1// polynomial for approximating log2(1+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 = 7; // poly degree
8// interval ~= 1/(2*N), where N is the table entries
9a= -0x1.f45p-8;
10b=  0x1.f45p-8;
11
12ln2 = evaluate(log(2),0);
13invln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits
14invln2lo = double(1/ln2 - invln2hi);
15
16// find log2(1+x) polynomial with minimal absolute error
17f = log(1+x)/ln2;
18
19// return p that minimizes |f(x) - poly(x) - x^d*p(x)|
20approx = proc(poly,d) {
21  return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10);
22};
23
24// first coeff is fixed, iteratively find optimal double prec coeffs
25poly = x*(invln2lo + invln2hi);
26for i from 2 to deg do {
27  p = roundcoefficients(approx(poly,i), [|D ...|]);
28  poly = poly + x^i*coeff(p,0);
29};
30
31display = hexadecimal;
32print("invln2hi:", invln2hi);
33print("invln2lo:", invln2lo);
34print("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30));
35//// relative error computation fails if f(0)==0
36//// g = f(x)/x = log2(1+x)/x; using taylor series
37//g = 0;
38//for i from 0 to 60 do { g = g + (-x)^i/(i+1)/ln2; };
39//print("rel error:", accurateinfnorm(1-(poly(x)/x)/g(x), [a;b], 30));
40print("in [",a,b,"]");
41print("coeffs:");
42for i from 0 to deg do coeff(poly,i);
43