Better code for two variable fractal interpolation functions

Question

I am trying to write code for 2-variable fractal interpolation functions using two iterated function systems and two starting functions (both $y(x)=x$) which creates a sequence of piecewise defined functions using iterated functions systems(IFS) that converges to an element of the single superfractal. There is randomness involved so each run generates a different attractor.

I am not familiar with most of Mathematica's built in features, so the best I could do involved too many if-statements. The code is very long due to the fact that at each stage of the iteration process (and for each function in the chosen IFS) one of the two previous stage's functions is used. The code plots the first four stages of the two sequences of functions. There must be an easier way and any help would be much appreciated.

{a1, b1} = {.25, .4}; {a2, b2} = {.28, .22}; {a3, 
  b3} = {.3, .26}; {a4, b4} = {.32, .1}; {a5, b5} = {.7, .8}; {a6, 
  b6} = {.73, .6}; {a7, b7} = {.78, .8}; {a8, b8} = {.8, .6};
{c1, d1} = {.20, .45}; {c2, d2} = {.25, .3}; {c3, 
  d3} = {.28, .45}; {c4, d4} = {.32, .2}; {c5, d5} = {.6, .75}; {c6, 
  d6} = {.7, .65}; {c7, d7} = {.8, .7}; {c8, d8} = {.83, .6};
y0[x_] := x; z0[x_] := x;

If[RandomInteger[] == 0,
If[RandomInteger[] == 0,
  f1[x_] := b1*y0[x/a1],
  f1[x_] := b1*z0[x/a1]];
 If[RandomInteger[] == 0,
  g1[x_] := (b2 - b1)*y0[(x - a1)/(a2 - a1)] + b1,
  g1[x_] := (b2 - b1)*z0[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h1[x_] := (b3 - b2)*y0[(x - a2)/(a3 - a2)] + b2,
  h1[x_] := (b3 - b2)*z0[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i1[x_] := (b4 - b3)*y0[(x - a3)/(a4 - a3)] + b3,
  i1[x_] := (b4 - b3)*z0[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j1[x_] := (b5 - b4)*y0[(x - a4)/(a5 - a4)] + b4,
  j1[x_] := (b5 - b4)*z0[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k1[x_] := (b6 - b5)*y0[(x - a5)/(a6 - a5)] + b5,
  k1[x_] := (b6 - b5)*z0[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l1[x_] := (b7 - b6)*y0[(x - a6)/(a7 - a6)] + b6,
  l1[x_] := (b7 - b6)*z0[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m1[x_] := (b8 - b7)*y0[(x - a7)/(a8 - a7)] + b7,
  m1[x_] := (b8 - b7)*z0[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n1[x_] := (1 - b8)*y0[(x - a8)/(1 - a8)] + b8,
  n1[x_] := (1 - b8)*z0[(x - a8)/(1 - a8)] + b8];
  y1[x_] := 
  If[0 <= x < a1, f1[x], 
   If[a1 <= x < a2, g1[x], 
    If[a2 <= x < a3, h1[x], 
     If[a3 <= x < a4, i1[x], 
      If[a4 <= x < a5, j1[x], 
       If[a5 <= x < a6, k1[x], 
        If[a6 <= x < a7, l1[x], If[a7 <= x < a8, m1[x], n1[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o1[x_] := d1*y0[1/c1*x],
   o1[x_] := d1*z0[1/c1*x]];
 If[RandomInteger[] == 0,
   p1[x_] := (d2 - d1)*y0[(x - c1)/(c2 - c1)] + d1,
   p1[x_] := (d2 - d1)*z0[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q1[x_] := (d3 - d2)*y0[(x - c2)/(c3 - c2)] + d2,
   q1[x_] := (d3 - d2)*z0[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r1[x_] := (d4 - d3)*y0[(x - c3)/(c4 - c3)] + d3,
   r1[x_] := (d4 - d3)*z0[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s1[x_] := (d5 - d4)*y0[(x - c4)/(c5 - c4)] + d4,
   s1[x_] := (d5 - d4)*z0[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t1[x_] := (d6 - d5)*y0[(x - c5)/(c6 - c5)] + d5,
   t1[x_] := (d6 - d5)*z0[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u1[x_] := (d7 - d6)*y0[(x - c6)/(c7 - c6)] + d6,
   u1[x_] := (d7 - d6)*z0[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v1[x_] := (d8 - d7)*y0[(x - c7)/(c8 - c7)] + d7,
   v1[x_] := (d8 - d7)*z0[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w1[x_] := (1 - d8)*y0[(x - c8)/(1 - c8)] + d8,
   w1[x_] := (1 - d8)*z0[(x - c8)/(1 - c8)] + d8];
  y1[x_] := 
  If[0 <= x < c1, o1[x], 
   If[c1 <= x < c2, p1[x], 
    If[c2 <= x < c3, q1[x], 
     If[c3 <= x < c4, r1[x], 
      If[c4 <= x < c5, s1[x], 
       If[c5 <= x < c6, t1[x], 
        If[c6 <= x < c7, u1[x], If[c7 <= x < c8, v1[x], w1[x]]]]]]]]]]

