Fourth attempt

We thus compute with signed-digit, arbitrary large binary expansions, processing digits from left to right using a lazy language. As discussed above, computations are performed by demand. For example, if one wants 5 correct digits of y=h(g(x)), one may need

1.

49 digits of the intermediate value g(x),

2.

7 digits of the input x.

Six correct decimal digits of f⁶⁰(x₀) can be computed in a fraction of a second. It is effective up to f²⁵⁰(x₀) in my PC, with a not very careful implementation in the Haskell functional programming language, using an interpreter rather than a compiler. The following is the output produced by a Haskell implementation. The first column displays six correct decimal digits. Internally, the program first computes 9#9 correct signed binary digits, which are then converted to decimal. For comparison, I have also produced the results with floating-point computations, using the two different, but mathematically equivalent, expressions f₁(x)=4x(1-x) and f₂(x)=1-(2x-1)² of the logistic map. The floating-point numbers are computed in simple precision and displayed with six digits too. Notice that the floating-point answers differ from those produced by the C implementation, although both languages use the same floating-point standard--I guess that this is due to ``optimizations'' implemented by the C compiler based on the field properties of real numbers.

n       exact           Float 1         Float 2
-----------------------------------------------------------------
0       0.671875        0.671875        0.671875        
1       0.881836        0.881836        0.881836        
2       0.416805        0.416805        0.416805        
3       0.972315        0.972315        0.972315        
4       0.107676        0.107676        0.107676        
5       0.384327        0.384327        0.384327        
6       0.946479        0.946479        0.946479        
7       0.202625        0.202625        0.202625        
8       0.646273        0.646273        0.646273        
9       0.914416        0.914417        0.914417        
10      0.313037        0.313035        0.313033        
11      0.860179        0.860177        0.860174        
12      0.481084        0.481091        0.481099        
13      0.998569        0.998570        0.998571        
14      0.005716        0.005712        0.005707        
15      0.022735        0.022720        0.022700        
16      0.088875        0.088815        0.088740        
17      0.323907        0.323710        0.323462        
18      0.875965        0.875688        0.875338        
19      0.434601        0.435436        0.436486        
20      0.982892        0.983326        0.983864        
21      0.067261        0.065585        0.063503        
22      0.250949        0.245135        0.237881        
23      0.751894        0.740176        0.725175        
24      0.746197        0.769262        0.797184        
25      0.757549        0.709991        0.646726        
26      0.734675        0.823615        0.913886        
27      0.779711        0.581093        0.314793        
28      0.687047        0.973695        0.862793        
29      0.860054        0.102451        0.473525        
30      0.481445        0.367818        0.997196        
31      0.998623        0.930112        0.011183        
32      0.005501        0.260015        0.044233        
33      0.021884        0.769629        0.169108        
34      0.085620        0.709201        0.562043        
35      0.313159        0.824940        0.984603        
36      0.860362        0.577656        0.060641        
37      0.480558        0.975878        0.227856        
38      0.998488        0.094160        0.703751        
39      0.006038        0.341177        0.833942        
40      0.024009        0.899100        0.553930        
41      0.093730        0.362875        0.988366        
42      0.339781        0.924787        0.045993        
43      0.897320        0.278223        0.175511        
44      0.368548        0.803259        0.578828        
45      0.930881        0.632135        0.975145        
46      0.257366        0.930161        0.096950        
47      0.764515        0.259846        0.350203        
48      0.720128        0.769305        0.910244        
49      0.806175        0.709899        0.326801        
50      0.625028        0.823770        0.880008        
51      0.937472        0.580692        0.422376        
52      0.234472        0.973955        0.975898        
53      0.717980        0.101467        0.094084        
54      0.809939        0.364686        0.340930        
55      0.615752        0.926760        0.898787        
56      0.946406        0.271503        0.363877        
57      0.202886        0.791157        0.925882        
58      0.646894        0.660911        0.274497        
59      0.913689        0.896431        0.796593        
60      0.315445        0.371371        0.648129        
61      0.863758        0.933818        0.912231        
62      0.470720        0.247207        0.320263        
63      0.996571        0.744383        0.870779