparam
  func relax3_func
    caption="relaxation 3 function"
    default=ident()
    visible=(@solntype>2)
  endfunc
  complex param relax4
    caption="relaxation 4"
    default=(1,0)
    visible=((@pixeltype!="relaxation 4")&&(@pixeltype!="all relaxations")&&(@solntype>4))
  endparam
  func relax4_func
    caption="relaxation 4 function"
    default=ident()
    visible=(@solntype>4)
  endfunc
  complex param relax5
    caption="relaxation 5"
    default=(1,0)
    visible=((@pixeltype!="relaxation 5")&&(@pixeltype!="all relaxations")&&(@solntype>6))
  endparam
  func relax5_func
    caption="relaxation 5 function"
    default=ident()
    visible=(@solntype>6)
  endfunc
  complex param relax6
    caption="relaxation 6"
    default=(1,0)
    visible=((@pixeltype!="relaxation 6")&&(@pixeltype!="all relaxations")&&(@solntype>8))
  endparam
  func relax6_func
    caption="relaxation 6 function"
    default=ident()
    visible=(@solntype>8)
  endfunc
}

extra-foam { ; Kerry Mitchell 12 June 2021
;
; Generalization of "Mandelbrot Foam" from fractal forums
; https://fractalforums.org/fractal-mathematics-and-new-theories/28/mandelbrot-foam-math/1665/msg8427#msg8427
; original:
;   w = a*w/z
;   z = z^2 + w^2 + c
;
; powers of w and z in w and z formulas are parameters
;
; added parallel/series modes:
;   parallel: compute new w and new z independently
;   series: compute new w and use that for new z
;
; pixel_type parameter allows the pixel to represent either complex parameter (or both) or the initial values
;   of either complex variable (or both).
;
init:
  complex q=(0,0)
  complex c=(0,0)
  complex w=(0,0)
  complex wnew=(0,0)
  complex z=(0,0)
  complex znew=(0,0)
;
; Use pixel value to set parameters (like a Mandelbrot set) and/or
;   initialize variables (like a Julia set)
;
  if(@pixel_type=="q")
    q=#pixel
    c=@cinit
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="c")
    q=@qinit
    c=#pixel
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="q = c")
    q=#pixel
    c=#pixel
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="initial w")
    q=@qinit
    c=@cinit
    w=#pixel
    if(@zc==true)
      z=c
    else
      z=@zinit
    endif
  elseif(@pixel_type=="initial z")
    q=@qinit
    c=@cinit
    if(@wq==true)
      w=q
    else
      w=@winit
    endif
    z=#pixel
  elseif(@pixel_type=="w = z")
    q=@qinit
    c=@cinit
    w=#pixel
    z=#pixel
  elseif(@pixel_type=="all in")
    q=#pixel
    c=#pixel
    w=#pixel
    z=#pixel
  endif
loop:
;
; Series: compute new w and use that for new z.
; Parallel: compute new w and new z independently.
