1// polynomial for approximating log2(1+x) 2// 3// Copyright (c) 2019, Arm Limited. 4// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 5 6deg = 7; // poly degree 7// interval ~= 1/(2*N), where N is the table entries 8a= -0x1.f45p-8; 9b= 0x1.f45p-8; 10 11ln2 = evaluate(log(2),0); 12invln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits 13invln2lo = double(1/ln2 - invln2hi); 14 15// find log2(1+x) polynomial with minimal absolute error 16f = log(1+x)/ln2; 17 18// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 19approx = proc(poly,d) { 20 return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 21}; 22 23// first coeff is fixed, iteratively find optimal double prec coeffs 24poly = x*(invln2lo + invln2hi); 25for i from 2 to deg do { 26 p = roundcoefficients(approx(poly,i), [|D ...|]); 27 poly = poly + x^i*coeff(p,0); 28}; 29 30display = hexadecimal; 31print("invln2hi:", invln2hi); 32print("invln2lo:", invln2lo); 33print("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 34//// relative error computation fails if f(0)==0 35//// g = f(x)/x = log2(1+x)/x; using taylor series 36//g = 0; 37//for i from 0 to 60 do { g = g + (-x)^i/(i+1)/ln2; }; 38//print("rel error:", accurateinfnorm(1-(poly(x)/x)/g(x), [a;b], 30)); 39print("in [",a,b,"]"); 40print("coeffs:"); 41for i from 0 to deg do coeff(poly,i); 42