1*31914882SAlex Richardson// polynomial for approximating log2(1+x) 2*31914882SAlex Richardson// 3*31914882SAlex Richardson// Copyright (c) 2019, Arm Limited. 4*31914882SAlex Richardson// SPDX-License-Identifier: MIT 5*31914882SAlex Richardson 6*31914882SAlex Richardsondeg = 7; // poly degree 7*31914882SAlex Richardson// interval ~= 1/(2*N), where N is the table entries 8*31914882SAlex Richardsona= -0x1.f45p-8; 9*31914882SAlex Richardsonb= 0x1.f45p-8; 10*31914882SAlex Richardson 11*31914882SAlex Richardsonln2 = evaluate(log(2),0); 12*31914882SAlex Richardsoninvln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits 13*31914882SAlex Richardsoninvln2lo = double(1/ln2 - invln2hi); 14*31914882SAlex Richardson 15*31914882SAlex Richardson// find log2(1+x) polynomial with minimal absolute error 16*31914882SAlex Richardsonf = log(1+x)/ln2; 17*31914882SAlex Richardson 18*31914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 19*31914882SAlex Richardsonapprox = proc(poly,d) { 20*31914882SAlex Richardson return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 21*31914882SAlex Richardson}; 22*31914882SAlex Richardson 23*31914882SAlex Richardson// first coeff is fixed, iteratively find optimal double prec coeffs 24*31914882SAlex Richardsonpoly = x*(invln2lo + invln2hi); 25*31914882SAlex Richardsonfor i from 2 to deg do { 26*31914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 27*31914882SAlex Richardson poly = poly + x^i*coeff(p,0); 28*31914882SAlex Richardson}; 29*31914882SAlex Richardson 30*31914882SAlex Richardsondisplay = hexadecimal; 31*31914882SAlex Richardsonprint("invln2hi:", invln2hi); 32*31914882SAlex Richardsonprint("invln2lo:", invln2lo); 33*31914882SAlex Richardsonprint("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 34*31914882SAlex Richardson//// relative error computation fails if f(0)==0 35*31914882SAlex Richardson//// g = f(x)/x = log2(1+x)/x; using taylor series 36*31914882SAlex Richardson//g = 0; 37*31914882SAlex Richardson//for i from 0 to 60 do { g = g + (-x)^i/(i+1)/ln2; }; 38*31914882SAlex Richardson//print("rel error:", accurateinfnorm(1-(poly(x)/x)/g(x), [a;b], 30)); 39*31914882SAlex Richardsonprint("in [",a,b,"]"); 40*31914882SAlex Richardsonprint("coeffs:"); 41*31914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 42