comment { Copyright © 1999-2018 by Mark Townsend.

  Fractal Formulas for Ultra Fractal 2.0 and above
  
  Last update 24 October 2018
  
; This work is licensed under a Creative Commons
; Attribution-NonCommercial-ShareAlike 4.0 International License.
; http://creativecommons.org/licenses/by-nc-sa/4.0/

}

mt-fractalia-1 { ; Mark Townsend, 9 Feb 1999
;
; This formula is from Clifford A. Pickover's
; book "Keys to Infinity". It's hard to find
; interesting spots just by entering random
; parameters, so I've implemented six versions
; of the formula in a loop of switches. 
;
init:
  z = 0
  float xx = 0
  float x  = real(#pixel)
  float y  = imag(#pixel)
  float a  = real(@ag)
  float g  = imag(@ag)
  float b  = real(@bh)
  float h  = imag(@bh)
  float c  = real(@ci)
  float i  = imag(@ci)
  float d  = real(@dj)
  float j  = imag(@dj)
  float e1 = real(@ek)
  float k  = imag(@ek)
  float f  = real(@fl)
  float l  = imag(@fl)
loop:
  xx = x
  x = a + (b * x) + (c * x^2) + (d * x * y) + (e1 * y) + (f * y^2)
  y = g + (h * xx) + (i * xx^2) + (j * xx * y) + (k * y) + (l * y^2)
  z = x + flip(y)
bailout:
  |z| < @bailout
default:
  title = "Fractalia I"
  maxiter = 1000
  param ag
    caption = "a and g"
    default = (0,0)
  endparam    
  param bh
    caption = "b and h"
    default = (1,-0.2)
  endparam    
  param ci
    caption = "c and i"
    default = (0.5,0.8)
  endparam    
  param dj
    caption = "d and j"
    default = (0.4,0.9)
  endparam    
  param ek
    caption = "e and k"
    default = (-0.7,-0.7)
  endparam    
  param fl
    caption = "f and l"
    default = (0.9,-0.2)
  endparam    
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-fractalia-2"
  bailout = bailout
  ag = ag
  bh = #pixel
  ci = ci
  dj = dj
  ek = ek
  fl = fl
}

mt-fractalia-2 { ; Mark Townsend, 9 Feb 1999
;
; This formula is from Clifford A. Pickover's
; book "Keys to Infinity".
;
init:
  z = 0
  float xx = 0
  float x  = real(#pixel)
  float y  = imag(#pixel)
  float a  = real(@ag)
  float g  = imag(@ag)
  float b  = real(@bh)
  float h  = imag(@bh)
  float c  = real(@ci)
  float i  = imag(@ci)
  float d  = real(@dj)
  float j  = imag(@dj)
  float e1 = real(@ek)
  float k  = imag(@ek)
  float f  = real(@fl)
  float l  = imag(@fl)
loop:
  xx = x
  x = a + (b * x) + (c * x^2) + (d * x * y) + (e1 * y) + (f * y^2)
  y = g + (h * xx) + (i * xx^2) + (j * xx * y) + (k * y) + (l * y^2)
  z = x + flip(y)
bailout:
  |z| < @bailout
default:
  title = "Fractalia II"
  maxiter = 1000
  param ag
    caption = "a and g"
    default = (0,0)
  endparam    
  param bh
    caption = "b and h"
    default = (1,-0.2)
  endparam    
  param ci
    caption = "c and i"
    default = (0.5,0.8)
  endparam    
  param dj
    caption = "d and j"
    default = (0.4,0.9)
  endparam    
  param ek
    caption = "e and k"
    default = (-0.7,-0.7)
  endparam    
  param fl
    caption = "f and l"
    default = (0.9,-0.2)
  endparam    
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-fractalia-3"
  bailout = bailout
  ag = ag
  bh = bh
  ci = #pixel
  dj = dj
  ek = ek
  fl = fl
}

mt-fractalia-3 { ; Mark Townsend, 9 Feb 1999
;
; This formula is from Clifford A. Pickover's
; book "Keys to Infinity".
;
init:
  z = 0
  float xx = 0
  float x  = real(#pixel)
  float y  = imag(#pixel)
  float a  = real(@ag)
  float g  = imag(@ag)
  float b  = real(@bh)
  float h  = imag(@bh)
  float c  = real(@ci)
  float i  = imag(@ci)
  float d  = real(@dj)
  float j  = imag(@dj)
  float e1 = real(@ek)
  float k  = imag(@ek)
  float f  = real(@fl)
  float l  = imag(@fl)
loop:
  xx = x
  x = a + (b * x) + (c * x^2) + (d * x * y) + (e1 * y) + (f * y^2)
  y = g + (h * xx) + (i * xx^2) + (j * xx * y) + (k * y) + (l * y^2)
  z = x + flip(y)
bailout:
  |z| < @bailout
default:
  title = "Fractalia III"
  maxiter = 1000
  param ag
    caption = "a and g"
    default = (0,0)
  endparam    
  param bh
    caption = "b and h"
    default = (1,-0.2)
  endparam    
  param ci
    caption = "c and i"
    default = (0.5,0.8)
  endparam    
  param dj
    caption = "d and j"
    default = (0.4,0.9)
  endparam    
  param ek
    caption = "e and k"
    default = (-0.7,-0.7)
  endparam    
  param fl
    caption = "f and l"
    default = (0.9,-0.2)
  endparam    
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-fractalia-4"
  bailout = bailout
  ag = ag
  bh = bh
  ci = ci
  dj = #pixel
  ek = ek
  fl = ll
}

mt-fractalia-4 { ; Mark Townsend, 9 Feb 1999
;
; This formula is from Clifford A. Pickover's
; book "Keys to Infinity".
;
init:
  z = 0
  float xx = 0
  float x  = real(#pixel)
  float y  = imag(#pixel)
  float a  = real(@ag)
  float g  = imag(@ag)
  float b  = real(@bh)
  float h  = imag(@bh)
  float c  = real(@ci)
  float i  = imag(@ci)
  float d  = real(@dj)
  float j  = imag(@dj)
  float e1 = real(@ek)
  float k  = imag(@ek)
  float f  = real(@fl)
  float l  = imag(@fl)
loop:
  xx = x
  x = a + (b * x) + (c * x^2) + (d * x * y) + (e1 * y) + (f * y^2)
  y = g + (h * xx) + (i * xx^2) + (j * xx * y) + (k * y) + (l * y^2)
  z = x + flip(y)
bailout:
  |z| < @bailout
default:
  title = "Fractalia IV"
  maxiter = 1000
  param ag
    caption = "a and g"
    default = (0,0)
  endparam    
  param bh
    caption = "b and h"
    default = (1,-0.2)
  endparam    
  param ci
    caption = "c and i"
    default = (0.5,0.8)
  endparam    
  param dj
    caption = "d and j"
    default = (0.4,0.9)
  endparam    
  param ek
    caption = "e and k"
    default = (-0.7,-0.7)
  endparam    
  param fl
    caption = "f and l"
    default = (0.9,-0.2)
  endparam    
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-fractalia-5"
  bailout = bailout
  ag = ag
  bh = bh
  ci = ci
  dj = dj
  ek = #pixel
  fl = fl
}

mt-fractalia-5 { ; Mark Townsend, 9 Feb 1999
;
; This formula is from Clifford A. Pickover's
; book "Keys to Infinity".
;
init:
  z = 0
  float xx = 0
  float x  = real(#pixel)
  float y  = imag(#pixel)
  float a  = real(@ag)
  float g  = imag(@ag)
  float b  = real(@bh)
  float h  = imag(@bh)
  float c  = real(@ci)
  float i  = imag(@ci)
  float d  = real(@dj)
  float j  = imag(@dj)
  float e1 = real(@ek)
  float k  = imag(@ek)
  float f  = real(@fl)
  float l  = imag(@fl)
loop:
  xx = x
  x = a + (b * x) + (c * x^2) + (d * x * y) + (e1 * y) + (f * y^2)
  y = g + (h * xx) + (i * xx^2) + (j * xx * y) + (k * y) + (l * y^2)
  z = x + flip(y)
bailout:
  |z| < @bailout
default:
  title = "Fractalia V"
  maxiter = 1000
  param ag
    caption = "a and g"
    default = (0,0)
  endparam    
  param bh
    caption = "b and h"
    default = (1,-0.2)
  endparam    
  param ci
    caption = "c and i"
    default = (0.5,0.8)
  endparam    
  param dj
    caption = "d and j"
    default = (0.4,0.9)
  endparam    
  param ek
    caption = "e and k"
    default = (-0.7,-0.7)
  endparam    
  param fl
    caption = "f and l"
    default = (0.9,-0.2)
  endparam    
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-fractalia-6"
  bailout = bailout
  ag = ag
  bh = bh
  ci = ci
  dj = dj
  ek = ek
  fl = #pixel
}

mt-fractalia-6 { ; Mark Townsend, 9 Feb 1999
;
; This formula is from Clifford A. Pickover's
; book "Keys to Infinity".
;
init:
  z = 0
  float xx = 0
  float x  = real(#pixel)
  float y  = imag(#pixel)
  float a  = real(@ag)
  float g  = imag(@ag)
  float b  = real(@bh)
  float h  = imag(@bh)
  float c  = real(@ci)
  float i  = imag(@ci)
  float d  = real(@dj)
  float j  = imag(@dj)
  float e1 = real(@ek)
  float k  = imag(@ek)
  float f  = real(@fl)
  float l  = imag(@fl)
loop:
  xx = x
  x = a + (b * x) + (c * x^2) + (d * x * y) + (e1 * y) + (f * y^2)
  y = g + (h * xx) + (i * xx^2) + (j * xx * y) + (k * y) + (l * y^2)
  z = x + flip(y)
bailout:
  |z| < @bailout
default:
  title = "Fractalia VI"
  maxiter = 1000
  param ag
    caption = "a and g"
    default = (0,0)
  endparam    
  param bh
    caption = "b and h"
    default = (1,-0.2)
  endparam    
  param ci
    caption = "c and i"
    default = (0.5,0.8)
  endparam    
  param dj
    caption = "d and j"
    default = (0.4,0.9)
  endparam    
  param ek
    caption = "e and k"
    default = (-0.7,-0.7)
  endparam    
  param fl
    caption = "f and l"
    default = (0.9,-0.2)
  endparam    
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-fractalia-1"
  bailout = bailout
  ag = #pixel
  bh = bh
  ci = ci
  dj = dj
  ek = ek
  fl = fl
}

