Re: centers on the octave [2/3, 4/3] but the error is actually more balanced across [1/sqrt(2), sqrt(2)] For the computation of the natural logarithm using bin

Re: centers on the octave [2/3, 4/3] but the error is actually more balanced across [1/sqrt(2), sqrt(2)] For the computation of the natural logarithm using binary floating-point arithmetic there is no practical difference, i.e. the overall accuracy that can be achieved is basically the same. Here is a worked example based on minimax core approximations extracted from my personal code collection (design and implementation are my own; no guarantee of fitness for any particular purpose). /* natural logarithm */ float my_logf (float a) { float m, r, s, t, i, f; int32_t e; #if LOG_VARIANT == 1 // max ulp err = 0.86280 const float cutoff_f = 0.707106781f; #elif LOG_VARIANT == 3 // max ulp err = 0.85089 const float cutoff_f = 0.666666667f; #endif // LOG_VARIANT if ((a > 0.0f) && (a <= 0x1.fffffep+127f)) { // 3.40282347e+38f m = frexpf (a, &e); if (m < cutoff_f) { m = m + m; e = e - 1; } i = (float)e; f = m - 1.0f; s = f * f; #if LOG_VARIANT == 1 /* Compute log1p(m) for m in [sqrt(0.5)-1, sqrt(2.0)-1] */ r = -0x1.384000p-4f; // -0.076232910 t = 0x1.084000p-3f; // 0.129028320 r = fmaf (r, s, -0x1.0f218cp-3f); // -0.132388204 t = fmaf (t, s, 0x1.226b04p-3f); // 0.141805679 r = fmaf (r, s, -0x1.5427a4p-3f); // -0.166091233 t = fmaf (t, s, 0x1.99a35ap-3f); // 0.200018600 r = fmaf (r, s, -0x1.000416p-2f); // -0.250015587 r = fmaf (t, f, r); r = fmaf (r, f, 0x1.55555cp-2f); // 0.333333433 r = fmaf (r, f, -0x1.fffffap-2f); // -0.499999911 #elif LOG_VARIANT == 3 /* Compute log1p(f) for f in [-1/3, 1/3] */ r = -0x1.0ae000p-3f; // -0.130310059 t = 0x1.208000p-3f; // 0.140869141 r = fmaf (r, s, -0x1.f1988ap-4f); // -0.121483363 t = fmaf (t, s, 0x1.1e5740p-3f); // 0.139814854 r = fmaf (r, s, -0x1.55b36ep-3f); // -0.166846141 t = fmaf (t, s, 0x1.99d8b2p-3f); // 0.200120345 r = fmaf (r, s, -0x1.fffe02p-3f); // -0.249996200 r = fmaf (t, f, r); r = fmaf (r, f, 0x1.5554fap-2f); // 0.333331972 r = fmaf (r, f, -0x1.000000p-1f); // -0.500000000 #endif // LOG_VARIANT r = fmaf (r, s, f); r = fmaf (i, 0x1.62e430p-01f, r); // 0.693147182 // log(2) } else { r = a + a; // silence NaNs if necessary if (a < 0.0f) r = 0.0f/0.0f; // QNaN INDEFINITE if (a == 0.0f) r = -INFINITY; // -INF } return r; }