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//===-- Half-precision acospi 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/acospif16.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 {
LLVM_LIBC_FUNCTION(float16, acospif16, (float16 x)) {
using FPBits = fputil::FPBits<float16>;
FPBits xbits(x);
uint16_t x_u = xbits.uintval();
uint16_t x_abs = x_u & 0x7fff;
uint16_t x_sign = x_u >> 15;
// |x| > 0x1p0, |x| > 1, or x is NaN.
if (LIBC_UNLIKELY(x_abs > 0x3c00)) {
// acospif16(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();
}
// |x| == 0x1p0, x is 1 or -1
// if x is (-)1, return 1
// if x is (+)1, return 0
if (LIBC_UNLIKELY(x_abs == 0x3c00))
return fputil::cast<float16>(x_sign ? 1.0f : 0.0f);
float xf = x;
float xsq = xf * xf;
// Degree-6 minimax polynomial coefficients of asin(x) generated by Sollya
// with: > P = fpminimax(asin(x)/(pi * x), [|0, 2, 4, 6, 8|], [|SG...|], [0,
// 0.5]);
constexpr float POLY_COEFFS[5] = {0x1.45f308p-2f, 0x1.b2900cp-5f,
0x1.897e36p-6f, 0x1.9efafcp-7f,
0x1.06d884p-6f};
// |x| <= 0x1p-1, |x| <= 0.5
if (x_abs <= 0x3800) {
// if x is 0, return 0.5
if (LIBC_UNLIKELY(x_abs == 0))
return fputil::cast<float16>(0.5f);
// Note that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x), then
// acospi(x) = 0.5 - asin(x)/pi
float interm =
fputil::polyeval(xsq, POLY_COEFFS[0], POLY_COEFFS[1], POLY_COEFFS[2],
POLY_COEFFS[3], POLY_COEFFS[4]);
return fputil::cast<float16>(fputil::multiply_add(-xf, interm, 0.5f));
}
// 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 - 2 * 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))
// 11: Recall that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x)
// Therefore:
// acos(x) = pi/2 - (pi/2 - 2 * asin(sqrt(u)))
// acos(x) = 2 * asin(sqrt(u))
// acospi(x) = 2 * (asin(sqrt(u)) / pi)
//
// THE RANGE REDUCTION, HOW?
// 12: Recall [7], u = (1 - x)/2
// 13: 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 [11] as `sqrt(u)` is in range.
// When -1 < x <= -0.5, the identity:
// acos(x) = pi - acos(-x)
// acospi(x) = 1 - acos(-x)/pi
// allows us to compute for the negative x value (lhs)
// with a positive x value instead (rhs).
float xf_abs = (xf < 0 ? -xf : xf);
float u = fputil::multiply_add(-0.5f, xf_abs, 0.5f);
float sqrt_u = fputil::sqrt<float>(u);
float asin_sqrt_u =
sqrt_u * fputil::polyeval(u, POLY_COEFFS[0], POLY_COEFFS[1],
POLY_COEFFS[2], POLY_COEFFS[3], POLY_COEFFS[4]);
// Same as acos(x), but devided the expression with pi
return fputil::cast<float16>(
x_sign ? fputil::multiply_add(-2.0f, asin_sqrt_u, 1.0f)
: 2.0f * asin_sqrt_u);
}
} // namespace LIBC_NAMESPACE_DECL
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