mt-newton-fz-m { ; Mark Townsend, 1 Mar 1999
;
; Relaxed Newton's method for f(z)+z^n-c
;
init:
  c = #pixel
  float h = 0.0001
  z = fz = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = @f(z) + z^@n - c
  z = z - @r * fz / (1 / h * (@f(zh) + zh^@n - c - fz))
bailout:
; I could test fz since we're trying to
; find the solution to fz = 0, but this
; works better with my log smoothing.
  |z - oldz| > @epsilon
default:
  title = "Newton Function I"
  center = (-1.5,0)
  periodicity = 0
  param n
    caption = "Exponent"
    default = 0.0
  endparam
  param r
    caption = "Relaxation"
    default = (1,0)
    hint = "The suggested range is 0<r<2."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
  func f
    caption = "Function"
    default = sin()
  endfunc  
switch:
  type = "mt-newton-fz-j"
  c = #pixel
  n = n
  r = r
  f = f
  epsilon = epsilon
}

mt-newton-fz-j { ; Mark Townsend, 1 Mar 1999
;
; Relaxed Newton's method for f(z)+z^n-c
;
init:
  z = #pixel
  float h = 0.0001
  fz = oldz = zh = 0
loop:
  oldz = z
  zh = z + h
  fz = @f(z) + z^@n - @c
  z = z -@r * fz / (1/h * (@f(zh) + zh^@n - @c - fz))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton Function I Julia"
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (0,-1)
  endparam
  param n
    caption = "Exponent"
    default = 3.0
  endparam  
  param r
    caption = "Relaxation"
    default = (1,0)
    hint = "The suggested range is 0<r<2."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
  func f
    caption = "Function"
    default = sin()
  endfunc  
switch:
  type = "mt-newton-fz-m"
  n = n
  r = r
  f = f
  epsilon = epsilon
}

mt-newton-error-m { ; Mark Townsend, 5 Mar 1999
;
; This formula is the result of an error while
; trying to implement Newton's method for
; z+(z*c-1)^3
;
init:
  z = @fn(1 - #pixel)
  c = #pixel
  float h = 0.0001
  fz = fzd = oldz = 0
loop:
  oldz = z
  fz = z + (z * c - 1)^3
  fzd = 1 / h * (z + ((z + h) * c - 1)^3 - fz) 
  z = z - fz / fzd 
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton Error"
  maxiter = 1000
  periodicity = 0
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
  func fn
    caption = "Transform"
    default = sqr()
    hint = "This determines the starting point for z."
  endfunc  
switch:
  type = "mt-newton-error-j"
  c = #pixel
  epsilon = epsilon
}

mt-newton-error-j { ; Mark Townsend, 5 Mar 1999
;
; This formula is the result of an error while
; trying to implement Newton's method for
; z+(z*c-1)^3
;
init:
  z = #pixel
  float h = 0.0001
  fz = fzd = oldz = 0
loop:
  oldz = z
  fz = z + (z * @c - 1)^3
  fzd = 1 / h * (z + ((z + h) * @c - 1)^3 - fz)
  z = z - fz / fzd
bailout:
 |z - oldz| > @epsilon
default:
  title = "Newton Error Julia"
  maxiter = 1000
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (-0.5,0)
  endparam
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
switch:
  type = "mt-newton-error-m"
  epsilon = epsilon
}

mt-secant-zncz { ; Mark Townsend, 6 Mar 1999
;
; I think I have the Secant method right,
; thanks to an anonymous formula called
; "Secant_z4-1" in the Orgform collection.
; Here I use it on the same expression that
; the Cayley formulas use.
;
init:
  z = #pixel
  oz = @f(z)
  fz = fzd = oldz = 0
loop:
  oldz = z
  fzd = oz ^ @n - @c * oz + @c - 1
  fz = z^@n - @c * z + @c - 1
  z = z - fz * (z - oz) / (fz - fzd)
  oz = oldz
bailout:  
 |fz| > @epsilon
default:
  title = "Secant I"
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam    
  param n
    caption = "Exponent"
    default = 3.0
  endparam    
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value"
  endparam  
  func f
    caption = "Transform"
    hint = "This determines how the second initital value \
           is derived from the first."
    default = sin()
  endfunc
switch:
  type = "mt-secant-zncz"
  epsilon = epsilon
  n = n
  c = #pixel
  f = f
}

mt-magnet-II-m { ; Mark Townsend, 16 Mar 1999
;
; Magnetism Model II from "The Beauty of
; Fractals". This implementation allows you
; to select the "critical value", in other
; words the starting point for z. When z
; starts at sqr(1-q) the resulting set seems
; to be a better "catalogue" of the Julia sets.
; 
init:
  q = #pixel
  if @v == 0
    z = 0
  else
    z = sqr(1 - q)
  endif
loop:
  z = ((z^3+3*(q-1)*z+(q-1)*(q-2))/(3*z^2+3*(q-2)*z+q^2-3*q+3))^2
bailout:
  |z| < @bailout && |z - 1| > @epsilon
default:
  title = "Magnetism Model II"
  magn = 0.75
  center = (1,0)
  maxiter = 1000
  param v
    caption = "Critical value"
    enum = "0" "Sqr(1-q)"
    default = 1
    hint = "This determines the starting point for z."
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
    hint = "This is the bailout value for divergent orbits."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value for convergent orbits."
  endparam
switch:
  type = "mt-magnet-II-j"
  q = #pixel  
  bailout = bailout
  epsilon = epsilon
}  

mt-magnet-II-j { ; Mark Townsend, 16 Mar 1999
;
; Magnetism Model II from "The Beauty of
; Fractals"
;
init:
  z = #pixel
  oldz = 0
loop:
  oldz = z
  z = ((z^3+3*(@q-1)*z+(@q-1)*(@q-2))/(3*z^2+3*(@q-2)*z+@q^2-3*@q+3))^2
bailout:
  |z| < @bailout && |z - oldz| > @epsilon
default:
  title = "Magnetism Model II Julia"
  maxiter = 1000
  magn = 0.125
  center = (-1.7,0)
  periodicity = 0
  param q
    caption = "Julia seed"
    default = (0,0)
  endparam 
  param bailout
    caption = "Bailout value"
    default = 1000.0
    hint = "This is the bailout value for divergent orbits."
  endparam   
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value for convergent orbits."
  endparam  
switch:
  type = "mt-magnet-II-m"
  bailout = bailout
  epsilon = epsilon
}  

mt-bof-newton-I-m { ; Mark Townsend, 18 Mar 1999
;
; Modified Newton's method for 
; (z-1)*(z+c)*(z^2+1), from "The Beauty of
; Fractals". 
;
init:
  z = 0
  c = #pixel
  float h = 0.0001
  fz = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = (z - 1) * (z + c) * (z^2 + 1)
  fzd = 1 / h *((zh - 1) * (zh + c) *(zh^2 + 1) - fz) 
  z = z - fz /(fzd + flip(@s))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Bof Newton I"
  magn = 0.75
  maxiter = 1000
  periodicity = 0
  param s
    caption = "Parameter (s)"
    default = 0.0
    hint = "The suggested range is 0<s<1."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
switch:
  type = "mt-bof-newton-I-j"
  s = s
  c = #pixel
  bailout = bailout
}
mt-bof-newton-I-j { ; Mark Townsend, 18 Mar 1999
;
; Modified Newton's method for 
; (z-1)*(z+c)*(z^2+1), from "The Beauty of
; Fractals". The default parameters are those
; used in the book to produce the Julia set
; featured in Maps 61 to 65.
;
init:
  z = #pixel
  float h = 0.0001
  fz = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = (z - 1) * (z + @c) * (z^2 + 1)
  fzd = 1 / h * ((zh - 1) * (zh + @c) * (zh^2 + 1) - fz)
  z = z - fz / (fzd + flip(@s))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Bof Newton I Julia"
  maxiter = 1000
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (0.5,0)
  endparam
  param s
    caption = "Parameter (s)"
    default = -0.5
    hint = "The suggested range is 0<s<1."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
switch:
  type = "mt-bof-newton-I-m"
  s = s
  epsilon = epsilon
}

mt-bof-newton-II-m { ; Mark Townsend, 18 Mar 1999
;
; Modified Newton's method for (z-1)*(z^2+z+c),
; from "The Beauty of Fractals".
;
init:
  z = 0
  c = #pixel
  float h = 0.0001
  fz = fzd = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = (z - 1) * (z^2 + z + c)
  fzd = 1 / h * ((zh - 1) * (zh^2 + zh + c) - fz)
  z = z - fz / (fzd + flip(@s))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Bof Newton II"
  magn = 0.7
  maxiter = 1000
  periodicity = 0
  param s
    caption = "Parameter (s)"
    default = 0.0
    hint = "The suggested range is 0<s<1."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
switch:
  type = "mt-bof-newton-II-j"
  c = #pixel
  s = s
  epsilon = epsilon
}

