libc/src/math/generic/asinf16.cpp

//===-- Half-precision asinf16(x) 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/asinf16.h"
#include "hdr/errno_macros.h"
#include "hdr/fenv_macros.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/cast.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/sqrt.h"
#include "src/__support/macros/optimization.h"

namespace LIBC_NAMESPACE_DECL {

// Generated by Sollya using the following command:
// > round(pi/2, D, RN);
static constexpr float PI_2 = 0x1.921fb54442d18p0f;

LLVM_LIBC_FUNCTION(float16, asinf16, (float16 x)) {
  using FPBits = fputil::FPBits<float16>;
  FPBits xbits(x);

  uint16_t x_u = xbits.uintval();
  uint16_t x_abs = x_u & 0x7fff;
  float xf = x;

  // |x| > 0x1p0, |x| > 1, or x is NaN.
  if (LIBC_UNLIKELY(x_abs > 0x3c00)) {
    // asinf16(NaN) = NaN
    if (xbits.is_nan()) {
      if (xbits.is_signaling_nan()) {
        fputil::raise_except_if_required(FE_INVALID);
        return FPBits::quiet_nan().get_val();
      }

      return x;
    }

    // 1 < |x| <= +/-inf
    fputil::raise_except_if_required(FE_INVALID);
    fputil::set_errno_if_required(EDOM);

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

  float xsq = xf * xf;

  // |x| <= 0x1p-1, |x| <= 0.5
  if (x_abs <= 0x3800) {
    // asinf16(+/-0) = +/-0
    if (LIBC_UNLIKELY(x_abs == 0))
      return x;

    // Exhaustive tests show that,
    // for |x| <= 0x1.878p-9, when:
    // x > 0, and rounding upward, or
    // x < 0, and rounding downward, then,
    // asin(x) = x * 2^-11 + x
    // else, in other rounding modes,
    // asin(x) = x
    if (LIBC_UNLIKELY(x_abs <= 0x1a1e)) {
      int rounding = fputil::quick_get_round();

      if ((xbits.is_pos() && rounding == FE_UPWARD) ||
          (xbits.is_neg() && rounding == FE_DOWNWARD))
        return fputil::cast<float16>(fputil::multiply_add(xf, 0x1.0p-11f, xf));
      return x;
    }

    // Degree-6 minimax odd polynomial of asin(x) generated by Sollya with:
    // > P = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8|], [|SG...|], [0, 0.5]);
    float result =
        fputil::polyeval(xsq, 0x1.000002p0f, 0x1.554c2ap-3f, 0x1.3541ccp-4f,
                         0x1.43b2d6p-5f, 0x1.a0d73ep-5f);
    return fputil::cast<float16>(xf * result);
  }

  // When |x| > 0.5, assume that 0.5 < |x| <= 1,
  //
  // Step-by-step range-reduction proof:
  // 1:  Let y = asin(x), such that, x = sin(y)
  // 2:  From complimentary angle identity:
  //       x = sin(y) = cos(pi/2 - y)
  // 3:  Let z = pi/2 - y, such that x = cos(z)
  // 4:  From double angle formula; cos(2A) = 1 - sin^2(A):
  //       z = 2A, z/2 = A
  //       cos(z) = 1 - 2 * sin^2(z/2)
  // 5:  Make sin(z/2) subject of the formula:
  //       sin(z/2) = sqrt((1 - cos(z))/2)
  // 6:  Recall [3]; x = cos(z). Therefore:
  //       sin(z/2) = sqrt((1 - x)/2)
  // 7:  Let u = (1 - x)/2
  // 8:  Therefore:
  //       asin(sqrt(u)) = z/2
  //       2 * asin(sqrt(u)) = z
  // 9:  Recall [3], z = pi/2 - y. Therefore:
  //       y = pi/2 - z
  //       y = pi/2 - 2 * asin(sqrt(u))
  // 10: Recall [1], y = asin(x). Therefore:
  //       asin(x) = pi/2 - 2 * asin(sqrt(u))
  //
  // WHY?
  // 11: Recall [7], u = (1 - x)/2
  // 12: Since 0.5 < x <= 1, therefore:
  //       0 <= u <= 0.25 and 0 <= sqrt(u) <= 0.5
  //
  // Hence, we can reuse the same [0, 0.5] domain polynomial approximation for
  // Step [10] as `sqrt(u)` is in range.

  // 0x1p-1 < |x| <= 0x1p0, 0.5 < |x| <= 1.0
  float xf_abs = (xf < 0 ? -xf : xf);
  float sign = (xbits.uintval() >> 15 == 1 ? -1.0 : 1.0);
  float u = fputil::multiply_add(-0.5f, xf_abs, 0.5f);
  float u_sqrt = fputil::sqrt<float>(u);

  // Degree-6 minimax odd polynomial of asin(x) generated by Sollya with:
  // > P = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8|], [|SG...|], [0, 0.5]);
  float asin_sqrt_u =
      u_sqrt * fputil::polyeval(u, 0x1.000002p0f, 0x1.554c2ap-3f,
                                0x1.3541ccp-4f, 0x1.43b2d6p-5f, 0x1.a0d73ep-5f);

  return fputil::cast<float16>(sign *
                               fputil::multiply_add(-2.0f, asin_sqrt_u, PI_2));
}

} // namespace LIBC_NAMESPACE_DECL