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1// polynomial for approximating log2(1+x)2//3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.4// See https://llvm.org/LICENSE.txt for license information.5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception6 7deg = 11; // poly degree8// |log2(1+x)| > 0x1p-4 outside the interval9a = -0x1.5b51p-5;10b = 0x1.6ab2p-5;11 12ln2 = evaluate(log(2),0);13invln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits14invln2lo = double(1/ln2 - invln2hi);15 16// find log2(1+x)/x polynomial with minimal relative error17// (minimal relative error polynomial for log2(1+x) is the same * x)18deg = deg-1; // because of /x19 20// f = log(1+x)/x; using taylor series21f = 0;22for i from 0 to 60 do { f = f + (-x)^i/(i+1); };23f = f/ln2;24 25// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|26approx = proc(poly,d) {27 return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);28};29 30// first coeff is fixed, iteratively find optimal double prec coeffs31poly = invln2hi + invln2lo;32for i from 1 to deg do {33 p = roundcoefficients(approx(poly,i), [|D ...|]);34 poly = poly + x^i*coeff(p,0);35};36 37display = hexadecimal;38print("invln2hi:", invln2hi);39print("invln2lo:", invln2lo);40print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));41print("in [",a,b,"]");42print("coeffs:");43for i from 0 to deg do coeff(poly,i);44