mt-bof-newton-II-j { ; Mark Townsend, 18 Mar 1999
;
; Modified Newton's method for (z-1)*(z^2+z+c),
; from "The Beauty of Fractals".
;
init:
  z = #pixel
  float h = 0.0001
  fz = fzd = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = (z - 1) * (z^2 + z + @c)
  fzd = 1 / h *((zh - 1) * (zh^2 + zh + @c) - fz)
  z = z - fz / (fzd + flip(@s))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Bof Newton II Julia"
  maxiter = 1000
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (0.5,0)
  endparam
  param s
    caption = "Parameter (s)"
    default = 0.0
    hint = "The suggested range is 0<s<1."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
switch:
  type = "mt-bof-newton-II-m"
  s = s
  epsilon = epsilon
}

mt-newton-twist { ; Mark Townsend, 27 Mar 1999
;
; Modified Newton's method for z^n-z. I've
; allowed the "s" parameter from the Bof
; Newton formulas to become complex and
; used it as the seed for the Julia sets.
;
init:
  z = #pixel
  float h = 0.0001
  fz = oldz = zh = 0
loop:
  oldz = z
  zh = z + h
  fz = z^@n - z
  z = z - fz / (1 / h * ((zh^@n - zh) - fz) + @c)
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton Twist" 
  periodicity = 0
  param c
    caption = "Disturbance"
    default = (1,1)
    hint = "The best way to set this value is with \
           the Switch feature."
  endparam
  param n
    caption = "Exponent"
    default = 4.0
    hint = "The practical effect of this parameter is \
           to change the number of spiral arms." 
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
switch:
  type = "mt-newton-twist"
  n = n
  c = #pixel
  epsilon = epsilon
}

mt-attractor-julia { ; Mark Townsend, 29 Mar 1999
; 
; Julia sets that bail out when the orbit reaches
; (hopefully) a periodic attractor. I iterate the
; critical point z = (0,0) before the main loop
; and test against the resulting value. In the
; hope of speeding the process up I include
; the option to give up the search when z
; grows too large.
;
init:
  w = 0
  u = #pixel
  int iter = 0
  if @no_bail
    while iter < @max
      w = w^2 + @c
      iter = iter + 1
    endwhile
  else
    while (iter <= @max) && (|u| < 10)
      w = w^2 + @c
      u = u^2 + @c
      iter = iter + 1
    endwhile
  endif  
  z = #pixel
loop:
  z = z^2 + @c  
bailout:
  |w - z| > @epsilon && |z| < @bailout
default:
  title = "Attractor Julia"
  maxiter = 1000
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (-0.12,0.74)
  endparam
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value for convergent orbits."
  endparam
  param bailout
    caption = "Bailout value"
    default = 100.0
    hint = "This is the bailout value for divergent orbits."
  endparam
  param max
    caption = "Search maxiter"
    default = 100
    hint = "The is the maxiter value for the attractor \
           search."
  endparam  
  param no_bail
    caption = "Every pixel"
    default = false
    hint = "This determines whether to bailout of the attractor \
           search when z grows too large. Enable this when the \
           image is mainly inside."
  endparam  
switch:
  type = "mt-attractor-m"
  max = max
  epsilon = epsilon
  bailout = bailout
  no_bail = no_bail
}

mt-attractor-m { ; Mark Townsend, 29 Mar 1999
;
; This is just the Mandelbrot set to use as a
; switch to my "Attractor Julia" formula. It 
; includes dummy parameters to retain the
; values for the other formula.
;
init:
  z = 0
  c = #pixel
loop:
  z = z^2 + c 
bailout:
  |z| < @bailout
default:
  title = "Attractor Mandelbrot"
  center = (-0.5,0)
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam
  param epsilon
    caption = "Epsilon"
    default = 0.0001
    hint = "This is a dummy parameter so that the value is not \
           lost when switching to and from the Julia sets"
  endparam
  param maxiter
    caption = "Search maxiter"
    default = 100
    hint = "This is a dummy parameter so that the value is not \
           lost when switching to and from the Julia sets"
  endparam          
  param no_bail
    caption = "Every pixel"
    default = false
    hint = "This is a dummy parameter so that the value is not \
           lost when switching to and from the Julia sets"
  endparam  
switch:
  type = "mt-attractor-julia"
  max = max
  c = #pixel
  epsilon = epsilon
  bailout = bailout
  no_bail = no_bail
}


mt-newton-onef-m { ; Mark Townsend, 21 Mar 1999
;
; Newton's method for 1+f(c*z)-c^2
;
init:
  z = 0
  c =  #pixel
  float h = 0.0001
  fz = oldz = 0
loop:
  oldz = z
  fz = 1 + @f(c * z) - c^2
  z = z - fz / (1 / h * ((1 + @f(c * (z + h)) - c^2) - fz))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton Function II"
  periodicity = 0
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
  func f
    caption = "Function"
    default = tanh()
  endfunc  
switch:
  type = "mt-newton-onef-j"
  c = #pixel
  f = f
  epsilon = epsilon
}

mt-newton-onef-j { ; Mark Townsend, 21 Mar 1999
;
; Newton's method for 1+f(c*z)-c^2
;
init:
  z =  #pixel
  float h = 0.0001
  fz = oldz = 0
loop:
  oldz = z
  fz = 1 + @f(@c * z) - @c^2
  z = z - fz / (1 / h*((1 + @f(@c * (z + h)) - @c^2) - fz))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton Function II Julia"
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (0,-1)
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam  
  func f
    caption = "Function"
    default = sin()
  endfunc  
switch:
  type = "mt-newton-onef-m"
  f = f
  epsilon = epsilon
}

mt-pseudo-magnet { ; Mark Townsend, 8 Apr 1999
;
; This started out as an attempt to write a
; Newton method fractal from memory. The
; result seems more like a Magnet fractal
;
init:
; As ugly as it is, this is about the best
; map of the Julia sets that I could find.
  z = (@n-1)^(1/@n)
  c = #pixel
loop:
  z = z - (z^@n - c) / (@n * z)
bailout:
  |z| < @bailout 
default:
  title = "Pseudo Magnet"
  magn = 0.12
  center = (1.5,0)
  maxiter = 1000
  param n
    caption = "Power"
    default = 3.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
    hint = "This is the bailout value for divergent orbits."
  endparam  
switch:
  type = "mt-pseudo-magnet-j"
  c = #pixel
  n =n
  bailout = bailout
  epsilon = epsilon
}

mt-pseudo-magnet-j { ; Mark Townsend, 8 Apr 1999
;
; This started out as an attempt to write a
; Newton method fractal from memory. The
; result seems more like a Magnet fractal
;
init:
  z = #pixel
  oldz = 0
loop:
  oldz = z
  z = z - (z^@n - @c) / (@n * z)
bailout:
  |z - oldz| > @epsilon && |z| < @bailout 
default:
  title = "Pseudo Magnet Julia"
  magn = 0.4
  center = (1.5,0)
  periodicity = 0
  maxiter = 1000
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param n
    caption = "Power"
    default = 3.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
    hint = "This is the bailout value for divergent orbits."
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value for convergent orbits."
  endparam  
switch:
  type = "mt-pseudo-magnet"
  n = n
  bailout = bailout
  epsilon = epsilon
}

mt-newton-9904a-m { ; Mark Townsend, 5 Apr 1999
;
; Modified Newton's method for (z-1)*(z+c)*(z^2+c).
;
init:
  z = 0
  c = #pixel
  float h = 0.0001
  fz = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = (z - 1) * (z + c) * (z^2 + c)
  fzd = 1 / h *((zh - 1) * (zh + c) *(zh^2 + c) - fz) 
  z = z - fz /(fzd + flip(@s))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton 9904a"
  magn = 0.6
  maxiter = 1000
  periodicity = 0
  param s
    caption = "Parameter (s)"
    default = 0.0
    hint = "Suggested range is 0<s<1"
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "Determines the convergence required for \
           the orbit to bail out"
  endparam  
switch:
  type = "mt-newton-9904a-j"
  s = s
  c = #pixel
  bailout = bailout
}

mt-newton-9904a-j { ; Mark Townsend, 5 Apr Mar 1999
;
; Modified Newton's method for (z-1)*(z+c)*(z^2+c).
;
init:
  z = #pixel
  float h = 0.0001
  fz = zh = oldz = 0
loop:
  oldz = z
  zh = z + h
  fz = (z - 1) * (z + @c) * (z^2 + @c)
  fzd = 1 / h *((zh - 1) * (zh + @c) *(zh^2 + @c) - fz) 
  z = z - fz / (fzd + flip(@s))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Newton 9904a Julia"
  maxiter = 1000
  periodicity = 0
  param c
    caption = "Julia seed"
    default = (0.42,.36)
  endparam
  param s
    caption = "Parameter (s)"
    default = 0.0
    hint = "Suggested range is 0<s<1"
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "Determines the convergence required for \
           the orbit to bail out"
  endparam  
switch:
  type = "mt-newton-9904a-m"
  s = s
  epsilon = epsilon
}

mt-latoocarfian { ; Mark Townsend, 11 Apr 1999
;
; The Latoocarfian equations from Clifford A.
; Pickover's "Chaos in Wonderland".
;
init:
  if @map == 0
    float x = 0.01
    float y = 0.01
    float a = real(#pixel)
    float b = imag(#pixel)
    float c = real(@cd)
    float d = imag(@cd)
  elseif @map == 1
    float x = 0.01
    float y = 0.01
    float a = real(@ab)
    float b = imag(@ab)
    float c = real(#pixel)
    float d = imag(#pixel)
  else
    float x = real(#pixel)
    float y = imag(#pixel)
    float a = real(@ab)
    float b = imag(@ab)
    float c = real(@cd)
    float d = imag(@cd)
  endif
  float xx = 0
  z = 0    
loop:
  xx = x
  x = sin(y * b) + c * sin(x * b)
  y = sin(xx * a) + d * sin(y * a)
  z = x + flip(y)
bailout:
; no bailout  
  true
default:
  title = "Latoocarfian"
  param map
    caption = "Mapping"
    enum = "a, b" "c, d" "x, y"
  endparam  
  param ab
    caption = "a, b"
    default = (-0.966918,2.879879)
  endparam  
  param cd
    caption = "c, d"
    default = (0.765145,0.744728)
  endparam  
}

mt-pixel { ; Mark Townsend, 3 Apr 1999
; 
; Returns the unaltered pixel. 
;
init:
  z = #pixel
loop:
; Nothing here
bailout:
  @inside
default:
  title = "Pixel"
  param inside
    caption = "Inside"
    hint = "If this is enabled all of the pixels become inside."
    default = false
  endparam
}

mt-990425a { ; Mark Townsend, 25 Apr 1999
init:
   z = 0
   c = #pixel 
loop:
  z = z^2 + flip(real(c))
  z = z^2 + flip(imag(c))
bailout:
  |z| < @bailout
default:
  title = "990425a"
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam  
switch:
  type = "mt-990425a-j"
  bailout = bailout
  c = #pixel 
}

mt-990425a-j { ; Mark Townsend, 25 Apr 1999
init:
   z = #pixel
loop:
  z = z^2 + flip(real(@c))
  z = z^2 + flip(imag(@c))
bailout:
  |z| < @bailout
default:
  title = "990425a Julia" 
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam  
switch:
  type = "mt-990425a"
  bailout = bailout
}