If[RandomInteger[] == 0,
 If[RandomInteger[] == 0,
  f1[x_] := b1*y0[x/a1],
  f1[x_] := b1*z0[x/a1]];
 If[RandomInteger[] == 0,
  g1[x_] := (b2 - b1)*y0[(x - a1)/(a2 - a1)] + b1,
  g1[x_] := (b2 - b1)*z0[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h1[x_] := (b3 - b2)*y0[(x - a2)/(a3 - a2)] + b2,
  h1[x_] := (b3 - b2)*z0[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i1[x_] := (b4 - b3)*y0[(x - a3)/(a4 - a3)] + b3,
  i1[x_] := (b4 - b3)*z0[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j1[x_] := (b5 - b4)*y0[(x - a4)/(a5 - a4)] + b4,
  j1[x_] := (b5 - b4)*z0[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k1[x_] := (b6 - b5)*y0[(x - a5)/(a6 - a5)] + b5,
  k1[x_] := (b6 - b5)*z0[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l1[x_] := (b7 - b6)*y0[(x - a6)/(a7 - a6)] + b6,
  l1[x_] := (b7 - b6)*z0[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m1[x_] := (b8 - b7)*y0[(x - a7)/(a8 - a7)] + b7,
  m1[x_] := (b8 - b7)*z0[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n1[x_] := (1 - b8)*y0[(x - a8)/(1 - a8)] + b8,
  n1[x_] := (1 - b8)*z0[(x - a8)/(1 - a8)] + b8];
  z1[x_] := 
  If[0 <= x < a1, f1[x], 
   If[a1 <= x < a2, g1[x], 
    If[a2 <= x < a3, h1[x], 
     If[a3 <= x < a4, i1[x], 
      If[a4 <= x < a5, j1[x], 
       If[a5 <= x < a6, k1[x], 
        If[a6 <= x < a7, l1[x], If[a7 <= x < a8, m1[x], n1[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o1[x_] := d1*y0[1/c1*x],
   o1[x_] := d1*z0[1/c1*x]];
 If[RandomInteger[] == 0,
   p1[x_] := (d2 - d1)*y0[(x - c1)/(c2 - c1)] + d1,
   p1[x_] := (d2 - d1)*z0[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q1[x_] := (d3 - d2)*y0[(x - c2)/(c3 - c2)] + d2,
   q1[x_] := (d3 - d2)*z0[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r1[x_] := (d4 - d3)*y0[(x - c3)/(c4 - c3)] + d3,
   r1[x_] := (d4 - d3)*z0[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s1[x_] := (d5 - d4)*y0[(x - c4)/(c5 - c4)] + d4,
   s1[x_] := (d5 - d4)*z0[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t1[x_] := (d6 - d5)*y0[(x - c5)/(c6 - c5)] + d5,
   t1[x_] := (d6 - d5)*z0[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u1[x_] := (d7 - d6)*y0[(x - c6)/(c7 - c6)] + d6,
   u1[x_] := (d7 - d6)*z0[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v1[x_] := (d8 - d7)*y0[(x - c7)/(c8 - c7)] + d7,
   v1[x_] := (d8 - d7)*z0[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w1[x_] := (1 - d8)*y0[(x - c8)/(1 - c8)] + d8,
   w1[x_] := (1 - d8)*z0[(x - c8)/(1 - c8)] + d8];
  z1[x_] := 
  If[0 <= x < c1, o1[x], 
   If[c1 <= x < c2, p1[x], 
    If[c2 <= x < c3, q1[x], 
     If[c3 <= x < c4, r1[x], 
      If[c4 <= x < c5, s1[x], 
       If[c5 <= x < c6, t1[x], 
        If[c6 <= x < c7, u1[x], If[c7 <= x < c8, v1[x], w1[x]]]]]]]]]]

