1// polynomial for approximating log2(1+x) 2// 3// Copyright (c) 2019, Arm Limited. 4// SPDX-License-Identifier: MIT 5 6deg = 11; // poly degree 7// |log2(1+x)| > 0x1p-4 outside the interval 8a = -0x1.5b51p-5; 9b = 0x1.6ab2p-5; 10 11ln2 = evaluate(log(2),0); 12invln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits 13invln2lo = double(1/ln2 - invln2hi); 14 15// find log2(1+x)/x polynomial with minimal relative error 16// (minimal relative error polynomial for log2(1+x) is the same * x) 17deg = deg-1; // because of /x 18 19// f = log(1+x)/x; using taylor series 20f = 0; 21for i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 22f = f/ln2; 23 24// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 25approx = proc(poly,d) { 26 return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 27}; 28 29// first coeff is fixed, iteratively find optimal double prec coeffs 30poly = invln2hi + invln2lo; 31for i from 1 to deg do { 32 p = roundcoefficients(approx(poly,i), [|D ...|]); 33 poly = poly + x^i*coeff(p,0); 34}; 35 36display = hexadecimal; 37print("invln2hi:", invln2hi); 38print("invln2lo:", invln2lo); 39print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 40print("in [",a,b,"]"); 41print("coeffs:"); 42for i from 0 to deg do coeff(poly,i); 43