sinpi.sollya

// polynomial for approximating sinpi(x)
//
// Copyright (c) 2023-2024, Arm Limited.
// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception

deg = 19;  // polynomial degree
a = -1/2; // interval
b = 1/2;

// find even polynomial with minimal abs error compared to sinpi(x)

// f = sin(pi* x);
f = pi*x;
c = 1;
for i from 1 to 80 do { c = 2*i*(2*i + 1)*c; f = f + (-1)^i*(pi*x)^(2*i+1)/c; };

// return p that minimizes |f(x) - poly(x) - x^d*p(x)|
approx = proc(poly,d) {
  return remez(f(x)-poly(x), deg-d, [a;b], x^d, 1e-10);
};

// first coeff is predefine, iteratively find optimal double prec coeffs
poly = pi*x;
for i from 0 to (deg-1)/2 do {
  p = roundcoefficients(approx(poly,2*i+1), [|D ...|]);
  poly = poly + x^(2*i+1)*coeff(p,0);
};

display = hexadecimal;
print("abs error:", accurateinfnorm(sin(pi*x)-poly(x), [a;b], 30));
print("in [",a,b,"]");
print("coeffs:");
for i from 0 to deg do coeff(poly,i);