CNewtonMset1 { ; Mark Townsend, 29 Apr 1999
;
; This is Paul Carlson's TbNewtMset1 stripped
; of the coloring method.
;
init:
  c = #pixel
  #z = sqrt((1 - c)/6)
loop:
  z2 = z * z
  z3 = z * z2
  z4 = z * z3
  Err = (z4 + (c - 1)*z2 - c) / (4 * z3 + 2*(c - 1) * z)
  z = z - Err
bailout:
  |Err| > @epsilon
default:
  title = "Newton Mset 1"
  maxiter = 3000
  periodicity = 0
  param epsilon
    caption = "Epsilon"
    default = 1e-6
  endparam  
switch:  
  type = "mt-newton4-julia"
  epsilon = epsilon
  c = #pixel
}

mt-martin { ; Mark Townsend, 7 May 1999
; 
; Formulas devised by Barry Martin. I found
; them in A. K. Dewdney's "The Armchair
; Universe". They're usually mapped in a
; different way (look at the Fractint
; "Martin" fractal type).
;
init:
  z = 0    
  p = @fn1(#pixel)
  float x = real(p) 
  float y = imag(p) 
  float a = @a
  float xx = 0
loop:
  xx = x
  x = y - sin(x)
  y = a - xx
  z = @fn2(x + flip(y))
bailout:
; All inside
  true
default:
  title = "Martin"
  maxiter = 20 
  magn = 0.1
  param a
    caption = "Parameter (a)"
    default = 3.14159
  endparam  
  func fn1
    caption = "Pre function"
    default = ident()
  endfunc  
  func fn2
    caption = "Post function"
    default = ident()
  endfunc  
}

mt-hopalong { ; Mark Townsend, 8 May 1999
; 
; Formulas devised by Barry Martin. I found
; them in A. K. Dewdney's "The Armchair 
; Universe."
;
init:
  z = 0    
  p = @fn1(#pixel)
  float x = real(p) 
  float y = imag(p) 
  float xx = 0
  int iter = 0
  float sgnx = 0
loop:
  iter = iter + 1
  if x < 0
    sgnx = -1
  else
    sgnx = 1
  endif    
  xx = x
  x = y - sgnx * (abs(@b * x - @c))^0.5
  y = @a - xx
  z = @fn2(x + flip(y))
bailout:
; All outside but only after a
; certain number of iterations
  iter < #maxiter
default:
  title = "Hopalong"
  maxiter = 20 
  magn = 0.5
  param a
    caption = "Parameter (a)"
    default = -3.14
  endparam  
  param b
    caption = "Parameter (b)"
    default = 0.3
  endparam  
  param c
    caption = "Parameter (c)"
    default = 0.3
  endparam  
  func fn1
    caption = "Pre function"
    default = ident()
  endfunc  
  func fn2
    caption = "Post function"
    default = ident()
  endfunc  
}

mt-bifunctional-m { ; Mark Townsend, 11 Jun 1999
init:
  z = 0 
  c = #pixel
  float x = 0
  float y = 0
loop:
  x = real(z)
  y = imag(z)
  z = @fn1(x) + flip(@fn2(y))
  z = z^@n + c
bailout:
  |z| < @bailout
default:
  title = "Bifunctional Mset"
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam
  func fn1
    caption = "Real function"
    default = sinh()
  endfunc  
  func fn2
    caption = "Imaginary function"
    default = ident()
  endfunc  
switch:
  type = "mt-bifunctional-j"
  bailout = bailout
  c = #pixel
  fn1 = fn1
  fn2 = fn2
  n = n
}

mt-bifunctional-j { ; Mark Townsend, 11 Jun 1999
init:
  z = #pixel
  float x = 0
  float y = 0
loop:
  float x = real(z)
  float y = imag(z)
  z = @fn1(x) + flip(@fn2(y))
  z = z^@n + @c
bailout:
  |z| < @bailout
default:
  title = "Bifunctional Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam   
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam 
  func fn1
    caption = "Real function"
    default = sinh()
  endfunc  
  func fn2
    caption = "Imaginary function"
    default = ident()
  endfunc  
switch:
  type = "mt-bifunctional-m"
  bailout = bailout
  fn1 = fn1
  fn2 = fn2
  n = n
}

mt-gen-celtic-j { ; Mark Townsend, 24 Jun 1999
;
; From a formula by Paul Carlson
;
init:
  z = #pixel
loop:
  z = z^@n 
  z = z - @fn1(real(z)) + @fn2(real(z)) - @c
bailout:
  |z| < @bailout
default:
  title = "Generalized Celtic Julia"
  param @c
    caption = "Julia seed"
    default = (0,0)
  endparam
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam
  func fn1
    caption = "First function"
    default = ident()
  endfunc
  func fn2
    caption = "Second function"
    default = abs()
  endfunc
switch:
  type = "mt-gen-celtic-m"
  bailout = bailout  
  n = n
  fn1 = fn1
  fn2 = fn2
}

mt-gen-celtic-m { ; Mark Townsend, 24 Jun 1999
;
; From a formula by Paul Carlson
;
init:
  c = #pixel
  z = 0
loop:
  z = z^@n
  z = z - @fn1(real(z)) + @fn2(real(z)) - c
bailout:
  |z| < @bailout
default:
  title = "Generalized Celtic Mset"
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam
  func fn1
    caption = "First function"
    default = ident()
  endfunc
  func fn2
    caption = "Second function"
    default = abs()
  endfunc  
switch:
  type = "mt-gen-celtic-j"
  c = #pixel
  n = n
  bailout = bailout 
  fn1 = fn1 
  fn2 = fn2
}

mt-990625a-m { ; Mark Townsend, 25 Jun 1999
init:
  z = 0
  c = #pixel
loop:
  z = z - z^@n + c
bailout:
  |z| < @bailout
default:
  title = "990625a Mset"
  param n
    caption = "Exponent"
    default = 4.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam  
switch: 
  type = "mt-990625a-j"  
  c = #pixel
  bailout = bailout
  n = n
}

mt-990625a-j { ; Mark Townsend, 25 Jun 1999
init:
  z = #pixel
loop:
  z = z - z^@n + @c
bailout:
  |z| < @bailout
default:
  title = "990625a Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param n
    caption = "Exponent"
    default = 4.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam  
switch:  
  type = "mt-990625a-m"  
  bailout = bailout
  n = n
}

mt-990625b-j { ; Mark Townsend, 25 Jun 1999
init:
  z = #pixel
loop:
  z = z^@n + @fn1(real(z) / imag(z)) - @c
bailout:
  |z| < @bailout
default:
  title = "990625b Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam  
  func fn1
    caption = "Function"
    default = ident()
  endfunc  
switch:
  type = "mt-990625b-m"
  bailout = bailout
  n = n
  fn1 = fn1
}

mt-990625b-m { ; Mark Townsend, 25 Jun 1999
init:
  z = #pixel
  c = #pixel
loop:
  z = z^@n + @fn1(real(z) / imag(z)) - c
bailout:
  |z| < @bailout
default:
  title = "990625b Mset"
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 100.0
  endparam  
  func fn1
    caption = "Function"
    default = ident()
  endfunc  
switch:
  type = "mt-990625b-j"
  bailout = bailout
  n = n
  c = #pixel
  fn1 = fn1
}

mt-ifs-dragon-j { ; Mark Townsend, 3 Jun 1999
;
; Escape time IFS fractal - this one is from
; Fractint.ifs. Modified from a Fractint formula
; created with IfsInv (program by Ramiro Perez).
;
init:
  a0 = 0.82407, b0 = 0.28148, c0 = -0.21235, d0 = 0.86420
  a1 = 0.08827, b1 = 0.52099, c1 = -0.46389, d1 = -0.37778
  al0 = 0.77193, k0 = 1.59554, l0 = 0.49085
  al1 = 0.20833, k1 = 4.51450, l1 = -1.07895
  float op0 = 0
  float op1 = 0
  z = #pixel
loop:
  x = real(z)
  y = imag(z)
  o0 = (d0 *x - b0 * y + k0) / al0 + flip((-c0 * x + a0 * y + l0) / al0)
  op0 = |o0 - @c|
  o1 = (d1 * x - b1 * y + k1) / al1 + flip((-c1 * x + a1 * y + l1) / al1)
  op1 = |o1 - @c|
  if (op0 <= op1)
    z = o0
  else
    z = o1
  endif
bailout:  
  |z| <= @bailout
default:
  title = "IFS Dragon Julia"
  center = (0,4)
  magn = 0.2
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
switch:
  type = "mt-ifs-dragon-m"
  bailout = bailout
}

