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//===-- Single-precision sincos 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/sincosf.h"
#include "sincosf_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
#include <errno.h>
namespace __llvm_libc {
// Exceptional values
static constexpr int N_EXCEPTS = 6;
static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = {
0x46199998, // x = 0x1.33333p13 x
0x55325019, // x = 0x1.64a032p43 x
0x5922aa80, // x = 0x1.4555p51 x
0x5f18b878, // x = 0x1.3170fp63 x
0x6115cb11, // x = 0x1.2b9622p67 x
0x7beef5ef, // x = 0x1.ddebdep120 x
};
static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = {
{0xbeb1fa5d, 0, 1, 0}, // x = 0x1.33333p13, sin(x) = -0x1.63f4bap-2 (RZ)
{0xbf171adf, 0, 1, 1}, // x = 0x1.64a032p43, sin(x) = -0x1.2e35bep-1 (RZ)
{0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ)
{0x3dad60f6, 1, 0, 1}, // x = 0x1.3170fp63, sin(x) = 0x1.5ac1ecp-4 (RZ)
{0xbe7cc1e0, 0, 1, 1}, // x = 0x1.2b9622p67, sin(x) = -0x1.f983cp-3 (RZ)
{0xbf587d1b, 0, 1, 1}, // x = 0x1.ddebdep120, sin(x) = -0x1.b0fa36p-1 (RZ)
};
static constexpr uint32_t EXCEPT_OUTPUTS_COS[N_EXCEPTS][4] = {
{0xbf70090b, 0, 1, 0}, // x = 0x1.33333p13, cos(x) = -0x1.e01216p-1 (RZ)
{0x3f4ea5d2, 1, 0, 0}, // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)
{0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)
{0x3f7f14bb, 1, 0, 0}, // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)
{0x3f78142e, 1, 0, 1}, // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)
{0x3f08a21c, 1, 0, 0}, // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)
};
LLVM_LIBC_FUNCTION(void, sincosf, (float x, float *sinp, float *cosp)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
double xd = static_cast<double>(x);
// Range reduction:
// For |x| >= 2^-12, we perform range reduction as follows:
// Find k and y such that:
// x = (k + y) * pi/32
// k is an integer
// |y| < 0.5
// For small range (|x| < 2^45 when FMA instructions are available, 2^22
// otherwise), this is done by performing:
// k = round(x * 32/pi)
// y = x * 32/pi - k
// For large range, we will omit all the higher parts of 32/pi such that the
// least significant bits of their full products with x are larger than 63,
// since:
// sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x), and
// cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x).
//
// When FMA instructions are not available, we store the digits of 32/pi in
// chunks of 28-bit precision. This will make sure that the products:
// x * THIRTYTWO_OVER_PI_28[i] are all exact.
// When FMA instructions are available, we simply store the digits of326/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 and cosine
// of sum formulas:
// sin(x) = sin((k + y)*pi/32)
// = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
// cos(x) = cos((k + y)*pi/32)
// = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
// The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed
// and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
// computed using degree-7 and degree-6 minimax polynomials generated by
// Sollya respectively.
// |x| < 0x1.0p-12f
if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
if (LIBC_UNLIKELY(x_abs == 0U)) {
// For signed zeros.
*sinp = x;
*cosp = 1.0f;
return;
}
// When |x| < 2^-12, the relative errors of the approximations
// sin(x) ~ x, cos(x) ~ 1
// are:
// |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
// = x^2 / 6
// < 2^-25
// < epsilon(1)/2.
// |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.
// So the correctly rounded values of sin(x) and cos(x) are:
// sin(x) = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
// or (rounding mode = FE_UPWARD and x is
// negative),
// = x otherwise.
// cos(x) = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,
// = 1 otherwise.
// To simplify the rounding decision and make it more efficient and to
// prevent compiler to perform constant folding, we use
// sin(x) = fma(x, -2^-25, x),
// cos(x) = fma(x*0.5f, -x, 1)
// instead.
// 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)
*sinp = fputil::multiply_add(x, -0x1.0p-25f, x);
*cosp = fputil::multiply_add(FPBits(x_abs).get_val(), -0x1.0p-25f, 1.0f);
#else
*sinp = static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
*cosp = static_cast<float>(fputil::multiply_add(
static_cast<double>(FPBits(x_abs).get_val()), -0x1.0p-25, 1.0));
#endif // LIBC_TARGET_CPU_HAS_FMA
return;
}
// x is inf or nan.
if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
if (x_abs == 0x7f80'0000U) {
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
}
*sinp =
x + FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1));
*cosp = *sinp;
return;
}
// Check exceptional values.
for (int i = 0; i < N_EXCEPTS; ++i) {
if (LIBC_UNLIKELY(x_abs == EXCEPT_INPUTS[i])) {
uint32_t s = EXCEPT_OUTPUTS_SIN[i][0]; // FE_TOWARDZERO
uint32_t c = EXCEPT_OUTPUTS_COS[i][0]; // FE_TOWARDZERO
bool x_sign = x < 0;
switch (fputil::quick_get_round()) {
case FE_UPWARD:
s += x_sign ? EXCEPT_OUTPUTS_SIN[i][2] : EXCEPT_OUTPUTS_SIN[i][1];
c += EXCEPT_OUTPUTS_COS[i][1];
break;
case FE_DOWNWARD:
s += x_sign ? EXCEPT_OUTPUTS_SIN[i][1] : EXCEPT_OUTPUTS_SIN[i][2];
c += EXCEPT_OUTPUTS_COS[i][2];
break;
case FE_TONEAREST:
s += EXCEPT_OUTPUTS_SIN[i][3];
c += EXCEPT_OUTPUTS_COS[i][3];
break;
}
*sinp = x_sign ? -FPBits(s).get_val() : FPBits(s).get_val();
*cosp = FPBits(c).get_val();
return;
}
}
// Combine the results with the sine and cosine of sum formulas:
// sin(x) = sin((k + y)*pi/32)
// = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
// = sin_y * cos_k + (1 + cosm1_y) * sin_k
// = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
// cos(x) = cos((k + y)*pi/32)
// = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
// = cosm1_y * cos_k + sin_y * sin_k
// = (cosm1_y * cos_k + cos_k) + sin_y * sin_k
double sin_k, cos_k, sin_y, cosm1_y;
sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
*sinp = static_cast<float>(fputil::multiply_add(
sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
*cosp = static_cast<float>(fputil::multiply_add(
sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k)));
}
} // namespace __llvm_libc
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