1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
|
// polynomial for approximating e^x
//
// Copyright (c) 2019, Arm Limited.
// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
deg = 5; // poly degree
N = 128; // table entries
b = log(2)/(2*N); // interval
b = b + b*0x1p-16; // increase interval for non-nearest rounding (TOINT_NARROW)
a = -b;
// find polynomial with minimal abs error
// return p that minimizes |exp(x) - poly(x) - x^d*p(x)|
approx = proc(poly,d) {
return remez(exp(x)-poly(x), deg-d, [a;b], x^d, 1e-10);
};
// first 2 coeffs are fixed, iteratively find optimal double prec coeffs
poly = 1 + x;
for i from 2 to deg do {
p = roundcoefficients(approx(poly,i), [|D ...|]);
poly = poly + x^i*coeff(p,0);
};
display = hexadecimal;
print("rel error:", accurateinfnorm(1-poly(x)/exp(x), [a;b], 30));
print("abs error:", accurateinfnorm(exp(x)-poly(x), [a;b], 30));
print("in [",a,b,"]");
// double interval error for non-nearest rounding
print("rel2 error:", accurateinfnorm(1-poly(x)/exp(x), [2*a;2*b], 30));
print("abs2 error:", accurateinfnorm(exp(x)-poly(x), [2*a;2*b], 30));
print("in [",2*a,2*b,"]");
print("coeffs:");
for i from 0 to deg do coeff(poly,i);
|