mt-ifs-dragon-m { ; Mark Townsend, 3 Jun 1999 
;
; Escape time IFS fractal - this one is from
; Fractint.ifs. Modified from a Fractint formula
; created with IfsInv (program by Ramiro Perez).
;
init:
  a0 = 0.82407, b0 = 0.28148, c0 = -0.21235, d0 = 0.86420
  a1 = 0.08827, b1 = 0.52099, c1 = -0.46389, d1 = -0.37778
  al0 = 0.77193, k0 = 1.59554, l0 = 0.49085
  al1 = 0.20833, k1 = 4.51450, l1 = -1.07895
  z = #pixel
loop:
  x = real(z)
  y = imag(z)
  o0 = (d0 *x - b0 * y + k0) / al0 + flip((-c0 * x + a0 * y + l0) / al0)
  op0 = |o0 - #pixel|
  o1 = (d1 * x - b1 * y + k1) / al1 + flip((-c1 * x + a1 * y + l1) / al1)
  op1 = |o1 - #pixel|
  if (op0 <= op1)
    z = o0
  else
    z = o1
  endif
bailout:  
  |z| <= @bailout
default:
  title = "IFS Dragon Mset"
  center = (-2.5,3)
  magn = 0.2
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ifs-dragon-j"
  bailout = bailout
  c = #pixel    
}

mt-ifs-gills-j { ; Mark Townsend, 31 May 1999
init:
  a0 = -0.21053, b0 = -0.93117, c0 = -0.31579, d0 = 0.21862  
  a1 = 0.89474, b1 = -0.23482, d1=0.84615, al0 = -0.34008
  k0 = -6.00400, l0 = -2.53258, al1 = 0.75709, k1 = -0.82044
  l1 = -0.99777
  z = #pixel
loop:
  x = real(z)
  y = imag(z)
  o0 = (d0 * x - b0 * y + k0) / al0 + flip((-c0 * x + a0 * y + l0) / al0)
  op0 = |o0 - @c|
  o1 = (d1 * x - b1 * y + k1) / al1 + flip((a1 * y + l1) / al1)
  op1 = |o1 - @c|
  if (op0 <= op1)
    z = o0
  else
    z = o1
  endif
bailout:  
  |z| <= @bailout
default:
  title = "IFS Gills Julia"
  center = (-2,4)
  magn = 0.1
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ifs-03-m"  
  bailout = bailout
}

mt-ifs-gills-m { ; Mark Townsend, 31 May 1999
init:
  a0 = -0.21053, b0 = -0.93117, c0 = -0.31579, d0 = 0.21862
  a1 = 0.89474, b1 = -0.23482, d1 = 0.84615, al0 = -0.34008
  k0 = -6.00400, l0 = -2.53258, al1 = 0.75709, k1 = -0.82044
  l1=-0.99777
  z = #pixel
loop:
  x = real(z)
  y  = imag(z)
  o0 = (d0 * x - b0 * y + k0) / al0 + flip((-c0 * x + a0 * y + l0) / al0)
  op0 = |o0 - #pixel|
  o1 = (d1 * x - b1 * y + k1) / al1 + flip((a1 * y + l1) / al1)
  op1 = |o1 - #pixel|
  if (op0 <= op1)
    z = o0
  else
    z = o1
  endif
bailout:  
  |z| <= @bailout
default:
  title = "IFS Gills Mset"
  center = (-2,4)
  magn = 0.1
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ifs-03-j"  
  bailout = bailout
  c = #pixel
}

mt-ifs-leaf-j { ; Mark Townsend, 31 May 1999
;
; Escape time IFS fractal - designed with Fdesign
; (program by Doug Nelson) and modified from a
; Fractint formula created with IfsInv (program by
; Ramiro Perez).
; 
init:
  a0 = 0.74546, b0 = -0.36460, c0 = 0.40606, d0 = 0.51996
  a1 = -0.42424, b1 = 0.65969, c1 = -0.17576, d1 = -0.41768
  al0 = 0.53566, k0 = -0.89714, l0 = -2.47193
  al1 = 0.29314, k1 = 4.62115, l1 = 2.78720
  z = #pixel
loop:
  x = real(z)
  y = imag(z)
  o0 = (d0 * x - b0 * y + k0) / al0 + flip((-c0 * x + a0 * y + l0) / al0)
  op0 = |o0 - @c|
  o1 = (d1 * x - b1 * y + k1) / al1 + flip((-c1 * x + a1 * y + l1) / al1)
  op1 = |o1 - @c|
  if (op0 <= op1)
    z = o0
  else
    z = o1
  endif
bailout:  
  |z| <= @bailout
default:
  title = "IFS Leaf Julia"  
  center = (-0.5,5)
  magn = 0.2
  param @c
    caption = "Julia seed"
    default = (0,7)
  endparam  
  param @bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ifs-leaf-m"
  bailout = bailout
}

mt-ifs-leaf-m { ; Mark Townsend, 31 May 1999
;
; Escape time IFS fractal - designed with Fdesign
; (program by Doug Nelson) and modified from a
; Fractint formula created with IfsInv (program by
; Ramiro Perez).
; 
init:
  a0 = 0.74546, b0 = -0.36460, c0 = 0.40606, d0 = 0.51996
  a1 = -0.42424, b1 = 0.65969, c1 = -0.17576, d1 = -0.41768
  al0 = 0.53566, k0 = -0.89714, l0 = -2.47193
  al1 = 0.29314, k1 = 4.62115, l1 = 2.78720
  z = #pixel
loop:
  x = real(z)
  y = imag(z)
  o0 = (d0 * x - b0 * y + k0) / al0 + flip((-c0 * x + a0 * y + l0) / al0)
  op0 = |o0 - #pixel|
  o1 = (d1 * x - b1 * y + k1) / al1 + flip((-c1 * x + a1 * y + l1) / al1)
  op1 = |o1 - #pixel|
  if (op0 <= op1)
    z = o0
  else
    z = o1
  endif
bailout:  
  |z| <= @bailout
default:
  title = "IFS Leaf Mset"  
  center = (-0.5,5)
  magn = 0.2
  param @bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ifs-leaf-j"
  bailout = bailout
  c = #pixel
}

mt-vine { ; Mark Townsend, 26 Jun 1999
; 
; More variations of Pickover's Popcorn. I took
; the formulas from the sorce code of Rudy Rucker's
; program Vine.
;
init:
  z = 0    
  float x = real(#pixel) 
  float y = imag(#pixel) 
  float xx = 0
  float newx = 0
  float newy = 0
  int iter = 0
  int i = 0
loop:
  iter = iter + 1
  xx = x
  if @flavor == 0
    x = x - @h * sin(y + sin(@a * y ))
    y = y + @h * sin(xx + sin(@a * xx))
  elseif @flavor == 1
    x = x - @h * sin(y^@b + sin(@a * y))
    y = y + @h * sin(xx^@b + sin(@a * xx))
  elseif @flavor == 2
   x = x - @h * sin(y + sin(@a * (y + sin(@a * y))))
   y = y + @h * sin(xx + sin(@a * (xx + sin(@a * xx))))  
  else
    newx = y
    newy = x
    i = 0
    while i < @flavor
      i = i + 1
      newx = y + sin(@a * newx)
      newy = x + sin(@a * newy)
    endwhile
    x = x - @h * sin(newx)
    y = y + @h * sin(newy)
  endif
  z = x + flip(y)
bailout:
  iter < #maxiter
default:
  title = "Vine"
  maxiter = 20 
  magn = 0.15
  param flavor
    caption = "Flavor"
    default = 0
    min = 0
  endparam  
  param a
    caption = "Alpha"
    default = 2.7
  endparam  
  param b
    caption = "Beta"
    default = 2.0
    hint = "This parameter has an effect only with a \
           flavor of 2"
  endparam  
  param h
    caption = "Step size"
    default = 0.1
  endparam  
}

mt-sign-i-j { ; Mark Townsend, 26 Jun 1999
init:
  z = #pixel
loop:
  if real(z) < 0
    signx = -1
  else
    signx = 1
  endif
  z = z^2 + signx * @c
bailout:
  |z| < @bailout
default:
  title = "Sign I Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-sign-i-m"
  bailout = bailout
}

mt-sign-i-m { ; Mark Townsend, 26 Jun 1999
init:
  z = 0
loop:
  if real(z) < 0
    signx = -1
  else
    signx = 1
  endif
  z = z^2 + signx * #pixel
bailout:
  |z| < @bailout
default:
  title = "Sign I Mset"
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-sign-i-j"
  bailout = bailout
  c = #pixel
}

mt-sign-ii-j { ; Mark Townsend, 26 Jun 1999
init:
  z = #pixel
loop:
  if real(z) < 0
    signx = -1
  else
    signx = 1
  endif
  z = signx * z^2 + @c
bailout:
  |z| < @bailout
default:
  title = "Sign II Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-sign-ii-m"
  bailout = bailout
}

mt-sign-ii-m { ; Mark Townsend, 26 Jun 1999
init:
  z = 0
loop:
  if real(z) < 0
    signx = -1
  else
    signx = 1
  endif
  z = signx * z^2 + #pixel
bailout:
  |z| < @bailout
default:
  title = "Sign II Mset"
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-sign-ii-j"
  bailout = bailout
  c = #pixel
}

mt-biomorph-I-j { ; Mark Townsend, 13 Jul 1999
init:
  z = #pixel
loop:
  z = @fn1(z) + e^z + @c
bailout:
  |real(z)| < @bailout
default:
  title = "Biomorph I Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-biomorph-I-m"
  fn1 = fn1 
  bailout = bailout
}

mt-biomorph-I-m { ; Mark Townsend, 13 Jul 1999
init:
  z = 0
loop:
  z = @fn1(z) + e^z + #pixel
bailout:
  |real(z)| < @bailout
default:
  title = "Biomorph I Mset"
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-biomorph-I-j"
  fn1 = fn1 
  c = #pixel
  bailout = bailout
}