;
  if(@circuit_type=="series")
    w=q*w^@wpowerinw*z^@zpowerinw
    z=z^@zpowerinz+w^@wpowerinz+c
  elseif(@circuit_type=="parallel")
    wnew=q*w^@wpowerinw*z^@zpowerinw
    znew=z^@zpowerinz+w^@wpowerinz+c
    w=wnew, z=znew
  endif
bailout:
  |z|<@bailout
default:
  title="Extra Foam"
;
; General Parameters
;
  heading
    caption="General Parameters"
  endheading
  param pixel_type
    caption="pixel type"
    default=6
    enum="q" "c" "q = c" "initial w" "initial z" "w = z" "all in"
  endparam
  bool param wq
    caption="w = q"
    default=true
    visible=((@pixel_type=="q")||(@pixel_type=="c")||(@pixel_type=="q = c")||(@pixel_type=="initial z"))
  endparam
  bool param zc
    caption="z = c"
    default=true
    visible=((@pixel_type=="q")||(@pixel_type=="c")||(@pixel_type=="q = c")||(@pixel_type=="initial w"))
  endparam
  param circuit_type
    caption="circuit type"
    default=0
    enum="series" "parallel"
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; Initializations
;
  heading
    caption="Initializations"
  endheading
  complex param qinit
    caption="q"
    default=(1,0)
    visible=((@pixel_type=="c")||(@pixel_type=="initial w")||(@pixel_type=="initial z")||(@pixel_type=="w = z"))
  endparam
  complex param cinit
    caption="c"
    default=(1,0)
    visible=((@pixel_type=="q")||(@pixel_type=="initial w")||(@pixel_type=="initial z")||(@pixel_type=="w = z"))
  endparam
  complex param winit
    caption="initial w"
    default=(0,0)
    visible=((@wq==false)&&(@pixel_type!="initial w")&&(@pixel_type!="w = z"))
  endparam
  complex param zinit
    caption="initial z"
    default=(0,0)
    visible=((@zc==false)&&(@pixel_type!="initial z")&&(@pixel_type!="w = z"))
  endparam
;
; Variable Powers
;
  heading
    caption="Variable Powers"
  endheading
  complex param wpowerinw
    caption="w power in w"
    default=1
  endparam
  complex param zpowerinw
    caption="z power in w"
    default=-1
  endparam
  complex param wpowerinz
    caption="w power in z"
    default=2
  endparam
  complex param zpowerinz
    caption="z power in z"
    default=2
  endparam
}

round-and-transform { ; Kerry Mitchell 27Nov2021
;
; Each iteration, round z_old to the nearest mulitple of round_base and then
; transform the rounded result to z_new. Alternatively, transform the residual,
; z_rounded - z_old. Optionally, the round_base can also be transformed each
; iteration.
;
$define debug
init:
  complex base_amp=(0,0)
  complex base_offset=(0,0)
  complex base_power=(0,0)
  complex roundbase=(0,0)
  complex z=(0,0)
  complex zinc=(0,0)
  complex zold=(0,0)
  complex znew=(0,0)
  complex zround=(0,0)
  complex zresidual=(0,0)
  complex z_amp=(0,0)
  complex z_offset=(0,0)
  complex z_power=(0,0)
  float dx=0
  float dy=0
;
; initializations
;
  roundbase=@q_roundbase
  base_amp=@q_base_amp
  base_freq=@q_base_freq
  base_power=@q_base_power
  base_offset=@q_base_offset
  znew=@q_z_initial
  z_amp=@q_z_amp
  z_freq=@q_z_freq
  z_power=@q_z_power
  z_offset=@q_z_offset
  if(@pixel_type=="initial base")
    roundbase=#pixel
  elseif(@pixel_type=="base amplitude")
    base_amp=#pixel
  elseif(@pixel_type=="base frequency")
    base_freq=#pixel
  elseif(@pixel_type=="base power")
    base_power=#pixel
  elseif(@pixel_type=="base offset")
    base_offset=#pixel
  elseif(@pixel_type=="initial z")
    znew=#pixel
  elseif(@pixel_type=="z amplitude")
    z_amp=#pixel
  elseif(@pixel_type=="z frequency")
    z_freq=#pixel
  elseif(@pixel_type=="z power")
    z_power=#pixel
  elseif(@pixel_type=="z offset")
    z_offset=#pixel
  endif
loop:
;
; transform the rounding base
;
  if(@base_transform==true)
    roundbase=base_amp*@basefn(base_freq*roundbase)^base_power+base_offset
  endif
;
; round zold: zround=round(zold/roundbase)*roundbase
;
  zold=znew
  zround=round(zold/roundbase)*roundbase
;
; transform zround into znew
;
  if(@use_mod=="rounded")
    znew=zround
    znew=z_amp*@zfn(z_freq*znew)^z_power+z_offset
  elseif(@use_mod=="residual")