If[RandomInteger[] == 0,
 If[RandomInteger[] == 0,
  f2[x_] := b1*y1[x/a1],
  f2[x_] := b1*z1[x/a1]];
 If[RandomInteger[] == 0,
  g2[x_] := (b2 - b1)*y1[(x - a1)/(a2 - a1)] + b1,
  g2[x_] := (b2 - b1)*z1[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h2[x_] := (b3 - b2)*y1[(x - a2)/(a3 - a2)] + b2,
  h2[x_] := (b3 - b2)*z1[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i2[x_] := (b4 - b3)*y1[(x - a3)/(a4 - a3)] + b3,
  i2[x_] := (b4 - b3)*z1[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j2[x_] := (b5 - b4)*y1[(x - a4)/(a5 - a4)] + b4,
  j2[x_] := (b5 - b4)*z1[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k2[x_] := (b6 - b5)*y1[(x - a5)/(a6 - a5)] + b5,
  k2[x_] := (b6 - b5)*z1[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l2[x_] := (b7 - b6)*y1[(x - a6)/(a7 - a6)] + b6,
  l2[x_] := (b7 - b6)*z1[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m2[x_] := (b8 - b7)*y1[(x - a7)/(a8 - a7)] + b7,
  m2[x_] := (b8 - b7)*z1[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n2[x_] := (1 - b8)*y1[(x - a8)/(1 - a8)] + b8,
  n2[x_] := (1 - b8)*z1[(x - a8)/(1 - a8)] + b8];
  y2[x_] := 
  If[0 <= x < a1, f2[x], 
   If[a1 <= x < a2, g2[x], 
    If[a2 <= x < a3, h2[x], 
     If[a3 <= x < a4, i2[x], 
      If[a4 <= x < a5, j2[x], 
       If[a5 <= x < a6, k2[x], 
        If[a6 <= x < a7, l2[x], If[a7 <= x < a8, m2[x], n2[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o2[x_] := d1*y1[1/c1*x],
   o2[x_] := d1*z1[1/c1*x]];
 If[RandomInteger[] == 0,
   p2[x_] := (d2 - d1)*y1[(x - c1)/(c2 - c1)] + d1,
   p2[x_] := (d2 - d1)*z1[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q2[x_] := (d3 - d2)*y1[(x - c2)/(c3 - c2)] + d2,
   q2[x_] := (d3 - d2)*z1[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r2[x_] := (d4 - d3)*y1[(x - c3)/(c4 - c3)] + d3,
   r2[x_] := (d4 - d3)*z1[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s2[x_] := (d5 - d4)*y1[(x - c4)/(c5 - c4)] + d4,
   s2[x_] := (d5 - d4)*z1[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t2[x_] := (d6 - d5)*y1[(x - c5)/(c6 - c5)] + d5,
   t2[x_] := (d6 - d5)*z1[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u2[x_] := (d7 - d6)*y1[(x - c6)/(c7 - c6)] + d6,
   u2[x_] := (d7 - d6)*z1[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v2[x_] := (d8 - d7)*y1[(x - c7)/(c8 - c7)] + d7,
   v2[x_] := (d8 - d7)*z1[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w2[x_] := (1 - d8)*y1[(x - c8)/(1 - c8)] + d8,
   w2[x_] := (1 - d8)*z1[(x - c8)/(1 - c8)] + d8];
  y2[x_] := 
  If[0 <= x < c1, o2[x], 
   If[c1 <= x < c2, p2[x], 
    If[c2 <= x < c3, q2[x], 
     If[c3 <= x < c4, r2[x], 
      If[c4 <= x < c5, s2[x], 
       If[c5 <= x < c6, t2[x], 
        If[c6 <= x < c7, u2[x], If[c7 <= x < c8, v2[x], w2[x]]]]]]]]]]

If[RandomInteger[] == 0,
 If[RandomInteger[] == 0,
  f2[x_] := b1*y1[x/a1],
  f2[x_] := b1*z1[x/a1]];
 If[RandomInteger[] == 0,
  g2[x_] := (b2 - b1)*y1[(x - a1)/(a2 - a1)] + b1,
  g2[x_] := (b2 - b1)*z1[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h2[x_] := (b3 - b2)*y1[(x - a2)/(a3 - a2)] + b2,
  h2[x_] := (b3 - b2)*z1[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i2[x_] := (b4 - b3)*y1[(x - a3)/(a4 - a3)] + b3,
  i2[x_] := (b4 - b3)*z1[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j2[x_] := (b5 - b4)*y1[(x - a4)/(a5 - a4)] + b4,
  j2[x_] := (b5 - b4)*z1[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k2[x_] := (b6 - b5)*y1[(x - a5)/(a6 - a5)] + b5,
  k2[x_] := (b6 - b5)*z1[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l2[x_] := (b7 - b6)*y1[(x - a6)/(a7 - a6)] + b6,
  l2[x_] := (b7 - b6)*z1[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m2[x_] := (b8 - b7)*y1[(x - a7)/(a8 - a7)] + b7,
  m2[x_] := (b8 - b7)*z1[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n2[x_] := (1 - b8)*y1[(x - a8)/(1 - a8)] + b8,
  n2[x_] := (1 - b8)*z1[(x - a8)/(1 - a8)] + b8];
  z2[x_] := 
  If[0 <= x < a1, f2[x], 
   If[a1 <= x < a2, g2[x], 
    If[a2 <= x < a3, h2[x], 
     If[a3 <= x < a4, i2[x], 
      If[a4 <= x < a5, j2[x], 
       If[a5 <= x < a6, k2[x], 
        If[a6 <= x < a7, l2[x], If[a7 <= x < a8, m2[x], n2[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o2[x_] := d1*y1[1/c1*x],
   o2[x_] := d1*z1[1/c1*x]];
 If[RandomInteger[] == 0,
   p2[x_] := (d2 - d1)*y1[(x - c1)/(c2 - c1)] + d1,
   p2[x_] := (d2 - d1)*z1[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q2[x_] := (d3 - d2)*y1[(x - c2)/(c3 - c2)] + d2,
   q2[x_] := (d3 - d2)*z1[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r2[x_] := (d4 - d3)*y1[(x - c3)/(c4 - c3)] + d3,
   r2[x_] := (d4 - d3)*z1[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s2[x_] := (d5 - d4)*y1[(x - c4)/(c5 - c4)] + d4,
   s2[x_] := (d5 - d4)*z1[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t2[x_] := (d6 - d5)*y1[(x - c5)/(c6 - c5)] + d5,
   t2[x_] := (d6 - d5)*z1[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u2[x_] := (d7 - d6)*y1[(x - c6)/(c7 - c6)] + d6,
   u2[x_] := (d7 - d6)*z1[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v2[x_] := (d8 - d7)*y1[(x - c7)/(c8 - c7)] + d7,
   v2[x_] := (d8 - d7)*z1[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w2[x_] := (1 - d8)*y1[(x - c8)/(1 - c8)] + d8,
   w2[x_] := (1 - d8)*z1[(x - c8)/(1 - c8)] + d8];
  z2[x_] := 
  If[0 <= x < c1, o2[x], 
   If[c1 <= x < c2, p2[x], 
    If[c2 <= x < c3, q2[x], 
     If[c3 <= x < c4, r2[x], 
      If[c4 <= x < c5, s2[x], 
       If[c5 <= x < c6, t2[x], 
        If[c6 <= x < c7, u2[x], If[c7 <= x < c8, v2[x], w2[x]]]]]]]]]]