mt-biomorph-II-j { ; Mark Townsend, 13 Jul 1999
init:
  z = #pixel
loop:
  z = z^z + z^(@n) + @c
bailout:
  |real(z)| < @bailout
default:
  title = "Biomorph II Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param n
    caption = "Coefficient"
    default = 5.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 10.0
  endparam  
switch:
  type = "mt-biomorph-II-m"
  n = n
  bailout = bailout
}

mt-biomorph-II-m { ; Mark Townsend, 13 Jul 1999
init:
  z = 0
loop:
  z = z^z + z^(@n) + #pixel
bailout:
  |z| < @bailout 
default:
  title = "Biomorph II Mset"
  param n
    caption = "Coefficient"
    default = 6.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 10.0
  endparam  
switch:
  type = "mt-biomorph-II-j"
  c = #pixel
  n = n
  bailout = bailout
}

mt-dynamic { ; Mark Townsend, 6 Jul 1999
init:
  x = real(#pixel) 
  y = imag(#pixel)
  a = real(@a)
  b = imag(@a) 
  xx = 0
  int iter = 0
loop:
  iter = iter + 1
  xx = x
  x = x - a * @fn1(y)
  y = y - b * @fn2(xx)
  z = x + flip(y)
bailout:
  iter < #maxiter
default:
  title = "Dynamic"
  maxiter = 20 
  magn = 0.15
  param a
    caption = "Parameter"
    default = (1,1)
  endparam  
}

mt-ring-j { ; Mark Townsend, 11 Jul 1999
init:
  z = #pixel
  s = 0
loop:
  z = z^@n + @c + s
  s = |cabs(z)-@rad|
bailout:
  |z| < @bailout
default:
  title = "Ring Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param rad
    caption = "Radius"
    default = 0.5
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ring-m"
  rad = rad
  n = n
  bailout = bailout
}

mt-ring-m { ; Mark Townsend, 11 Jul 1999
init:
  z = 0
  s = 0
loop:
  z = z^@n + #pixel + s
  s = |cabs(z)-@rad|
bailout:
  |z| < @bailout
default:
  title = "Ring Mset"
  param n
    caption = "Exponent"
    default = 2.0
  endparam  
  param rad
    caption = "Radius"
    default = 0.5
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-ring-j"
  rad = rad
  n = n
  bailout = bailout
  c = #pixel  
}

mt-newton4-julia { ; Mark Townsend, 20 Jul 1999
;
; Julia version of Newton Mset 1, from the Paul Carlson 
; formula.
;
init:
  c = @c
  #z = #pixel
loop:
  z2 = z * z
  z3 = z * z2
  z4 = z * z3
  err = (z4 + (c - 1)*z2 - c) / (4 * z3 + 2*(c - 1) * z)
  z = z - err
bailout:
  |Err| > @epsilon
default:
  title = "Newton Julia"
  maxiter = 3000
  periodicity = 0
  param c 
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1e-6
  endparam  
switch:
  type = "CNewtonMset1"
  epsilon = epsilon
}

mt-990725a { ; Mark Townsend 25 Jul 1999
init:
  z = 0
  int iter = 0
loop:
  z = z^iter + iter * #pixel
  iter = iter + 1
bailout:
 |z| < @bailout
default:
  title = "990725a"
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
}

mt-modulate-j { ; Mark Townsend, 25 Jul 1999
init:
  z = #pixel
  int iter = 0
loop:
  iter = iter + 1
  z = z^2 + ((iter % @m)* @c)
bailout:
  |z| < @bailout
default:
  title = "Modulate Julia"
  param c
    caption = "Julia seed"
    default = (-0.7,0)
  endparam  
  param m
    caption = "Modulate"
    default = 3
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-modulate-m"  
  m = m
  bailout = bailout
}

mt-990801a-j { ; Mark Townsend, 1 Aug 1999
init:
  z = #pixel
loop:
  z = z^@e1 - z^@e2 + @c
bailout:
  |z| < @bailout 
default:
  title = "990801a Julia"
  param c
    caption = "Julia seed"
    default = (0,0)
  endparam  
  param e1
    caption = "First exponent"
    default = 2.0
  endparam  
  param e2
    caption = "Second exponent"
    default = -2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-990801a-m"
  e1 = e1
  e2 = e2
  bailout = bailout
}

mt-990801a-m { ; Mark Townsend, 1 Aug 1999
init:
  z = 0
  c = #pixel
loop:
  z = z^@e1 - z^@e2 + c
bailout:
  |z| < @bailout 
default:
  title = "990801a Mset"
  param e1
    caption = "First exponent"
    default = 2.0
  endparam  
  param e2
    caption = "Second exponent"
    default = -2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam 
switch:
  type = "mt-990801a-j"
  e1 = e1
  e2 = e2
  bailout = bailout
  c = #pixel
}

mt-modulate-m { ; Mark Townsend, 25 Jul 1999
init:
  z = 0
  int iter = 0
loop:
  iter = iter + 1
  z = z^2 + ((iter % @m)* #pixel)
bailout:
  |z| < @bailout
default:
  title = "Modulate Mset"
  param m
    caption = "Modulate"
    default = 3
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
switch:
  type = "mt-modulate-j"  
  m = m
  c = #pixel
  bailout = bailout
}

mt-990807a-m { ; Mark Townsend, Aug 7 1999
init:
  z = 0
  c = #pixel
loop:
  z = @fn1(c*z^@e1) + @fn2(z^@e2) + c
bailout:
  |z| < @bailout
default:
  title = "990807a Mset"
  param e1
    caption = "First exponent"
    default = 4.0
  endparam  
  param e2
    caption = "Second exponent"
    default = 2.0
  endparam  
  func fn1
    default = ident()
  endfunc  
  func fn2
    default = ident()
  endfunc  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
  func fn1
    caption = "First Function"
    default = ident()
  endfunc  
  func fn2
    caption = "Second Function"
    default = ident()
  endfunc  
switch:
  type = "mt-990807a-j"
  e1 = e1
  e2 = e2  
  fn1 = fn1
  fn2 = fn2
  bailout = bailout
  c = #pixel
}

mt-990807a-j { ; Mark Townsend, Aug 7 1999
init:
  z = #pixel
  c = @c
loop:
  z = @fn1(c*z^@e1) + @fn2(z^@e2) + c
bailout:
  |z| < @bailout
default:
  title = "990807a Julia"
  param c
    caption = "Julia seed"
    default = (-0.6,0.43)
  endparam  
  param e1
    caption = "First exponent"
    default = 4.0
  endparam  
  param e2
    caption = "Second exponent"
    default = 2.0
  endparam  
  func fn1
    caption = "First Function"
    default = ident()
  endfunc  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
  func fn2
    caption = "Second Function"
    default = ident()
  endfunc  
switch:
  type = "mt-990807a-m"
  e1 = e1
  e2 = e2  
  fn1 = fn1
  fn2 = fn2
  bailout = bailout
}

mt-990807b-m { ; Mark Townsend, Aug 7 1999
init:
  z = 0
  c = #pixel
loop:
  z = @fn1(c*z^@e1) + @fn2(-c*z^@e2) + c
bailout:
  |z| < @bailout
default:
  title = "990807b Mset"
  param e1
    caption = "First exponent"
    default = 3.0
  endparam  
  param e2
    caption = "Second exponent"
    default = 1.0
  endparam  
  func fn1
    default = ident()
  endfunc  
  func fn2
    default = ident()
  endfunc  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
  func fn1
    caption = "First Function"
    default = ident()
  endfunc  
  func fn2
    caption = "Second Function"
    default = ident()
  endfunc  
switch:
  type = "mt-990807b-j"
  e1 = e1
  e2 = e2  
  fn1 = fn1
  fn2 = fn2
  bailout = bailout
  c = #pixel
}

mt-990807b-j { ; Mark Townsend, Aug 7 1999
init:
  z = #pixel
  c = @c
loop:
  z = @fn1(c*z^@e1) + @fn2(-c*z^@e2) + c
bailout:
  |z| < @bailout
default:
  title = "990807b Julia"
  param c
    caption = "Julia seed"
    default = (1.2,0)
  endparam  
  param e1
    caption = "First exponent"
    default = 3.0
  endparam  
  param e2
    caption = "Second exponent"
    default = 1.0
  endparam  
  func fn1
    caption = "First Function"
    default = ident()
  endfunc  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam  
  func fn2
    caption = "Second Function"
    default = ident()
  endfunc  
switch:
  type = "mt-990807b-m"
  e1 = e1
  e2 = e2  
  fn1 = fn1
  fn2 = fn2
  bailout = bailout
}

mt-druid-m { ; Mark Townsend, 29 Oct 1999
;
; Switching between iterations of the Celtic Julia
; and the Standard Julia
;
init:
  c = #pixel
  z = 0
  int iter = @first
loop:
  if iter % 2 == 0
    z = z^@m + c
  else
    z = z^@n
    z = z - real(z) + abs(real(z)) + c
  endif
  iter = iter + 1
bailout:
  |z| < @bailout
default:
  title = "Druid Mset"
  center = (-0.5, 0)
  param @first
    caption = "First set"
    enum = "Mandelbrot" "Celtic" 
  endparam  
  param m
    caption = "Standard Exponent"
    default = 2.0
  endparam  
  param n
    caption = "Celtic Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam
switch:
  type = "mt-druid-j"
  c = #pixel
  n = n
  m = m
  bailout = bailout 
  first = first
}

mt-druid-j { ; Mark Townsend, 29 Oct 1999
;
; Switching between iterations of the Celtic Julia
; and the Standard Julia
;
init:
  z = #pixel
  int iter = @first
loop:
  if iter % 2 == 0
    z = z^@m + @c
  else
    z = z^@n
    z = z - real(z) + abs(real(z)) + @c
  endif
  iter = iter + 1
bailout:
  |z| < @bailout
default:
  title = "Druid Julia"
  param first
    caption = "First set"
    enum = "Julia" "Celtic"
  endparam  
  param @c
    caption = "Julia seed"
    default = (0,0)
  endparam
  param m
    caption = "Standard Exponent"
    default = 2.0
  endparam  
  param n
    caption = "Celtic Exponent"
    default = 2.0
  endparam  
  param bailout
    caption = "Bailout value"
    default = 1000.0
  endparam
switch:
  type = "mt-druid-m"
  bailout = bailout  
  n = n
  m = m
  first = first
}

