1*f3087befSAndrew Turner// polynomial for approximating log10(1+x) 2*f3087befSAndrew Turner// 3*f3087befSAndrew Turner// Copyright (c) 2019-2024, Arm Limited. 4*f3087befSAndrew Turner// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 5*f3087befSAndrew Turner 6*f3087befSAndrew Turnerdeg = 6; // poly degree 7*f3087befSAndrew Turner// |log10(1+x)| > 0x1p-5 outside the interval 8*f3087befSAndrew Turnera = -0x1.p-5; 9*f3087befSAndrew Turnerb = 0x1.p-5; 10*f3087befSAndrew Turner 11*f3087befSAndrew Turnerln10 = evaluate(log(10),0); 12*f3087befSAndrew Turnerinvln10hi = double(1/ln10 + 0x1p21) - 0x1p21; // round away last 21 bits 13*f3087befSAndrew Turnerinvln10lo = double(1/ln10 - invln10hi); 14*f3087befSAndrew Turner 15*f3087befSAndrew Turner// find log10(1+x)/x polynomial with minimal relative error 16*f3087befSAndrew Turner// (minimal relative error polynomial for log10(1+x) is the same * x) 17*f3087befSAndrew Turnerdeg = deg-1; // because of /x 18*f3087befSAndrew Turner 19*f3087befSAndrew Turner// f = log(1+x)/x; using taylor series 20*f3087befSAndrew Turnerf = 0; 21*f3087befSAndrew Turnerfor i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 22*f3087befSAndrew Turnerf = f/ln10; 23*f3087befSAndrew Turner 24*f3087befSAndrew Turner// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 25*f3087befSAndrew Turnerapprox = proc(poly,d) { 26*f3087befSAndrew Turner return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 27*f3087befSAndrew Turner}; 28*f3087befSAndrew Turner 29*f3087befSAndrew Turner// first coeff is fixed, iteratively find optimal double prec coeffs 30*f3087befSAndrew Turnerpoly = invln10hi + invln10lo; 31*f3087befSAndrew Turnerfor i from 1 to deg do { 32*f3087befSAndrew Turner p = roundcoefficients(approx(poly,i), [|D ...|]); 33*f3087befSAndrew Turner poly = poly + x^i*coeff(p,0); 34*f3087befSAndrew Turner}; 35*f3087befSAndrew Turnerdisplay = hexadecimal; 36*f3087befSAndrew Turnerprint("invln10hi:", invln10hi); 37*f3087befSAndrew Turnerprint("invln10lo:", invln10lo); 38*f3087befSAndrew Turnerprint("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 39*f3087befSAndrew Turnerprint("in [",a,b,"]"); 40*f3087befSAndrew Turnerprint("coeffs:"); 41*f3087befSAndrew Turnerfor i from 0 to deg do coeff(poly,i); 42*f3087befSAndrew Turner 43*f3087befSAndrew Turnerdisplay = decimal; 44*f3087befSAndrew Turnerprint("in [",a,b,"]"); 45