If[RandomInteger[] == 0,
 If[RandomInteger[] == 0,
  f3[x_] := b1*y2[x/a1],
  f3[x_] := b1*z2[x/a1]];
 If[RandomInteger[] == 0,
  g3[x_] := (b2 - b1)*y2[(x - a1)/(a2 - a1)] + b1,
  g3[x_] := (b2 - b1)*z2[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h3[x_] := (b3 - b2)*y2[(x - a2)/(a3 - a2)] + b2,
  h3[x_] := (b3 - b2)*z2[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i3[x_] := (b4 - b3)*y2[(x - a3)/(a4 - a3)] + b3,
  i3[x_] := (b4 - b3)*z2[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j3[x_] := (b5 - b4)*y2[(x - a4)/(a5 - a4)] + b4,
  j3[x_] := (b5 - b4)*z2[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k3[x_] := (b6 - b5)*y2[(x - a5)/(a6 - a5)] + b5,
  k3[x_] := (b6 - b5)*z2[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l3[x_] := (b7 - b6)*y2[(x - a6)/(a7 - a6)] + b6,
  l3[x_] := (b7 - b6)*z2[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m3[x_] := (b8 - b7)*y2[(x - a7)/(a8 - a7)] + b7,
  m3[x_] := (b8 - b7)*z2[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n3[x_] := (1 - b8)*y2[(x - a8)/(1 - a8)] + b8,
  n3[x_] := (1 - b8)*z2[(x - a8)/(1 - a8)] + b8];
  y3[x_] := 
  If[0 <= x < a1, f3[x], 
   If[a1 <= x < a2, g3[x], 
    If[a2 <= x < a3, h3[x], 
     If[a3 <= x < a4, i3[x], 
      If[a4 <= x < a5, j3[x], 
       If[a5 <= x < a6, k3[x], 
        If[a6 <= x < a7, l3[x], If[a7 <= x < a8, m3[x], n3[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o3[x_] := d1*y2[1/c1*x],
   o3[x_] := d1*z2[1/c1*x]];
 If[RandomInteger[] == 0,
   p3[x_] := (d2 - d1)*y2[(x - c1)/(c2 - c1)] + d1,
   p3[x_] := (d2 - d1)*z2[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q3[x_] := (d3 - d2)*y2[(x - c2)/(c3 - c2)] + d2,
   q3[x_] := (d3 - d2)*z2[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r3[x_] := (d4 - d3)*y2[(x - c3)/(c4 - c3)] + d3,
   r3[x_] := (d4 - d3)*z2[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s3[x_] := (d5 - d4)*y2[(x - c4)/(c5 - c4)] + d4,
   s3[x_] := (d5 - d4)*z2[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t3[x_] := (d6 - d5)*y2[(x - c5)/(c6 - c5)] + d5,
   t3[x_] := (d6 - d5)*z2[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u3[x_] := (d7 - d6)*y2[(x - c6)/(c7 - c6)] + d6,
   u3[x_] := (d7 - d6)*z2[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v3[x_] := (d8 - d7)*y2[(x - c7)/(c8 - c7)] + d7,
   v3[x_] := (d8 - d7)*z2[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w3[x_] := (1 - d8)*y2[(x - c8)/(1 - c8)] + d8,
   w3[x_] := (1 - d8)*z2[(x - c8)/(1 - c8)] + d8];
  y3[x_] := 
  If[0 <= x < c1, o3[x], 
   If[c1 <= x < c2, p3[x], 
    If[c2 <= x < c3, q3[x], 
     If[c3 <= x < c4, r3[x], 
      If[c4 <= x < c5, s3[x], 
       If[c5 <= x < c6, t3[x], 
        If[c6 <= x < c7, u3[x], If[c7 <= x < c8, v3[x], w3[x]]]]]]]]]]

