1// polynomial for approximating 2^x 2// 3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4// See https://llvm.org/LICENSE.txt for license information. 5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 7// exp2f parameters 8deg = 3; // poly degree 9N = 32; // table entries 10b = 1/(2*N); // interval 11a = -b; 12 13//// exp2 parameters 14//deg = 5; // poly degree 15//N = 128; // table entries 16//b = 1/(2*N); // interval 17//a = -b; 18 19// find polynomial with minimal relative error 20 21f = 2^x; 22 23// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 24approx = proc(poly,d) { 25 return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 26}; 27// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 28approx_abs = proc(poly,d) { 29 return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 30}; 31 32// first coeff is fixed, iteratively find optimal double prec coeffs 33poly = 1; 34for i from 1 to deg do { 35 p = roundcoefficients(approx(poly,i), [|D ...|]); 36// p = roundcoefficients(approx_abs(poly,i), [|D ...|]); 37 poly = poly + x^i*coeff(p,0); 38}; 39 40display = hexadecimal; 41print("rel error:", accurateinfnorm(1-poly(x)/2^x, [a;b], 30)); 42print("abs error:", accurateinfnorm(2^x-poly(x), [a;b], 30)); 43print("in [",a,b,"]"); 44// double interval error for non-nearest rounding: 45print("rel2 error:", accurateinfnorm(1-poly(x)/2^x, [2*a;2*b], 30)); 46print("abs2 error:", accurateinfnorm(2^x-poly(x), [2*a;2*b], 30)); 47print("in [",2*a,2*b,"]"); 48print("coeffs:"); 49for i from 0 to deg do coeff(poly,i); 50