libc/AOR_v20.02/math/tools/exp.sollya

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