If[RandomInteger[] == 0,
 If[RandomInteger[] == 0,
  f3[x_] := b1*y2[x/a1],
  f3[x_] := b1*z2[x/a1]];
 If[RandomInteger[] == 0,
  g3[x_] := (b2 - b1)*y2[(x - a1)/(a2 - a1)] + b1,
  g3[x_] := (b2 - b1)*z2[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h3[x_] := (b3 - b2)*y2[(x - a2)/(a3 - a2)] + b2,
  h3[x_] := (b3 - b2)*z2[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i3[x_] := (b4 - b3)*y2[(x - a3)/(a4 - a3)] + b3,
  i3[x_] := (b4 - b3)*z2[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j3[x_] := (b5 - b4)*y2[(x - a4)/(a5 - a4)] + b4,
  j3[x_] := (b5 - b4)*z2[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k3[x_] := (b6 - b5)*y2[(x - a5)/(a6 - a5)] + b5,
  k3[x_] := (b6 - b5)*z2[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l3[x_] := (b7 - b6)*y2[(x - a6)/(a7 - a6)] + b6,
  l3[x_] := (b7 - b6)*z2[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m3[x_] := (b8 - b7)*y2[(x - a7)/(a8 - a7)] + b7,
  m3[x_] := (b8 - b7)*z2[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n3[x_] := (1 - b8)*y2[(x - a8)/(1 - a8)] + b8,
  n3[x_] := (1 - b8)*z2[(x - a8)/(1 - a8)] + b8];
  z3[x_] := 
  If[0 <= x < a1, f3[x], 
   If[a1 <= x < a2, g3[x], 
    If[a2 <= x < a3, h3[x], 
     If[a3 <= x < a4, i3[x], 
      If[a4 <= x < a5, j3[x], 
       If[a5 <= x < a6, k3[x], 
        If[a6 <= x < a7, l3[x], If[a7 <= x < a8, m3[x], n3[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o3[x_] := d1*y2[1/c1*x],
   o3[x_] := d1*z2[1/c1*x]];
 If[RandomInteger[] == 0,
   p3[x_] := (d2 - d1)*y2[(x - c1)/(c2 - c1)] + d1,
   p3[x_] := (d2 - d1)*z2[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q3[x_] := (d3 - d2)*y2[(x - c2)/(c3 - c2)] + d2,
   q3[x_] := (d3 - d2)*z2[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r3[x_] := (d4 - d3)*y2[(x - c3)/(c4 - c3)] + d3,
   r3[x_] := (d4 - d3)*z2[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s3[x_] := (d5 - d4)*y2[(x - c4)/(c5 - c4)] + d4,
   s3[x_] := (d5 - d4)*z2[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t3[x_] := (d6 - d5)*y2[(x - c5)/(c6 - c5)] + d5,
   t3[x_] := (d6 - d5)*z2[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u3[x_] := (d7 - d6)*y2[(x - c6)/(c7 - c6)] + d6,
   u3[x_] := (d7 - d6)*z2[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v3[x_] := (d8 - d7)*y2[(x - c7)/(c8 - c7)] + d7,
   v3[x_] := (d8 - d7)*z2[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w3[x_] := (1 - d8)*y2[(x - c8)/(1 - c8)] + d8,
   w3[x_] := (1 - d8)*z2[(x - c8)/(1 - c8)] + d8];
  z3[x_] := 
  If[0 <= x < c1, o3[x], 
   If[c1 <= x < c2, p3[x], 
    If[c2 <= x < c3, q3[x], 
     If[c3 <= x < c4, r3[x], 
      If[c4 <= x < c5, s3[x], 
       If[c5 <= x < c6, t3[x], 
        If[c6 <= x < c7, u3[x], If[c7 <= x < c8, v3[x], w3[x]]]]]]]]]]





