Re: [tim-janik/beast] BSE: bsemathsignal: add approximations: Bse::fast_log2 and Bse::fast_exp2 (#124)

[Tim Janik <notifications github com>]

Relative exp2(x) approximation errors in interval [-1:1]:

That's a miniscule range to approximate, given the valid range of this function. Here are the errors of
bse_approx3_exp2() at mostly integer steps with some omissions for brevity:

   -2.0, +0.00000000000000000
   -1.5, -0.00016133294256113
   -1.0, +0.00000000000000000
   -0.5, -0.00032266588512226
   +0.0, +0.00000000000000000
   +0.5, -0.00112351661969536
   +1.0, +0.00000000000000000
   +1.5, -0.00224703323939071
   +2.0, +0.00000000000000000
   +2.5, -0.00449406647878142
   +7.0, +0.00000000000000000
  +11.0, +0.00000000000000000
  +16.0, +0.00000000000000000
  +32.0, +0.00000000000000000
  +40.0, +0.00000000000000000
  +48.0, +0.00000000000000000
  +54.0, +0.00000000000000000
  +64.0, +0.00000000000000000
 +127.0, +0.00000000000000000

Now the same for fast_exp2<2>():

   -2.0, +0.00011078548906696
   -1.5, -0.00060979588254370
   -1.0, +0.00022157097813391
   -0.5, -0.00121959176508741
   +0.0, +0.00044314195626782
   +0.5, +0.00243918353017435
   +1.0, +0.00088628391253565
   +1.5, +0.00487836706034869
   +2.0, +0.00177256782507129
   +2.5, +0.00975673412069738
   +7.0, +0.05672217040228134
  +11.0, +0.90755472643650137
  +16.0, +29.04175124596804380
  +32.0, +1903280.20965576171875000
  +40.0, +487239733.671875
  +48.0, +124733371820
  +54.0, +7982935796480
  +64.0, +8174526255595520
 +127.0, +75396696880394894980022595129180160

I.e. a pure Remez approximation works allmost as well as approxX_exp2 while using one less addmul, but only within -1:+1. I've discarded that approach when I wrote approxX_exp2, because outside of that range the error becomes gigantic. Above, I'm comparing approx3_exp2 with fast_exp2<2>, to show that with just one extra addmul, those giant errors can be avoided (which even fast_exp2<9> cannot accomplish) if the integer part and the fractional part are approximated separately.
Considering that fast_exp2<>() are not even minimizing the error at exactly -1, +1, or at 0 at the very least, I'd at most add them as rough approximations to be used only when the input range is known to be -1..+1 (which is not the case for general audio signals in BSE). For that it might be named fast_narrow_exp2<>() or similar, but I think fast_exp2<>() is better used if we rename the current bse_approxX_exp2 functions.

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