131914882SAlex Richardson// polynomial for approximating log2(1+x) 231914882SAlex Richardson// 331914882SAlex Richardson// Copyright (c) 2019, Arm Limited. 4*072a4ba8SAndrew Turner// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 531914882SAlex Richardson 631914882SAlex Richardsondeg = 11; // poly degree 731914882SAlex Richardson// |log2(1+x)| > 0x1p-4 outside the interval 831914882SAlex Richardsona = -0x1.5b51p-5; 931914882SAlex Richardsonb = 0x1.6ab2p-5; 1031914882SAlex Richardson 1131914882SAlex Richardsonln2 = evaluate(log(2),0); 1231914882SAlex Richardsoninvln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits 1331914882SAlex Richardsoninvln2lo = double(1/ln2 - invln2hi); 1431914882SAlex Richardson 1531914882SAlex Richardson// find log2(1+x)/x polynomial with minimal relative error 1631914882SAlex Richardson// (minimal relative error polynomial for log2(1+x) is the same * x) 1731914882SAlex Richardsondeg = deg-1; // because of /x 1831914882SAlex Richardson 1931914882SAlex Richardson// f = log(1+x)/x; using taylor series 2031914882SAlex Richardsonf = 0; 2131914882SAlex Richardsonfor i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 2231914882SAlex Richardsonf = f/ln2; 2331914882SAlex Richardson 2431914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 2531914882SAlex Richardsonapprox = proc(poly,d) { 2631914882SAlex Richardson return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 2731914882SAlex Richardson}; 2831914882SAlex Richardson 2931914882SAlex Richardson// first coeff is fixed, iteratively find optimal double prec coeffs 3031914882SAlex Richardsonpoly = invln2hi + invln2lo; 3131914882SAlex Richardsonfor i from 1 to deg do { 3231914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 3331914882SAlex Richardson poly = poly + x^i*coeff(p,0); 3431914882SAlex Richardson}; 3531914882SAlex Richardson 3631914882SAlex Richardsondisplay = hexadecimal; 3731914882SAlex Richardsonprint("invln2hi:", invln2hi); 3831914882SAlex Richardsonprint("invln2lo:", invln2lo); 3931914882SAlex Richardsonprint("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 4031914882SAlex Richardsonprint("in [",a,b,"]"); 4131914882SAlex Richardsonprint("coeffs:"); 4231914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 43