If[RandomInteger[] == 0,
 If[RandomInteger[] == 0,
  f4[x_] := b1*y3[x/a1],
  f4[x_] := b1*z3[x/a1]];
 If[RandomInteger[] == 0,
  g4[x_] := (b2 - b1)*y3[(x - a1)/(a2 - a1)] + b1,
  g4[x_] := (b2 - b1)*z3[(x - a1)/(a2 - a1)] + b1];
 If[RandomInteger[] == 0,
  h4[x_] := (b3 - b2)*y3[(x - a2)/(a3 - a2)] + b2,
  h4[x_] := (b3 - b2)*z3[(x - a2)/(a3 - a2)] + b2];
 If[RandomInteger[] == 0,
  i4[x_] := (b4 - b3)*y3[(x - a3)/(a4 - a3)] + b3,
  i4[x_] := (b4 - b3)*z3[(x - a3)/(a4 - a3)] + b3];
 If[RandomInteger[] == 0,
  j4[x_] := (b5 - b4)*y3[(x - a4)/(a5 - a4)] + b4,
  j4[x_] := (b5 - b4)*z3[(x - a4)/(a5 - a4)] + b4];
 If[RandomInteger[] == 0,
  k4[x_] := (b6 - b5)*y3[(x - a5)/(a6 - a5)] + b5,
  k4[x_] := (b6 - b5)*z3[(x - a5)/(a6 - a5)] + b5];
 If[RandomInteger[] == 0,
  l4[x_] := (b7 - b6)*y3[(x - a6)/(a7 - a6)] + b6,
  l4[x_] := (b7 - b6)*z3[(x - a6)/(a7 - a6)] + b6];
 If[RandomInteger[] == 0,
  m4[x_] := (b8 - b7)*y3[(x - a7)/(a8 - a7)] + b7,
  m4[x_] := (b8 - b7)*z3[(x - a7)/(a8 - a7)] + b7];
 If[RandomInteger[] == 0,
  n4[x_] := (1 - b8)*y3[(x - a8)/(1 - a8)] + b8,
  n4[x_] := (1 - b8)*z3[(x - a8)/(1 - a8)] + b8];
  y4[x_] := 
  If[0 <= x < a1, f4[x], 
   If[a1 <= x < a2, g4[x], 
    If[a2 <= x < a3, h4[x], 
     If[a3 <= x < a4, i4[x], 
      If[a4 <= x < a5, j4[x], 
       If[a5 <= x < a6, k4[x], 
        If[a6 <= x < a7, l4[x], If[a7 <= x < a8, m4[x], n4[x]]]]]]]]],

 If[RandomInteger[] == 0,
   o4[x_] := d1*y3[1/c1*x],
   o4[x_] := d1*z3[1/c1*x]];
 If[RandomInteger[] == 0,
   p4[x_] := (d2 - d1)*y3[(x - c1)/(c2 - c1)] + d1,
   p4[x_] := (d2 - d1)*z3[(x - c1)/(c2 - c1)] + d1];
 If[RandomInteger[] == 0,
   q4[x_] := (d3 - d2)*y3[(x - c2)/(c3 - c2)] + d2,
   q4[x_] := (d3 - d2)*z3[(x - c2)/(c3 - c2)] + d2];
 If[RandomInteger[] == 0,
   r4[x_] := (d4 - d3)*y3[(x - c3)/(c4 - c3)] + d3,
   r4[x_] := (d4 - d3)*z3[(x - c3)/(c4 - c3)] + d3];
 If[RandomInteger[] == 0,
   s4[x_] := (d5 - d4)*y3[(x - c4)/(c5 - c4)] + d4,
   s4[x_] := (d5 - d4)*z3[(x - c4)/(c5 - c4)] + d4];
 If[RandomInteger[] == 0,
   t4[x_] := (d6 - d5)*y3[(x - c5)/(c6 - c5)] + d5,
   t4[x_] := (d6 - d5)*z3[(x - c5)/(c6 - c5)] + d5];
 If[RandomInteger[] == 0,
   u4[x_] := (d7 - d6)*y3[(x - c6)/(c7 - c6)] + d6,
   u4[x_] := (d7 - d6)*z3[(x - c6)/(c7 - c6)] + d6];
 If[RandomInteger[] == 0,
   v4[x_] := (d8 - d7)*y3[(x - c7)/(c8 - c7)] + d7,
   v4[x_] := (d8 - d7)*z3[(x - c7)/(c8 - c7)] + d7];
 If[RandomInteger[] == 0,
   w4[x_] := (1 - d8)*y3[(x - c8)/(1 - c8)] + d8,
   w4[x_] := (1 - d8)*z3[(x - c8)/(1 - c8)] + d8];
  y4[x_] := 
  If[0 <= x < c1, o4[x], 
   If[c1 <= x < c2, p4[x], 
    If[c2 <= x < c3, q4[x], 
     If[c3 <= x < c4, r4[x], 
      If[c4 <= x < c5, s4[x], 
       If[c5 <= x < c6, t4[x], 
        If[c6 <= x < c7, u4[x], If[c7 <= x < c8, v4[x], w4[x]]]]]]]]]]

