1// polynomial for approximating e^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
7deg = 5; // poly degree
8N = 128; // table entries
9b = log(2)/(2*N);  // interval
10b = b + b*0x1p-16; // increase interval for non-nearest rounding (TOINT_NARROW)
11a = -b;
12
13// find polynomial with minimal abs error
14
15// return p that minimizes |exp(x) - poly(x) - x^d*p(x)|
16approx = proc(poly,d) {
17  return remez(exp(x)-poly(x), deg-d, [a;b], x^d, 1e-10);
18};
19
20// first 2 coeffs are fixed, iteratively find optimal double prec coeffs
21poly = 1 + x;
22for i from 2 to deg do {
23  p = roundcoefficients(approx(poly,i), [|D ...|]);
24  poly = poly + x^i*coeff(p,0);
25};
26
27display = hexadecimal;
28print("rel error:", accurateinfnorm(1-poly(x)/exp(x), [a;b], 30));
29print("abs error:", accurateinfnorm(exp(x)-poly(x), [a;b], 30));
30print("in [",a,b,"]");
31// double interval error for non-nearest rounding
32print("rel2 error:", accurateinfnorm(1-poly(x)/exp(x), [2*a;2*b], 30));
33print("abs2 error:", accurateinfnorm(exp(x)-poly(x), [2*a;2*b], 30));
34print("in [",2*a,2*b,"]");
35print("coeffs:");
36for i from 0 to deg do coeff(poly,i);
37