mt-advanced-gnarl {
; Mark Townsend, 17 Nov 1999
;
; Last update 15 December 1999
;
init:
  z = 0  
  bool test = true
  float m = 590872612825179551336102196593.0 
  float a = 1322939
  float c = 7336919
  float ar1 = trunc(real(@a_seed))
  float ar2 = trunc(imag(@a_seed))
  float br1 = trunc(real(@b_seed))
  float br2 = trunc(imag(@b_seed))
  float gr1 = trunc(real(@g_seed))
  float gr2 = trunc(imag(@g_seed))
  float seqr = @seqseed
  float acoef1 = real(@a1)
  float bcoef1 = real(@b1)
  float gcoef1 = real(@g1)
  float acoef2 = real(@a2)
  float bcoef2 = real(@b2)
  float gcoef2 = real(@g2)
  float ra1 = imag(@a1)
  float rb1 = imag(@b1)
  float rg1 = imag(@g1)
  float ra2 = imag(@a2)
  float rb2 = imag(@b2)
  float rg2 = imag(@g2)
  int iter = 0
  float xx = 0
  float x = real(#pixel) 
  float y = imag(#pixel) 
loop:
; "Random" numbers.
  if imag(@a1) != 0
    ar1 = (a * ar1 + c) % m
    acoef1 = real(@a1) + (ar1 / m) * ra1
  endif
  if imag(@b1) != 0
    br1 = (a * br1 + c) % m
    bcoef1 = real(@b1) + (br1 / m) * rb1
  endif
  if imag(@g1) != 0
    gr1 = (a * gr1 + c) % m
    gcoef1 = real(@g1) + (gr1 / m) * rg1
  endif
  if imag(@a2) != 0
    ar2 = (a * ar2 + c) % m
    acoef2 = real(@a2) + (ar2 / m) * ra2
  endif
  if imag(@b2) != 0
    br2 = (a * br2 + c) % m
    bcoef2 = real(@b2) + (br2 / m) * rb2
  endif
  if imag(@g2) != 0
    gr2 = (a * gr2 + c) % m
    gcoef2 = real(@g2) + (gr2 / m) * rg2
  endif  
  if @seq == 0 
  ; Iteration 
    test = iter % real(@mv) < imag(@mv)
  elseif @seq == 1
  ; Random
    seqr = (a * seqr + c) % m
    if seqr % 2 == 0
      test = true
    else
      test = false
    endif
  elseif @seq == 2
  ; First formula
    test = true
  elseif @seq == 3
  ; Second formula
    test = false    
  endif  
  xx = x
  if test
    x = x - @h * real(@af1(acoef1 * y + \
        @bf1(bcoef1 * (y + @gf1(gcoef1 * y)))))
    y = y + @h * real(@af1(acoef1 * xx +\
        @bf1(bcoef1 * (xx + @gf1(gcoef1 * xx)))))
  else
    x = x - @h * real(@af2(acoef2 * y + \
        @bf2(bcoef2 * (y + @gf2(gcoef2 * y)))))
    y = y + @h * real(@af2(acoef2 * xx + \
        @bf2(bcoef2 * (xx + @gf2(gcoef2 * xx)))))
  endif    
  z = x + flip(y)
  iter = iter + 1
bailout:
  iter < #maxiter
default:
  title = "Advanced Gnarl"
  helpfile = "mt.hlp"
  helptopic = "Advanced Gnarl"
$IFDEF VER30
  precision = 12
$ENDIF
  maxiter = 20
  magn = 0.5 
  param seq
    caption = "Sequence"
    enum = "Modulation" "Random" "First formula" \
           "Second formula" 
  endparam  
  param mv
    caption = "Modulation"
    default = (10,5)
    min = (1,0)
  endparam  
  param a1
    caption = "A 1"
    default = (1.0,0)
  endparam  
  param b1
    caption = "B 1"
    default = (3.0,0)
  endparam  
  param g1
    caption = "G 1"
    default = (2.0,0)
  endparam  
  param a2
    caption = "A 2"
    default = (1.0,0)
  endparam  
  param b2
    caption = "B 2"
    default = (3.0,0)
  endparam  
  param g2
    caption = "G 2"
    default = (2.0,0)
  endparam  
  param h
    caption = "Step size"
    default = 0.1
  endparam  
  param a_seed
    caption = "A seed"
    default = (3539,9923)
  endparam  
  param b_seed
    caption = "B seed"
    default = (6373,8191)
  endparam  
  param g_seed
    caption = "G seed"
    default = (1277,4583)
  endparam  
  param seqseed
    caption = "Sequence seed"
    default = 2017
  endparam  
  func af1
    caption = "A 1"
    default = sin()
  endfunc  
  func bf1
    caption = "B 1"
    default = sin()
  endfunc  
  func gf1
    caption = "G 1"
    default = sin()
  endfunc  
  func af2
    caption = "A 2"
    default = cos()
  endfunc  
  func bf2
    caption = "B 2"
    default = cos()
  endfunc  
  func gf2
    caption = "G 2"
    default = cos()
  endfunc  
}

mt-latoocarfian-ii { ; Mark Townsend 18 Dec 1999
init:
  float xnew = 0
  float ynew = 0
  float a = real(@ab)
  float b = imag(@ab)
  float c = real(@cd)
  float d = imag(@cd)
  float x = real(#pixel)
  float y = imag(#pixel)
loop:
  xnew = sin(y * b) + c * sin(x * b)
  ynew = sin(x * a) + d * sin(y * a)
  x = xnew
  y = ynew
  #z = x + flip(y)
bailout:
  |#z - #pixel| > @epsilon
default:
  title = "Latööcarfian II"
  magn = 0.5
  param ab
    caption = "(a, b)"
    default = (-0.966918, 2.879879)
  endparam  
  param cd
    caption = "(c, d)"
    default = (0.765145, 0.744728)
  endparam  
  param epsilon
    caption = "Epsilon"
    default = 1.0
  endparam  
}

mt-latoocarfian-iii { ; Mark Townsend 19 Dec 1999
;
; The Latööcarfian equations from Pickover's book
; "Chaos in Wonderland". Use Damien M. Jones' "Sampling"
; for the Inside coloring method.
;
init:
  float xnew = 0
  float ynew = 0
  float a = real(@ab)
  float b = imag(@ab)
  float c = real(@cd)
  float d = imag(@cd)
  float x = 0.1
  float y = 0.1
loop:
  xnew = sin(y * b) + c * sin(x * b)
  ynew = sin(x * a) + d * sin(y * a)
  x = xnew
  y = ynew
  #z = x + flip(y)
bailout:
  true
default:
  title = "Latööcarfian III"
  param ab
    caption = "(a, b)"
    default = (-0.966918, 2.879879)
  endparam  
  param cd
    caption = "(c, d)"
    default = (0.765145, 0.744728)
  endparam  
}

mt-popcorn { ; Mark Townsend, 8 May 1999
; 
; This is another gift to us from Clifford 
; Pickover. I found the equation in A. K.
; Dewdney's "The Magic Machine." This formula
; is similar to Popcorn Julia, except that I
; don't impose a bailout condition.
;
; Last update on 30 December 1999
;
init:
  z = 0    
  p = @fn1(#pixel / @scale)
  float x = real(p) 
  float y = imag(p) 
  float xx = 0
  int iter = 0
loop:
  iter = iter + 1
  xx = x
  x = x - @h * sin(y + tan(3 * y))
  y = y - @h * sin(xx + tan(3 * xx))
  z = @fn2(x + flip(y))
bailout:
; All outside but only after a
; certain number of iterations
  iter < #maxiter
default:
  title = "Popcorn"
  maxiter = 20 
  magn = 0.5
  param h
    caption = "Step Size"
    default = 0.05
  endparam  
  param scale
    caption = "Scale"
    default = 1.0
  endparam  
  func fn1
    caption = "Pre function"
    default = ident()
  endfunc  
  func fn2
    caption = "Post function"
    default = ident()
  endfunc  
}

mt-gnarl { ; Mark Townsend, 15 Aug 1999
; 
; Yet more variations of Pickover's Popcorn.
;
; Last update on 31 December 1999
;
init:
  z = 0    
  p = #pixel / @scale
  float x = real(p) 
  float y = imag(p) 
  float xx = 0
  int iter = 0
loop:
  iter = iter + 1
  xx = x
  x = x - @h * real(@fn1(y + @fn2(@a * (y + @fn3(@b * y)))))
  y = y + @h * real(@fn1(xx + @fn2(@a * (xx + @fn3(@b * xx)))))
  z = x + flip(y)
bailout:
  iter < #maxiter
default:
  title = "Gnarl"
  maxiter = 20 
  magn = 0.15
  param a
    caption = "Alpha"
    default = 2.7
  endparam  
  param b
    caption = "Beta"
    default = 2.7
  endparam  
  param h
    caption = "Step size"
    default = 0.1
  endparam  
  param scale
    caption = "Scale"
    default = 1.0
  endparam  
  func fn1
    caption = "First Function"
    default = sin()
  endfunc  
  func fn2
    caption = "Second Function"
    default = sin()
  endfunc  
  func fn3
    caption = "Third Function"
    default = sin()
  endfunc  
}