If[RandomInteger[] == 0,
  If[RandomInteger[] == 0,
   f4[x_] := b1*y3[x/a1],
   f4[x_] := b1*z3[x/a1]];
  If[RandomInteger[] == 0,
   g4[x_] := (b2 - b1)*y3[(x - a1)/(a2 - a1)] + b1,
   g4[x_] := (b2 - b1)*z3[(x - a1)/(a2 - a1)] + b1];
  If[RandomInteger[] == 0,
   h4[x_] := (b3 - b2)*y3[(x - a2)/(a3 - a2)] + b2,
   h4[x_] := (b3 - b2)*z3[(x - a2)/(a3 - a2)] + b2];
  If[RandomInteger[] == 0,
   i4[x_] := (b4 - b3)*y3[(x - a3)/(a4 - a3)] + b3,
   i4[x_] := (b4 - b3)*z3[(x - a3)/(a4 - a3)] + b3];
  If[RandomInteger[] == 0,
   j4[x_] := (b5 - b4)*y3[(x - a4)/(a5 - a4)] + b4,
   j4[x_] := (b5 - b4)*z3[(x - a4)/(a5 - a4)] + b4];
  If[RandomInteger[] == 0,
   k4[x_] := (b6 - b5)*y3[(x - a5)/(a6 - a5)] + b5,
   k4[x_] := (b6 - b5)*z3[(x - a5)/(a6 - a5)] + b5];
  If[RandomInteger[] == 0,
   l4[x_] := (b7 - b6)*y3[(x - a6)/(a7 - a6)] + b6,
   l4[x_] := (b7 - b6)*z3[(x - a6)/(a7 - a6)] + b6];
  If[RandomInteger[] == 0,
   m4[x_] := (b8 - b7)*y3[(x - a7)/(a8 - a7)] + b7,
   m4[x_] := (b8 - b7)*z3[(x - a7)/(a8 - a7)] + b7];
  If[RandomInteger[] == 0,
   n4[x_] := (1 - b8)*y3[(x - a8)/(1 - a8)] + b8,
   n4[x_] := (1 - b8)*z3[(x - a8)/(1 - a8)] + b8];
   z4[x_] := 
   If[0 <= x < a1, f4[x], 
    If[a1 <= x < a2, g4[x], 
     If[a2 <= x < a3, h4[x], 
      If[a3 <= x < a4, i4[x], 
       If[a4 <= x < a5, j4[x], 
        If[a5 <= x < a6, k4[x], 
         If[a6 <= x < a7, l4[x], If[a7 <= x < a8, m4[x], n4[x]]]]]]]]],

  If[RandomInteger[] == 0,
    o4[x_] := d1*y3[1/c1*x],
    o4[x_] := d1*z3[1/c1*x]];
  If[RandomInteger[] == 0,
    p4[x_] := (d2 - d1)*y3[(x - c1)/(c2 - c1)] + d1,
    p4[x_] := (d2 - d1)*z3[(x - c1)/(c2 - c1)] + d1];
  If[RandomInteger[] == 0,
    q4[x_] := (d3 - d2)*y3[(x - c2)/(c3 - c2)] + d2,
    q4[x_] := (d3 - d2)*z3[(x - c2)/(c3 - c2)] + d2];
  If[RandomInteger[] == 0,
    r4[x_] := (d4 - d3)*y3[(x - c3)/(c4 - c3)] + d3,
    r4[x_] := (d4 - d3)*z3[(x - c3)/(c4 - c3)] + d3];
  If[RandomInteger[] == 0,
    s4[x_] := (d5 - d4)*y3[(x - c4)/(c5 - c4)] + d4,
    s4[x_] := (d5 - d4)*z3[(x - c4)/(c5 - c4)] + d4];
  If[RandomInteger[] == 0,
    t4[x_] := (d6 - d5)*y3[(x - c5)/(c6 - c5)] + d5,
    t4[x_] := (d6 - d5)*z3[(x - c5)/(c6 - c5)] + d5];
  If[RandomInteger[] == 0,
    u4[x_] := (d7 - d6)*y3[(x - c6)/(c7 - c6)] + d6,
    u4[x_] := (d7 - d6)*z3[(x - c6)/(c7 - c6)] + d6];
  If[RandomInteger[] == 0,
    v4[x_] := (d8 - d7)*y3[(x - c7)/(c8 - c7)] + d7,
    v4[x_] := (d8 - d7)*z3[(x - c7)/(c8 - c7)] + d7];
  If[RandomInteger[] == 0,
    w4[x_] := (1 - d8)*y3[(x - c8)/(1 - c8)] + d8,
    w4[x_] := (1 - d8)*z3[(x - c8)/(1 - c8)] + d8];
   z4[x_] := 
   If[0 <= x < c1, o4[x], 
    If[c1 <= x < c2, p4[x], 
     If[c2 <= x < c3, q4[x], 
      If[c3 <= x < c4, r4[x], 
       If[c4 <= x < c5, s4[x], 
        If[c5 <= x < c6, t4[x], 
         If[c6 <= x < c7, u4[x], 
          If[c7 <= x < c8, v4[x], w4[x]]]]]]]]]];

