libc/src/__support/math/asinf.h

//===-- Implementation header for asinf -------------------------*- C++ -*-===//
//
// 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
//
//===----------------------------------------------------------------------===//

#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ASINF_H
#define LLVM_LIBC_SRC___SUPPORT_MATH_ASINF_H

#include "inv_trigf_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/except_value_utils.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/sqrt.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h"            // LIBC_UNLIKELY
#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA

namespace LIBC_NAMESPACE_DECL {

namespace math {

LIBC_INLINE constexpr float asinf(float x) {
  using namespace inv_trigf_utils_internal;
  using FPBits = typename fputil::FPBits<float>;

  FPBits xbits(x);
  uint32_t x_uint = xbits.uintval();
  uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
  constexpr double TWO[2] = {-2.0, 2.0};
  uint32_t x_sign = x_uint >> 31;

  // |x| <= 0.5-ish
  if (x_abs < 0x3f04'471dU) {
    // |x| < 0x1.d12edp-12
    if (LIBC_UNLIKELY(x_abs < 0x39e8'9768U)) {
      // When |x| < 2^-12, the relative error of the approximation asin(x) ~ x
      // is:
      //   |asin(x) - x| / |asin(x)| < |x^3| / (6|x|)
      //                             = x^2 / 6
      //                             < 2^-25
      //                             < epsilon(1)/2.
      // So the correctly rounded values of asin(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.d12edp-12.
      // 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_CPU_HAS_FMA_FLOAT)
      return fputil::multiply_add(x, 0x1.0p-25f, x);
#else
      double xd = static_cast<double>(x);
      return static_cast<float>(fputil::multiply_add(xd, 0x1.0p-25, xd));
#endif // LIBC_TARGET_CPU_HAS_FMA_FLOAT
    }

    // For |x| <= 0.5, we approximate asinf(x) by:
    //   asin(x) = x * P(x^2)
    // Where P(X^2) = Q(X) is a degree-24 minimax even polynomial approximating
    // asin(x)/x on [0, 0.5] generated by Sollya with:
    // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,
    //                              22, 24|], [|1, D...|], [0, 0.5]);
    // An exhaustive test shows that this approximation works well up to a
    // little more than 0.5.
    double xd = static_cast<double>(x);
    double xsq = xd * xd;
    double x3 = xd * xsq;
    double r = asin_eval(xsq);
    return static_cast<float>(fputil::multiply_add(x3, r, xd));
  }

  // |x| > 1, return NaNs.
  if (LIBC_UNLIKELY(x_abs > 0x3f80'0000U)) {
    if (xbits.is_signaling_nan()) {
      fputil::raise_except_if_required(FE_INVALID);
      return FPBits::quiet_nan().get_val();
    }

    if (x_abs <= 0x7f80'0000U) {
      fputil::set_errno_if_required(EDOM);
      fputil::raise_except_if_required(FE_INVALID);
    }

    return FPBits::quiet_nan().get_val();
  }

  // When |x| > 0.5, we perform range reduction as follow:
  //
  // Assume further that 0.5 < x <= 1, and let:
  //   y = asin(x)
  // We will use the double angle formula:
  //   cos(2y) = 1 - 2 sin^2(y)
  // and the complement angle identity:
  //   x = sin(y) = cos(pi/2 - y)
  //              = 1 - 2 sin^2 (pi/4 - y/2)
  // So:
  //   sin(pi/4 - y/2) = sqrt( (1 - x)/2 )
  // And hence:
  //   pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) )
  // Equivalently:
  //   asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) )
  // Let u = (1 - x)/2, then:
  //   asin(x) = pi/2 - 2 * asin( sqrt(u) )
  // Moreover, since 0.5 < x <= 1:
  //   0 <= u < 1/4, and 0 <= sqrt(u) < 0.5,
  // And hence we can reuse the same polynomial approximation of asin(x) when
  // |x| <= 0.5:
  //   asin(x) ~ pi/2 - 2 * sqrt(u) * P(u),

  constexpr double M_PI_OVER_4 = -0x1.921fb54442d18p-1;

  xbits.set_sign(Sign::POS);
  double sign_two = TWO[x_sign]; // sign * (-2)
  double uf = fputil::multiply_add(-0.5f, xbits.get_val(), 0.5f);
  double u = static_cast<double>(uf);
  double c1 = sign_two * fputil::sqrt<double>(u);
  double c2 = fputil::multiply_add(sign_two, M_PI_OVER_4, c1);
  double c3 = c1 * u;

  double r = asin_eval(u);
  return static_cast<float>(fputil::multiply_add(c3, r, c2));
}

} // namespace math

} // namespace LIBC_NAMESPACE_DECL

#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ASINF_H