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50 lines
1.5 KiB
50 lines
1.5 KiB
// polynomial for approximating 2^x
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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// exp2f parameters
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deg = 3; // poly degree
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N = 32; // table entries
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b = 1/(2*N); // interval
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a = -b;
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//// exp2 parameters
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//deg = 5; // poly degree
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//N = 128; // table entries
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//b = 1/(2*N); // interval
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//a = -b;
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// find polynomial with minimal relative error
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f = 2^x;
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// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|
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approx = proc(poly,d) {
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return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);
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};
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// return p that minimizes |f(x) - poly(x) - x^d*p(x)|
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approx_abs = proc(poly,d) {
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return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10);
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};
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// first coeff is fixed, iteratively find optimal double prec coeffs
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poly = 1;
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for i from 1 to deg do {
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p = roundcoefficients(approx(poly,i), [|D ...|]);
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// p = roundcoefficients(approx_abs(poly,i), [|D ...|]);
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poly = poly + x^i*coeff(p,0);
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};
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display = hexadecimal;
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print("rel error:", accurateinfnorm(1-poly(x)/2^x, [a;b], 30));
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print("abs error:", accurateinfnorm(2^x-poly(x), [a;b], 30));
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print("in [",a,b,"]");
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// double interval error for non-nearest rounding:
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print("rel2 error:", accurateinfnorm(1-poly(x)/2^x, [2*a;2*b], 30));
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print("abs2 error:", accurateinfnorm(2^x-poly(x), [2*a;2*b], 30));
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print("in [",2*a,2*b,"]");
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print("coeffs:");
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for i from 0 to deg do coeff(poly,i);
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