GraphicsRow[{Show[Plot[y1[x], {x, 0, 1}], 
   PlotRange -> {{0, 1}, {0, 1}}], 
  Show[Plot[z1[x], {x, 0, 1}], PlotRange -> {{0, 1}, {0, 1}}]}, 
 ImageSize -> {Large}]
GraphicsRow[{Show[Plot[y2[x], {x, 0, 1}], 
   PlotRange -> {{0, 1}, {0, 1}}], 
  Show[Plot[z2[x], {x, 0, 1}], PlotRange -> {{0, 1}, {0, 1}}]}, 
 ImageSize -> {Large}]
GraphicsRow[{Show[Plot[y3[x], {x, 0, 1}], 
   PlotRange -> {{0, 1}, {0, 1}}], 
  Show[Plot[z3[x], {x, 0, 1}], PlotRange -> {{0, 1}, {0, 1}}]}, 
 ImageSize -> {Large}]
GraphicsRow[{Show[Plot[y4[x], {x, 0, 1}], 
   PlotRange -> {{0, 1}, {0, 1}}], 
  Show[Plot[z4[x], {x, 0, 1}], PlotRange -> {{0
      , 1}, {0, 1}}]}, ImageSize -> {Large}]

Please try to post a minimal working example showing your troubles. Going through all that code is too tedious. — Dr. belisarius, May 30 '14 at 05:54

score 2 · Answer 1 · edited Apr 13 '17 at 12:55

I notice some striking similarities between this question and the following two:

Presumably, the user is the same, which conceals information about your mathematical and programming background - information that's useful to potential answers. Nonetheless, the question is interesting, so here's a go.

As I understand it, you're simultaneously generating two functions $y(x)$ and $z(x)$ on $[0,1]$. The $n^{\text{th}}$ approximations $y_n(x)$ and $z_n(x)$ are piecewise linear functions starting with $y_0(x)=z_0(x)=x$. You have two partitions of $[0,1]$, namely $$0<a_1<a_2<\cdots<a_8<1$$ and $$0<c_1<c_2<\cdots<c_8<1.$$ Each of $y_n(x)$ and $z_n(x)$ will be constructed by first randomly choosing one of those partitions (one for $y$ and one for $z$) and then placing scaled copies of either $y_{n-1}$ or $z_{n-1}$ (also chosen randomly and independently for each sub-interval) over each of the nine sub-intervals in the partition.

The main problem with your code, is that you're repeating tremendous amounts of code, rather than encapsulating repetitive stuff into functions. For example, you define y0, y1, y2, y3, and y4 using the exact same code with suitable replacements. Why not define a function and then recursively define y[n]?

Here's one approach:

(* Your as and cs *)
(* One partition will be randomly chosen for each step *)
partitions = {
   Partition[{0.0, 0.25, 0.28, 0.3, 0.32, 0.7, 0.73, 0.78, 0.8, 1.0}, 2, 1],
   Partition[{0.0, 0.2, 0.25, 0.28, 0.32, 0.6, 0.7, 0.8, 0.83, 1.0}, 2, 1]
};
(* Your bs and ds *)
compressions = {
   Partition[{0, 0.4, 0.22, 0.26, 0.1, 0.8, 0.6, 0.8, 0.6, 1}, 2, 1],
   Partition[{0, 0.45, 0.3, 0.45, 0.2, 0.75, 0.65, 0.7, 0.6, 1}, 2, 1]
};
transform[f_, x_, {a1_, a2_}, {b1_, b2_}] := 
  (b2 - b1)*(f /. x -> (x - a1)/(a2 - a1)) + b1;
next[{y_, z_}, x_] := With[{
    part = RandomChoice[partitions],
    bs = RandomChoice[compressions]},
   Piecewise[Table[
     {transform[RandomChoice[{y, z}], x, part[[k]], bs[[k]]], 
      part[[k, 1]] <= x <= part[[k, 2]]},
     {k, 1, Length[partitions[[1]]]}]]];

Note that next accepts two expressions of x, rather than functions. It then returns a single Piecewise defined expression. The recursion can be implemented as follows:

Clear[y, z];
y[0] = z[0] = x;
y[n_] := y[n] = next[{y[n - 1], z[n - 1]}, x];
z[n_] := z[n] = next[{y[n - 1], z[n - 1]}, x];

We now graph it.

GraphicsGrid[Table[Plot[#, {x, 0, 1}] & /@ {y[n], z[n]}, {n, 1, 4}]]

An alternative approach is to use functional iteration.

step[{y_, z_}, x_] := {next[{y, z}, x], next[{y, z}, x]};
iterated = NestList[step[#, x] &, {x, x}, 4];
GraphicsGrid[Map[Plot[#, {x, 0, 1}] &, Rest[iterated], {2}]]

In either case, the graphs look like so:

enter image description here

Better code for two variable fractal interpolation functions

1 Answers1