mt-popcorn-mset { ; Mark Townsend, 30 January 2000
; 
; An "Mset" version of Pickover's Popcorn.
;
init:
  z = 0    
  float cr = real(#pixel) 
  float ci = imag(#pixel) 
  float x = real(@start)
  float y = imag(@start)
  float xx = 0
  int iter = 0
loop:
  iter = iter + 1
  xx = x
  x = x - @h * sin(y + tan(cr * y))
  y = y - @h * sin(xx + tan(ci * xx))
  z = @fn2(x + flip(y))
bailout:
; All outside but only after a
; certain number of iterations
  iter < #maxiter
default:
  title = "Popcorn Mset"
  maxiter = 20 
  magn = 0.25
  param start
    caption = "Starting Point"
    default = (1,0.75)
  endparam  
  param h
    caption = "Step Size"
    default = 0.05
  endparam  
}

mt-gnarl-mset { ; Mark Townsend, 30 January 1999
; 
; Mset version of Gnarl.
;
init:
  z = 0    
  float a = real(#pixel) 
  float b = imag(#pixel) 
  float x = real(@start)
  float y = imag(@start)
  float xx = 0
  int iter = 0
loop:
  iter = iter + 1
  xx = x
  x = x - @h * real(@fn1(y + @fn2(a * (y + @fn3(b * y)))))
  y = y + @h * real(@fn1(xx + @fn2(a * (xx + @fn3(b * xx)))))
  z = x + flip(y)
bailout:
  iter < #maxiter
default:
  title = "Gnarl Mset"
  maxiter = 20 
  magn = 0.25
  param start
    caption = "Starting Point"
    default = (2.7, 2.7)
  endparam  
  param h
    caption = "Step size"
    default = 0.1
  endparam  
  func fn1
    caption = "First Function"
    default = sin()
  endfunc  
  func fn2
    caption = "Second Function"
    default = sin()
  endfunc  
  func fn3
    caption = "Third Function"
    default = sin()
  endfunc  
}

mt-reverse { ; Mark Townsend, 7 May 2000
init:
  z = #pixel
  oldz = 0
loop:
  oldz = z
  z = sqrt(z - @c) 
bailout:
 |z - oldz| > @bailout
default:
  title = "Reverse"
  periodicity  = 0
  param c
     caption = "Julia Seed"
  endparam
  param bailout
    caption = "Bailout Value"
    default = 1e-6
  endparam       
switch:
  type = "mt-reverse"
  c = #pixel
  bailout = bailout
}

mt-zeepop-j { ; Mark Townsend, 7 May 2000
init:
  z = #pixel
  oldz = 0
loop:
  oldz = z
  z = z - @h * @fn1(z + @fn2(@c * z))
bailout:
  |z| < @bailout && |z - oldz| > @epsilon
default:
  title = "Zeepop Julia"
  periodicity  = 0
  param c
     caption = "Julia Seed"
     default = (3,0)
  endparam
  param h
    caption = "Step Size"
    default = (0.1,0)
  endparam       
  param bailout
    caption = "Bailout Value"
    default = 100.0
  endparam       
  param epsilon
    caption = "Epsilon"
    default = 1e-6
  endparam       
  func fn1
    caption = "Function #1"
    default = sin()
  endfunc  
  func fn2
    caption = "Function #1"
    default = tan()
  endfunc  
switch:
  type = "mt-zeepop-m"
  h = h
  fn1 = fn1
  fn2 = fn2
  bailout = bailout
  epsilon = epsilon
}

mt-zeepop-m { ; Mark Townsend, 7 May 2000
init:
  c = #pixel
  z = sqr(1 - #pixel)
  oldz = 0
loop:
  oldz = z
  z = z - @h * @fn1(z + @fn2(c * z))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Zeepop Mset"
  periodicity  = 0
  param h
    caption = "Step Size"
    default = (0.1,0)
  endparam       
  param bailout
    caption = "Bailout Value"
    default = 100
  endparam       
  param epsilon
    caption = "Epsilon"
    default = 1e-6
  endparam       
  func fn1
    caption = "Function #1"
    default = sin()
  endfunc  
  func fn2
    caption = "Function #1"
    default = tan()
  endfunc  
switch:
  type = "mt-zeepop-j"
  c = #pixel
  h = h
  fn1 = fn1
  fn2 = fn2
  bailout = bailout
  epsilon = epsilon
}

mt-barnsleyish-j { ; Mark Townsend, 12 May 2000
init:
  z = #pixel
loop:
  if real(z) >= @threshold
    z = @fn1(z + 1) * -@c
  else  
    z = @fn1(z - 1) * @c
  endif  
bailout:
  |z| < @bailout
default:
  title = "Barnsleyish Julia"
  param c
    caption = "Julia Seed"
    default = (0.8, 1.05)
  endparam  
  param threshold
    caption = "Threshold"
    default = 0.0
  endparam  
  param bailout
    caption = "Bailout Value"
    default = 4.0
  endparam  
  func fn1
    caption = "Function"
    default = cos()
  endfunc  
switch:
  type = "mt-barnsleyish-m"  
  bailout = bailout
  threshold = threshold
  fn1 = fn1
}

mt-barnsleyish-m { ; Mark Townsend, 12 May 2000
init:
  z = 0
loop:
  if real(z) >= @threshold
    z = @fn1(z + 1) * -#pixel
  else  
    z = @fn1(z - 1) * #pixel
  endif  
bailout:
  |z| < @bailout
default:
  title = "Barnsleyish Mset"
  param threshold
    caption = "Threshold"
    default = 0.0
  endparam  
  param bailout
    caption = "Bailout Value"
    default = 4.0
  endparam  
    func fn1
    caption = "Function"
    default = cos()
  endfunc  
switch:
  type = "mt-barnsleyish-j"  
  bailout = bailout
  threshold = threshold
  c = #pixel
  fn1 = fn1
}

mt-rudys-cubic {
; Mark Townsend, 17 May 2001
; From a Fractint formula that I wrote in January 1999
init:  
  #z = #pixel
  k3 = @p2 * 3
loop:
  #z = @fn1(#z^3) - k3 * #z + @c
bailout:    
  |#z|< @bailout
default:
  title = "Rudy's Cubic (Julia)"
  param c
    caption = "Julia Seed"
    default = (0.3, 0.6)
  endparam  
  param p2
    caption = "Coefficient"
    default = (0.36, 0.18)
  endparam  
  param bailout
    caption = "Bailout Value"
    default = 1e20
  endparam  
  func  fn1
    caption = "Function"
    default = ident()
  endfunc  
switch:  
  type = "mt-rudys-cubic-m"
  bailot = balout
  p2 = #pixel
  fn1 = fn1
  }

mt-rudys-cubic-m {
; Mark Townsend, 17 May 2001
; From a Fractint formula that I wrote in January 1999
init:  
  #z = #pixel
  k3 = @p2 * 3
loop:
  #z = @fn1(#z^3) - k3 * #z + #pixel
bailout:    
  |#z|< @bailout
default:
  title = "Rudy's Cubic (Mandelbrot)"
  param p2
    caption = "Coefficient"
  endparam  
  param bailout
    caption = "Bailout Value"
    default = 100.0
  endparam  
  func fn1
    caption = "Function"
    default = ident()
  endfunc   
switch:
  type = "mt-rudys-cubic"
  bailot = balout
  p2 = p2
  c = #pixel    
  fn1 = fn1
}

mt-jewel-newton-j {
 ; Mark Townsend, 30 April 2008
init:
  z = #pixel
  float h = 0.0001
  oldz = 0
loop:
  oldz = z
  zh = z + h
  z = z - (z^@p_power + z + @c) / (1 / h * ((zh^@p_power + zh + @c) - (z^@p_power + z + @c)))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Jewel Newton Julia"
  maxiter = 1000
  periodicity = 0
  param c
    caption = "Seed"
    default = (-0.83, 1.44)
   endparam
  param p_power
    caption = "Power"
    default = 4
    min = 3
  endparam
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam
switch:
  type = "mt-jewel-newton-m"
  p_power = p_power
  epsilon = epsilon
}

mt-jewel-newton-m {
 ; Mark Townsend, 30 April 2008
init:
  z = 0
  c = #pixel
  float h = 0.0001
  oldz = 0
loop:
  oldz = z
  zh = z + h
  z = z - (z^@p_power + z + c) / (1 / h * ((zh^@p_power + zh + c) - (z^@p_power + z + c)))
bailout:
  |z - oldz| > @epsilon
default:
  title = "Jewel Newton"
  magn = 0.7
  maxiter = 1000
  periodicity = 0
  param p_power
    caption = "Power"
    default = 4
    min = 3
  endparam
  param epsilon
    caption = "Epsilon"
    default = 1e-6
    hint = "This is the bailout value."
  endparam
switch:
  type = "mt-jewel-newton-j"
  c = #pixel
  p_power = p_power
  epsilon = epsilon
}

mt-dynamical-systems { ; Mark Townsend, September 2018
init:
  import "mt.ulb"
  MT_DynamicalFunction XFunc = new @XFuncParam
  MT_DynamicalFunction YFunc = new @YFuncParam
  z = 0
  float xx = real(#pixel)
  float yy = imag(#pixel)
  float xnew = 0
  float ynew = 0
  int iter = 0
  float n = 0 ; dummy for noise param
loop:
  iter = iter + 1
  if @dual_functions
    xnew = xx - @h * XFunc.f(yy, xx, n)
    ynew = yy + @h * YFunc.f(xx, yy, n)
  else
    xnew = xx - @h * XFunc.f(yy, xx, n)
    ynew = yy + @h * XFunc.f(xx, yy, n)
  endif
  xx = xnew
  yy = ynew
  z = xx + flip(yy)
bailout:
  iter < #maxiter
default:
  title = "Dynamical Systems"
  maxiter = 20
  magn = 0.5
  param h
    caption = "Step Size"
    default = 0.05
    min = 0.00001
  endparam
  bool param dual_functions
    caption = "Dual Functions"
    default = false
  endparam
  MT_DynamicalFunction param XFuncParam
    caption = "X Function"
  endparam
  MT_DynamicalFunction param YFuncParam
    caption = "Y Function"
    visible = @dual_functions
  endparam
}