1// polynomial for approximating log2(1+x)
2//
3// Copyright (c) 2019, Arm Limited.
4// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
5
6deg = 11; // poly degree
7// |log2(1+x)| > 0x1p-4 outside the interval
8a = -0x1.5b51p-5;
9b =  0x1.6ab2p-5;
10
11ln2 = evaluate(log(2),0);
12invln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits
13invln2lo = double(1/ln2 - invln2hi);
14
15// find log2(1+x)/x polynomial with minimal relative error
16// (minimal relative error polynomial for log2(1+x) is the same * x)
17deg = deg-1; // because of /x
18
19// f = log(1+x)/x; using taylor series
20f = 0;
21for i from 0 to 60 do { f = f + (-x)^i/(i+1); };
22f = f/ln2;
23
24// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|
25approx = proc(poly,d) {
26  return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);
27};
28
29// first coeff is fixed, iteratively find optimal double prec coeffs
30poly = invln2hi + invln2lo;
31for i from 1 to deg do {
32  p = roundcoefficients(approx(poly,i), [|D ...|]);
33  poly = poly + x^i*coeff(p,0);
34};
35
36display = hexadecimal;
37print("invln2hi:", invln2hi);
38print("invln2lo:", invln2lo);
39print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));
40print("in [",a,b,"]");
41print("coeffs:");
42for i from 0 to deg do coeff(poly,i);
43