src/math/generic/sinf.cpp

//===-- Single-precision sin function -------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#include "src/math/sinf.h"
#include "common_constants.h"
#include "src/__support/FPUtil/BasicOperations.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/except_value_utils.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/common.h"

#include <errno.h>

#if defined(LIBC_TARGET_HAS_FMA)
#include "range_reduction_fma.h"
// using namespace __llvm_libc::fma;
using __llvm_libc::fma::FAST_PASS_BOUND;
using __llvm_libc::fma::large_range_reduction;
using __llvm_libc::fma::N_EXCEPTS;
using __llvm_libc::fma::SinfExcepts;
using __llvm_libc::fma::small_range_reduction;
#else
#include "range_reduction.h"
// using namespace __llvm_libc::generic;
using __llvm_libc::generic::FAST_PASS_BOUND;
using __llvm_libc::generic::large_range_reduction;
using __llvm_libc::generic::N_EXCEPTS;
using __llvm_libc::generic::SinfExcepts;
using __llvm_libc::generic::small_range_reduction;
#endif

namespace __llvm_libc {

LLVM_LIBC_FUNCTION(float, sinf, (float x)) {
  using FPBits = typename fputil::FPBits<float>;
  FPBits xbits(x);

  uint32_t x_u = xbits.uintval();
  uint32_t x_abs = x_u & 0x7fff'ffffU;
  double xd = static_cast<double>(x);

  // Range reduction:
  // For |x| > pi/16, we perform range reduction as follows:
  // Find k and y such that:
  //   x = (k + y) * pi/16
  //   k is an integer
  //   |y| < 0.5
  // For small range (|x| < 2^46 when FMA instructions are available, 2^22
  // otherwise), this is done by performing:
  //   k = round(x * 16/pi)
  //   y = x * 16/pi - k
  // For large range, we will omit all the higher parts of 16/pi such that the
  // least significant bits of their full products with x are larger than 31,
  // since sin((k + y + 32*i) * pi/16) = sin(x + i * 2pi) = sin(x).
  //
  // When FMA instructions are not available, we store the digits of 16/pi in
  // chunks of 28-bit precision.  This will make sure that the products:
  //   x * SIXTEEN_OVER_PI_28[i] are all exact.
  // When FMA instructions are available, we simply store the digits of 16/pi in
  // chunks of doubles (53-bit of precision).
  // So when multiplying by the largest values of single precision, the
  // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80.  By the
  // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
  // us more than 40 bits of accuracy. For the worst-case estimation of range
  // reduction, see for instances:
  //   Elementary Functions by J-M. Muller, Chapter 11,
  //   Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
  //   Chapter 10.2.
  //
  // Once k and y are computed, we then deduce the answer by the sine of sum
  // formula:
  //   sin(x) = sin((k + y)*pi/16)
  //          = sin(y*pi/16) * cos(k*pi/16) + cos(y*pi/16) * sin(k*pi/16)
  // The values of sin(k*pi/16) and cos(k*pi/16) for k = 0..31 are precomputed
  // and stored using a vector of 32 doubles. Sin(y*pi/16) and cos(y*pi/16) are
  // computed using degree-7 and degree-8 minimax polynomials generated by
  // Sollya respectively.

  // |x| <= pi/16
  if (unlikely(x_abs <= 0x3e49'0fdbU)) {

    // |x| < 0x1.d12ed2p-12f
    if (unlikely(x_abs < 0x39e8'9769U)) {
      if (unlikely(x_abs == 0U)) {
        // For signed zeros.
        return x;
      }
      // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x
      // is:
      //   |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
      //                           = x^2 / 6
      //                           < 2^-25
      //                           < epsilon(1)/2.
      // So the correctly rounded values of sin(x) are:
      //   = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
      //                        or (rounding mode = FE_UPWARD and x is
      //                        negative),
      //   = x otherwise.
      // To simplify the rounding decision and make it more efficient, we use
      //   fma(x, -2^-25, x) instead.
      // An exhaustive test shows that this formula work correctly for all
      // rounding modes up to |x| < 0x1.c555dep-11f.
      // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
      // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
      // |x| < 2^-125. For targets without FMA instructions, we simply use
      // double for intermediate results as it is more efficient than using an
      // emulated version of FMA.
#if defined(LIBC_TARGET_HAS_FMA)
      return fputil::multiply_add(x, -0x1.0p-25f, x);
#else
      return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
#endif // LIBC_TARGET_HAS_FMA
    }

