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//===-- Single-precision tan 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/tanf.h"
#include "sincosf_utils.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/nearest_integer.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 cases for tanf.
constexpr size_t N_EXCEPTS = 6;
constexpr fputil::ExceptValues<float, N_EXCEPTS> TANF_EXCEPTS{{
// (inputs, RZ output, RU offset, RD offset, RN offset)
// x = 0x1.ada6aap27, tan(x) = 0x1.e80304p-3 (RZ)
{0x4d56d355, 0x3e740182, 1, 0, 0},
// x = 0x1.862064p33, tan(x) = -0x1.8dee56p-3 (RZ)
{0x50431032, 0xbe46f72b, 0, 1, 1},
// x = 0x1.af61dap48, tan(x) = 0x1.60d1c6p-2 (RZ)
{0x57d7b0ed, 0x3eb068e3, 1, 0, 1},
// x = 0x1.0088bcp52, tan(x) = 0x1.ca1edp0 (RZ)
{0x5980445e, 0x3fe50f68, 1, 0, 0},
// x = 0x1.f90dfcp72, tan(x) = 0x1.597f9cp-1 (RZ)
{0x63fc86fe, 0x3f2cbfce, 1, 0, 0},
// x = 0x1.a6ce12p86, tan(x) = -0x1.c5612ep-1 (RZ)
{0x6ad36709, 0xbf62b097, 0, 1, 0},
}};
LLVM_LIBC_FUNCTION(float, tanf, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
bool x_sign = xbits.uintval() >> 31;
uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
// |x| < pi/32
if (LIBC_UNLIKELY(x_abs <= 0x3dc9'0fdbU)) {
double xd = static_cast<double>(x);
// |x| < 0x1.0p-12f
if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
if (LIBC_UNLIKELY(x_abs == 0U)) {
// For signed zeros.
return x;
}
// When |x| < 2^-12, the relative error of the approximation tan(x) ~ x
// is:
// |tan(x) - x| / |tan(x)| < |x^3| / (3|x|)
// = x^2 / 3
// < 2^-25
// < epsilon(1)/2.
// So the correctly rounded values of tan(x) are:
// = x + sign(x)*eps(x) if rounding mode = FE_UPWARD and x is positive,
// or (rounding mode = FE_DOWNWARD and x is
// negative),
// = x otherwise.
// To simplify the rounding decision and make it more efficient, we use
// fma(x, 2^-25, x) 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)
return fputil::multiply_add(x, 0x1.0p-25f, x);
#else
return static_cast<float>(fputil::multiply_add(xd, 0x1.0p-25, xd));
#endif // LIBC_TARGET_CPU_HAS_FMA
}
// |x| < pi/32
double xsq = xd * xd;
// Degree-9 minimax odd polynomial of tan(x) generated by Sollya with:
// > P = fpminimax(tan(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/32]);
double result =
fputil::polyeval(xsq, 1.0, 0x1.555555553d022p-2, 0x1.111111ce442c1p-3,
0x1.ba180a6bbdecdp-5, 0x1.69c0a88a0b71fp-6);
return static_cast<float>(xd * result);
}
// Check for exceptional values
if (LIBC_UNLIKELY(x_abs == 0x3f8a1f62U)) {
// |x| = 0x1.143ec4p0
float sign = x_sign ? -1.0f : 1.0f;
// volatile is used to prevent compiler (gcc) from optimizing the
// computation, making the results incorrect in different rounding modes.
volatile float tmp = 0x1.ddf9f4p0f;
tmp = fputil::multiply_add(sign, tmp, sign * 0x1.1p-24f);
return tmp;
}
// |x| > 0x1.ada6a8p+27f
if (LIBC_UNLIKELY(x_abs > 0x4d56'd354U)) {
// 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);
}
return x + FPBits::build_quiet_nan(0);
}
// Other large exceptional values
if (auto r = TANF_EXCEPTS.lookup_odd(x_abs, x_sign);
LIBC_UNLIKELY(r.has_value()))
return r.value();
}
// For |x| >= pi/32, we use the definition of tan(x) function:
// tan(x) = sin(x) / cos(x)
// The we follow the same computations of sin(x) and cos(x) as sinf, cosf,
// and sincosf.
double xd = static_cast<double>(x);
double sin_k, cos_k, sin_y, cosm1_y;
sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
// tan(x) = sin(x) / cos(x)
// = (sin_y * cos_k + cos_y * sin_k) / (cos_y * cos_k - sin_y * sin_k)
using fputil::multiply_add;
return static_cast<float>(
multiply_add(sin_y, cos_k, multiply_add(cosm1_y, sin_k, sin_k)) /
multiply_add(sin_y, -sin_k, multiply_add(cosm1_y, cos_k, cos_k)));
}
} // namespace __llvm_libc
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