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 = 7; // poly degree 731914882SAlex Richardson// interval ~= 1/(2*N), where N is the table entries 831914882SAlex Richardsona= -0x1.f45p-8; 931914882SAlex Richardsonb= 0x1.f45p-8; 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) polynomial with minimal absolute error 1631914882SAlex Richardsonf = log(1+x)/ln2; 1731914882SAlex Richardson 1831914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 1931914882SAlex Richardsonapprox = proc(poly,d) { 2031914882SAlex Richardson return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 2131914882SAlex Richardson}; 2231914882SAlex Richardson 2331914882SAlex Richardson// first coeff is fixed, iteratively find optimal double prec coeffs 2431914882SAlex Richardsonpoly = x*(invln2lo + invln2hi); 2531914882SAlex Richardsonfor i from 2 to deg do { 2631914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 2731914882SAlex Richardson poly = poly + x^i*coeff(p,0); 2831914882SAlex Richardson}; 2931914882SAlex Richardson 3031914882SAlex Richardsondisplay = hexadecimal; 3131914882SAlex Richardsonprint("invln2hi:", invln2hi); 3231914882SAlex Richardsonprint("invln2lo:", invln2lo); 3331914882SAlex Richardsonprint("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 3431914882SAlex Richardson//// relative error computation fails if f(0)==0 3531914882SAlex Richardson//// g = f(x)/x = log2(1+x)/x; using taylor series 3631914882SAlex Richardson//g = 0; 3731914882SAlex Richardson//for i from 0 to 60 do { g = g + (-x)^i/(i+1)/ln2; }; 3831914882SAlex Richardson//print("rel error:", accurateinfnorm(1-(poly(x)/x)/g(x), [a;b], 30)); 3931914882SAlex Richardsonprint("in [",a,b,"]"); 4031914882SAlex Richardsonprint("coeffs:"); 4131914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 42