xref: /llvm-project/libc/src/math/generic/asinf.cpp (revision 46944b0cbc9a9d8daad0182c40fcd3560bc9ca35)

//===-- Single-precision asin 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/asinf.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/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

#include "inv_trigf_utils.h"

namespace LIBC_NAMESPACE_DECL {

static constexpr size_t N_EXCEPTS = 2;

// Exceptional values when |x| <= 0.5
static constexpr fputil::ExceptValues<float, N_EXCEPTS> ASINF_EXCEPTS_LO = {{
    // (inputs, RZ output, RU offset, RD offset, RN offset)
    // x = 0x1.137f0cp-5, asinf(x) = 0x1.138c58p-5 (RZ)
    {0x3d09bf86, 0x3d09c62c, 1, 0, 1},
    // x = 0x1.cbf43cp-4, asinf(x) = 0x1.cced1cp-4 (RZ)
    {0x3de5fa1e, 0x3de6768e, 1, 0, 0},
}};

// Exceptional values when 0.5 < |x| <= 1
static constexpr fputil::ExceptValues<float, N_EXCEPTS> ASINF_EXCEPTS_HI = {{
    // (inputs, RZ output, RU offset, RD offset, RN offset)
    // x = 0x1.107434p-1, asinf(x) = 0x1.1f4b64p-1 (RZ)
    {0x3f083a1a, 0x3f0fa5b2, 1, 0, 0},
    // x = 0x1.ee836cp-1, asinf(x) = 0x1.4f0654p0 (RZ)
    {0x3f7741b6, 0x3fa7832a, 1, 0, 0},
}};

LLVM_LIBC_FUNCTION(float, asinf, (float x)) {
  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 SIGN[2] = {1.0, -1.0};
  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)
      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
    }

    // Check for exceptional values
    if (auto r = ASINF_EXCEPTS_LO.lookup_odd(x_abs, x_sign);
        LIBC_UNLIKELY(r.has_value()))
      return r.value();

    // For |x| <= 0.5, we approximate asinf(x) by:
    //   asin(x) = x * P(x^2)
    // Where P(X^2) = Q(X) is a degree-20 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|],
    //                 [|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 (x_abs <= 0x7f80'0000U) {
      fputil::set_errno_if_required(EDOM);
      fputil::raise_except_if_required(FE_INVALID);
    }
    return FPBits::quiet_nan().get_val();
  }

  // Check for exceptional values
  if (auto r = ASINF_EXCEPTS_HI.lookup_odd(x_abs, x_sign);
      LIBC_UNLIKELY(r.has_value()))
    return r.value();

  // 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),

  xbits.set_sign(Sign::POS);
  double sign = SIGN[x_sign];
  double xd = static_cast<double>(xbits.get_val());
  double u = fputil::multiply_add(-0.5, xd, 0.5);
  double c1 = sign * (-2 * fputil::sqrt<double>(u));
  double c2 = fputil::multiply_add(sign, M_MATH_PI_2, c1);
  double c3 = c1 * u;

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

} // namespace LIBC_NAMESPACE_DECL