    znew=zround-zold
    znew=z_amp*@zfn(z_freq*znew)^z_power+z_offset
  elseif(@use_mod=="no transform")
  endif
;
; set z to send to coloring formula
;
  if(@z_type=="rounding base")
    z=roundbase
  elseif(@z_type=="rounded z")
    z=zround
  elseif(@z_type=="transformed z")
    z=znew
  elseif(@z_type=="last absolute residual")
    z=zround-zold
  elseif(@z_type=="sum absolute residuals")
    zresidual=zresidual+(zround-zold)
    z=zresidual
  elseif(@z_type=="last relative residual")
    z=(zround-zold)/roundbase
  elseif(@z_type=="sum relative residuals")
    zresidual=zresidual+(zround-zold)/roundbase
    z=zresidual
  elseif(@z_type=="last unit residual")
    zinc=zround-zold
    dx=real(zinc), dy=imag(zinc)
    zinc=(0,0)
    if(dx>0)
      zinc=1
    elseif(dx<0)
      zinc=-1
    endif
    if(dy>0)
      zinc=zinc+(0,1)
    elseif(dy<0)
      zinc=zinc-(0,1)
    endif
    z=zinc
  elseif(@z_type=="sum unit residuals")
    zinc=zround-zold
    dx=real(zinc), dy=imag(zinc)
    zinc=(0,0)
    if(dx>0)
      zinc=1
    elseif(dx<0)
      zinc=-1
    endif
    if(dy>0)
      zinc=zinc+(0,1)
    elseif(dy<0)
      zinc=zinc-(0,1)
    endif
    zresidual=zresidual+zinc
    z=zresidual
  endif
bailout:
  |z|<@bailout
default:
  title="Round and Transform"
;
; general parameters
;
  heading
    caption="General Parameters"
  endheading
  param pixel_type
    caption="pixel type"
    default=5
    enum="initial base" "base amplitude" "base frequency" "base power" "base offset"\
      "initial z" "z amplitude" "z frequency" "z power" "z offset"
  endparam
  param z_type
    caption="z type"
    default=8
    enum="rounding base" "rounded z" "transformed z" "last absolute residual"\
      "sum absolute residuals" "last relative residual" "sum relative residuals"\
      "last unit residual" "sum unit residuals"
  endparam
  float param bailout
    caption="bailout"
    default=1000
  endparam
;
; rounding base parameters
;
  heading
    caption="Rounding Base Parameters"
  endheading
  complex param q_roundbase
    caption="initial rounding base"
    default=(1,2)
    visible=(@pixel_type!="initial base")
  endparam
  bool param base_transform
    caption="transform base"
    default=true
  endparam
  func basefn
    caption="base function"
    default=ident()
    visible=(@base_transform==true)
  endfunc
  complex param q_base_amp
    caption="base amplitude"
    default=(1.6,-1.2)
    visible=((@base_transform==true)&&(@pixel_type!="base amplitude"))
  endparam
  complex param q_base_freq
    caption="base frequency"
    default=(1,0)
    visible=((@base_transform==true)&&(@pixel_type!="base frequency"))
  endparam
  complex param q_base_power
    caption="base power"
    default=(1,0)
    visible=((@base_transform==true)&&(@pixel_type!="base power"))
  endparam
  complex param q_base_offset
    caption="base offset"
    default=(0,0)
    visible=((@base_transform==true)&&(@pixel_type!="base offset"))
  endparam
;
; z transform parameters
;
  heading
    caption="z Transform Parameters"
  endheading
  complex param q_z_initial
    caption="initial z"
    default=(0,1)
    visible=(@pixel_type!="initial z")
  endparam
  param use_mod
    caption="what to transform"
    default=0
    enum="rounded" "residual" "no transform"
  endparam
  func zfn
    caption="z function"
    default=ident()
    visible=(@use_mod!="no transform")
  endfunc
  complex param q_z_amp
    caption="z amplitude"
    default=(1,0)
    visible=((@use_mod!="no transform")&&(@pixel_type!="z amplitude"))
  endparam
  complex param q_z_freq
    caption="z frequency"
    default=(1,0)
    visible=((@use_mod!="no transform")&&(@pixel_type!="z frequency"))
  endparam
  complex param q_z_power
    caption="z power"
    default=(1,0)
    visible=((@use_mod!="no transform")&&(@pixel_type!="z power"))
  endparam
  complex param q_z_offset
    caption="z offset"
    default=(0,0)
    visible=((@use_mod!="no transform")&&(@pixel_type!="z offset"))
  endparam
}