1*31914882SAlex Richardson// polynomial for approximating log(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 = 6; // poly degree 7*31914882SAlex Richardson// interval ~= 1/(2*N), where N is the table entries 8*31914882SAlex Richardsona = -0x1.fp-9; 9*31914882SAlex Richardsonb = 0x1.fp-9; 10*31914882SAlex Richardson 11*31914882SAlex Richardson// find log(1+x) polynomial with minimal absolute error 12*31914882SAlex Richardsonf = log(1+x); 13*31914882SAlex Richardson 14*31914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 15*31914882SAlex Richardsonapprox = proc(poly,d) { 16*31914882SAlex Richardson return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 17*31914882SAlex Richardson}; 18*31914882SAlex Richardson 19*31914882SAlex Richardson// first coeff is fixed, iteratively find optimal double prec coeffs 20*31914882SAlex Richardsonpoly = x; 21*31914882SAlex Richardsonfor i from 2 to deg do { 22*31914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 23*31914882SAlex Richardson poly = poly + x^i*coeff(p,0); 24*31914882SAlex Richardson}; 25*31914882SAlex Richardson 26*31914882SAlex Richardsondisplay = hexadecimal; 27*31914882SAlex Richardsonprint("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 28*31914882SAlex Richardson// relative error computation fails if f(0)==0 29*31914882SAlex Richardson// g = f(x)/x = log(1+x)/x; using taylor series 30*31914882SAlex Richardsong = 0; 31*31914882SAlex Richardsonfor i from 0 to 60 do { g = g + (-x)^i/(i+1); }; 32*31914882SAlex Richardsonprint("rel error:", accurateinfnorm(1-poly(x)/x/g(x), [a;b], 30)); 33*31914882SAlex Richardsonprint("in [",a,b,"]"); 34*31914882SAlex Richardsonprint("coeffs:"); 35*31914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 36