131914882SAlex Richardson// polynomial for approximating e^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 = 5; // poly degree 731914882SAlex RichardsonN = 128; // table entries 831914882SAlex Richardsonb = log(2)/(2*N); // interval 931914882SAlex Richardsonb = b + b*0x1p-16; // increase interval for non-nearest rounding (TOINT_NARROW) 1031914882SAlex Richardsona = -b; 1131914882SAlex Richardson 1231914882SAlex Richardson// find polynomial with minimal abs error 1331914882SAlex Richardson 1431914882SAlex Richardson// return p that minimizes |exp(x) - poly(x) - x^d*p(x)| 1531914882SAlex Richardsonapprox = proc(poly,d) { 1631914882SAlex Richardson return remez(exp(x)-poly(x), deg-d, [a;b], x^d, 1e-10); 1731914882SAlex Richardson}; 1831914882SAlex Richardson 1931914882SAlex Richardson// first 2 coeffs are fixed, iteratively find optimal double prec coeffs 2031914882SAlex Richardsonpoly = 1 + x; 2131914882SAlex Richardsonfor i from 2 to deg do { 2231914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 2331914882SAlex Richardson poly = poly + x^i*coeff(p,0); 2431914882SAlex Richardson}; 2531914882SAlex Richardson 2631914882SAlex Richardsondisplay = hexadecimal; 2731914882SAlex Richardsonprint("rel error:", accurateinfnorm(1-poly(x)/exp(x), [a;b], 30)); 2831914882SAlex Richardsonprint("abs error:", accurateinfnorm(exp(x)-poly(x), [a;b], 30)); 2931914882SAlex Richardsonprint("in [",a,b,"]"); 3031914882SAlex Richardson// double interval error for non-nearest rounding 3131914882SAlex Richardsonprint("rel2 error:", accurateinfnorm(1-poly(x)/exp(x), [2*a;2*b], 30)); 3231914882SAlex Richardsonprint("abs2 error:", accurateinfnorm(exp(x)-poly(x), [2*a;2*b], 30)); 3331914882SAlex Richardsonprint("in [",2*a,2*b,"]"); 3431914882SAlex Richardsonprint("coeffs:"); 3531914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 36