Implement reciprocal Fibonacci constant

[fredrik-johansson]

A nearly useless mathematical constant, but I couldn't not implement it, just to have as a benchmark, when I realized that the preposterous series by Bill Gosper leads to a quasilinear-time algorithm (did no one else notice this before?).

In terms of speed, we have, for example, pi < reciprocal_fibonacci < log2:

~/src/flint$ build/examples/pi 10000000 -constant reciprocal_fibonacci -threads 8
precision = 33219285 bits... cpu/wall(s): 8.159 1.468
virt/peak/res/peak(MB): 799.89 799.89 489.29 489.29
[3.35988566624317755317{...9999959 digits...}54369788228597414996 +/- 2.55e-10000000]

~/src/flint$ build/examples/pi 10000000 -constant pi -threads 8
precision = 33219285 bits... cpu/wall(s): 3.809 0.644
virt/peak/res/peak(MB): 687.07 687.07 252.21 252.21
[3.14159265358979323846{...9999959 digits...}63171948173534895590 +/- 3.20e-10000000]

~/src/flint$ build/examples/pi 10000000 -constant log2 -threads 8
precision = 33219285 bits... cpu/wall(s): 14.056 2.429
virt/peak/res/peak(MB): 806.46 937.12 475.55 475.55
[0.69314718055994530941{...9999960 digits...}80135652602465385883 +/- 3.19e-10000001]

Edit: 1 billion digits in 12 minutes; I need to drop down to 2 threads or I run out of memory:

~/src/flint$ build/examples/pi 1000000005 -constant reciprocal_fibonacci -threads 2
precision = 3321928116 bits... cpu/wall(s): 1009.31 718.785
virt/peak/res/peak(MB): 16534.32 18638.34 15483.07 18098.75
[3.35988566624317755317{...999999964 digits...}02927932123820488401 +/- 2.91e-1000000005]

[mezzarobba]

I realized that the preposterous series by Bill Gosper leads to a quasilinear-time algorithm

I wonder if this would generalize to a wider class of series using the ideas in https://epubs.siam.org/doi/abs/10.1137/1.9781611976496.14 (this is just wild speculation because of a vague formal analogy; I didn't try to understand what you are doing). Edit: probably not, I think I just imagined a link that doesn't really exist...

[fredrik-johansson]

Yeah, it's really interesting. Some more comments here: https://fredrikj.net/blog/2023/11/the-reciprocal-fibonacci-constant/