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 = 11; // poly degree
8// |log2(1+x)| > 0x1p-4 outside the interval
9a = -0x1.5b51p-5;
10b =  0x1.6ab2p-5;
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)/x polynomial with minimal relative error
17// (minimal relative error polynomial for log2(1+x) is the same * x)
18deg = deg-1; // because of /x
19
20// f = log(1+x)/x; using taylor series
21f = 0;
22for i from 0 to 60 do { f = f + (-x)^i/(i+1); };
23f = f/ln2;
24
25// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|
26approx = proc(poly,d) {
27  return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);
28};
29
30// first coeff is fixed, iteratively find optimal double prec coeffs
31poly = invln2hi + invln2lo;
32for i from 1 to deg do {
33  p = roundcoefficients(approx(poly,i), [|D ...|]);
34  poly = poly + x^i*coeff(p,0);
35};
36
37display = hexadecimal;
38print("invln2hi:", invln2hi);
39print("invln2lo:", invln2lo);
40print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));
41print("in [",a,b,"]");
42print("coeffs:");
43for i from 0 to deg do coeff(poly,i);
44