xref: /llvm-project-15.0.7/libc/src/math/generic/sinf.cpp (revision 83193a5e)

//===-- 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 "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::LargeExcepts;
using __llvm_libc::fma::N_EXCEPT_LARGE;
using __llvm_libc::fma::N_EXCEPT_SMALL;
using __llvm_libc::fma::small_range_reduction;
using __llvm_libc::fma::SmallExcepts;
#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::LargeExcepts;
using __llvm_libc::generic::N_EXCEPT_LARGE;
using __llvm_libc::generic::N_EXCEPT_SMALL;
using __llvm_libc::generic::small_range_reduction;
using __llvm_libc::generic::SmallExcepts;
#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, y;

  // Range reduction:
  // For |x| > pi/16, we perform range reduction as follows:
  // Find k and y such that:
  //   x = (k + y) * pi
  //   k is an integer
  //   |y| < 0.5
  // For small range (|x| < 2^50 when FMA instructions are available, 2^26
  // otherwise), this is done by performing:
  //   k = round(x * 1/pi)
  //   y = x * 1/pi - k
  // For large range, we will omit all the higher parts of 1/pi such that the
  // least significant bits of their full products with x are larger than 1,
  // since sin(x + i * 2pi) = sin(x).
  //
  // When FMA instructions are not available, we store the digits of 1/pi in
  // chunks of 28-bit precision.  This will make sure that the products:
  //   x * ONE_OVER_PI_28[i] are all exact.
  // When FMA instructions are available, we simply store the digits of 1/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)
  //          = sin(y*pi) * cos(k*pi) + cos(y*pi) * sin(k*pi)
  //          = (-1)^(k & 1) * sin(y*pi)
  //          ~ (-1)^(k & 1) * y * P(y^2)
  // where y*P(y^2) is a degree-15 minimax polynomial generated by Sollya
  // with: > Q = fpminimax(sin(x*pi)/x, [|0, 2, 4, 6, 8, 10, 12, 14|],
  // [|D...|], [0, 0.5]);

  // |x| <= pi/16
  if (x_abs <= 0x3e49'0fdbU) {
    xd = static_cast<double>(x);

    // |x| < 0x1.d12ed2p-12f
    if (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;
  }

  bool x_sign = xbits.get_sign();

  int64_t k;
  xd = static_cast<double>(x);

  if (x_abs < FAST_PASS_BOUND) {
    using ExceptChecker =
        typename fputil::ExceptionChecker<float, N_EXCEPT_SMALL>;
    {
      float result;
      if (ExceptChecker::check_odd_func(SmallExcepts, x_abs, x_sign, result)) {
        return result;
      }
    }

    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));
    }

    using ExceptChecker =
        typename fputil::ExceptionChecker<float, N_EXCEPT_LARGE>;
    {
      float result;
      if (ExceptChecker::check_odd_func(LargeExcepts, x_abs, x_sign, result))
        return result;
    }

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

  // After range reduction, k = round(x / pi) and y = (x/pi) - k.
  // So k is an integer and -0.5 <= y <= 0.5.
  // Then sin(x) = sin(y*pi + k*pi)
  //             = (-1)^(k & 1) * sin(y*pi)
  //             ~ (-1)^(k & 1) * y * P(y^2)
  // where y*P(y^2) is a degree-15 minimax polynomial generated by Sollya
  // with: > P = fpminimax(sin(x*pi)/x, [|0, 2, 4, 6, 8, 10, 12, 14|],
  // [|D...|], [0, 0.5]);

  constexpr double SIGN[2] = {1.0, -1.0};

  double ysq = y * y;
  double result =
      y * fputil::polyeval(ysq, 0x1.921fb54442d17p1, -0x1.4abbce625bd4bp2,
                           0x1.466bc67750a3fp1, -0x1.32d2cce1612b5p-1,
                           0x1.507832417bce6p-4, -0x1.e3062119b6071p-8,
                           0x1.e89c7aa14122dp-12, -0x1.625b1709dece6p-16);

  return SIGN[k & 1] * result;
  // }
}

} // namespace __llvm_libc