    // |x| < pi/16.
    double xsq = xd * xd;

    // Degree-9 polynomial approximation:
    //   sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9
    //          = x (1 + a_3 x^2 + ... + a_9 x^8)
    //          = x * P(x^2)
    // generated by Sollya with the following commands:
    // > display = hexadecimal;
    // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);
    double result =
        fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7,
                         -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19);
    return xd * result;
  }

  using ExceptChecker = typename fputil::ExceptionChecker<float, N_EXCEPTS>;
  {
    float result;
    if (ExceptChecker::check_odd_func(SinfExcepts, x_abs, xbits.get_sign(),
                                      result))
      return result;
  }

  int k;
  double y;

  if (likely(x_abs < FAST_PASS_BOUND)) {
    k = small_range_reduction(xd, y);
  } else {
    // x is inf or nan.
    if (unlikely(x_abs >= 0x7f80'0000U)) {
      if (x_abs == 0x7f80'0000U) {
        errno = EDOM;
        fputil::set_except(FE_INVALID);
      }
      return x +
             FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1));
    }

    k = large_range_reduction(xd, xbits.get_exponent(), y);
  }

  // After range reduction, k = round(x * 16 / pi) and y = (x * 16 / pi) - k.
  // So k is an integer and -0.5 <= y <= 0.5.
  // Then sin(x) = sin((k + y)*pi/16)
  //             = sin(y*pi/16) * cos(k*pi/16) + cos(y*pi/16) * sin(k*pi/16)

  double ysq = y * y;

  // Degree-6 minimax even polynomial for sin(y*pi/16)/y generated by Sollya
  // with:
  // > Q = fpminimax(sin(y*pi/16)/y, [|0, 2, 4, 6|], [|D...|], [0, 0.5]);
  double sin_y =
      fputil::polyeval(ysq, 0x1.921fb54442d17p-3, -0x1.4abbce6256adp-10,
                       0x1.466bc5a5ac6b3p-19, -0x1.32bdcb4207562p-29);
  // Degree-8 minimax even polynomial for cos(y*pi/16) generated by Sollya with:
  // > P = fpminimax(cos(x*pi/16), [|0, 2, 4, 6, 8|], [|1, D...|], [0, 0.5]);
  // Note that cosm1_y = cos(y*pi/16) - 1.
  double cosm1_y =
      ysq * fputil::polyeval(ysq, -0x1.3bd3cc9be45dcp-6, 0x1.03c1f081b08ap-14,
                             -0x1.55d3c6fb0fb6ep-24, 0x1.e1d3d60f58873p-35);

  double sin_k = SIN_K_PI_OVER_16[k & 31];
  // cos(k * pi/16) = sin(k * pi/16 + pi/2) = sin((k + 8) * pi/16).
  // cos_k = y * cos(k * pi/16)
  double cos_k = y * SIN_K_PI_OVER_16[(k + 8) & 31];

  // Combine the results with the sine of sum formula:
  //   sin(x) = sin((k + y)*pi/16)
  //          = sin(y*pi/16) * cos(k*pi/16) + cos(y*pi/16) * sin(k*pi/16)
  //          = sin_y * cos_k + (1 + cosm1_y) * sin_k
  //          = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
  return fputil::multiply_add(sin_y, cos_k,
                              fputil::multiply_add(cosm1_y, sin_k, sin_k));
}

} // namespace __llvm_libc