log2.c

#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

static uint32_t mulhi(uint32_t a, uint32_t b) {
  return (uint32_t)(((uint64_t)a * (uint64_t)b) >> 32);
}

// x in (0, 1). computes an approximation of -log2(x)
float approx_log2(uint32_t x) {
  uint32_t k;
  x |= 1;
  k = 1 + __builtin_clz(x);
  x <<= k;
  x = mulhi(mulhi(0x55555555, x), -x) + x;
  x = (((k-1) << 24) | ((-x) >> 8));
  x = x < 29350 ? 0 : x - 29350;

  return x * 5.960464477539063e-08f;
}

int main() {
  float maxerr = 0;
  float toterr = 0;
  uint32_t i = 0, maxbad=0;
  do {
    float g = (i|1)*2.3283064365386962890625e-10;
    float err = -logf(g)/logf(2) - approx_log2(i);
    toterr += err;
    if (fabs(err) > fabs(maxerr)) maxerr = err, maxbad = i;
    i++;
  } while (i);
  printf("max err: %.10f [%08x]\navg err: %.10f\n", fabs(maxerr), maxbad, toterr/(double)(1ull << 32));
}

output.txt

max err: 0.0079612732 [0000269e]
avg err: -0.0000610352