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//===-- Double-precision cos 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/cos.h"
#include "hdr/errno_macros.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/dyadic_float.h"
#include "src/__support/FPUtil/except_value_utils.h"
#include "src/__support/common.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 "src/math/generic/sincos_eval.h"
#ifdef LIBC_TARGET_CPU_HAS_FMA
#include "range_reduction_double_fma.h"
using LIBC_NAMESPACE::fma::FAST_PASS_EXPONENT;
using LIBC_NAMESPACE::fma::ONE_TWENTY_EIGHT_OVER_PI;
using LIBC_NAMESPACE::fma::range_reduction_small;
using LIBC_NAMESPACE::fma::SIN_K_PI_OVER_128;
LIBC_INLINE constexpr bool NO_FMA = false;
#else
#include "range_reduction_double_nofma.h"
using LIBC_NAMESPACE::nofma::FAST_PASS_EXPONENT;
using LIBC_NAMESPACE::nofma::ONE_TWENTY_EIGHT_OVER_PI;
using LIBC_NAMESPACE::nofma::range_reduction_small;
using LIBC_NAMESPACE::nofma::SIN_K_PI_OVER_128;
LIBC_INLINE constexpr bool NO_FMA = true;
#endif // LIBC_TARGET_CPU_HAS_FMA
// TODO: We might be able to improve the performance of large range reduction of
// non-FMA targets further by operating directly on 25-bit chunks of 128/pi and
// pre-split SIN_K_PI_OVER_128, but that might double the memory footprint of
// those lookup table.
#include "range_reduction_double_common.h"
#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0)
#define LIBC_MATH_COS_SKIP_ACCURATE_PASS
#endif
namespace LIBC_NAMESPACE_DECL {
using DoubleDouble = fputil::DoubleDouble;
using Float128 = typename fputil::DyadicFloat<128>;
LLVM_LIBC_FUNCTION(double, cos, (double x)) {
using FPBits = typename fputil::FPBits<double>;
FPBits xbits(x);
uint16_t x_e = xbits.get_biased_exponent();
DoubleDouble y;
unsigned k;
generic::LargeRangeReduction<NO_FMA> range_reduction_large{};
// |x| < 2^32 (with FMA) or |x| < 2^23 (w/o FMA)
if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT)) {
// |x| < 2^-27
if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 27)) {
// Signed zeros.
if (LIBC_UNLIKELY(x == 0.0))
return 1.0;
// For |x| < 2^-27, |cos(x) - 1| < |x|^2/2 < 2^-54 = ulp(1 - 2^-53)/2.
return fputil::round_result_slightly_down(1.0);
}
// // Small range reduction.
k = range_reduction_small(x, y);
} else {
// Inf or NaN
if (LIBC_UNLIKELY(x_e > 2 * FPBits::EXP_BIAS)) {
// sin(+-Inf) = NaN
if (xbits.get_mantissa() == 0) {
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
}
return x + FPBits::quiet_nan().get_val();
}
// Large range reduction.
k = range_reduction_large.compute_high_part(x);
y = range_reduction_large.fast();
}
DoubleDouble sin_y, cos_y;
generic::sincos_eval(y, sin_y, cos_y);
// Look up sin(k * pi/128) and cos(k * pi/128)
// Memory saving versions:
// Use 128-entry table instead:
// DoubleDouble sin_k = SIN_K_PI_OVER_128[k & 127];
// uint64_t sin_s = static_cast<uint64_t>((k + 128) & 128) << (63 - 7);
// sin_k.hi = FPBits(FPBits(sin_k.hi).uintval() ^ sin_s).get_val();
// sin_k.lo = FPBits(FPBits(sin_k.hi).uintval() ^ sin_s).get_val();
// DoubleDouble cos_k = SIN_K_PI_OVER_128[(k + 64) & 127];
// uint64_t cos_s = static_cast<uint64_t>((k + 64) & 128) << (63 - 7);
// cos_k.hi = FPBits(FPBits(cos_k.hi).uintval() ^ cos_s).get_val();
// cos_k.lo = FPBits(FPBits(cos_k.hi).uintval() ^ cos_s).get_val();
// Use 64-entry table instead:
// auto get_idx_dd = [](unsigned kk) -> DoubleDouble {
// unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);
// DoubleDouble ans = SIN_K_PI_OVER_128[idx];
// if (kk & 128) {
// ans.hi = -ans.hi;
// ans.lo = -ans.lo;
// }
// return ans;
// };
// DoubleDouble sin_k = get_idx_dd(k + 128);
// DoubleDouble cos_k = get_idx_dd(k + 64);
// Fast look up version, but needs 256-entry table.
// -sin(k * pi/128) = sin((k + 128) * pi/128)
// cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
DoubleDouble msin_k = SIN_K_PI_OVER_128[(k + 128) & 255];
DoubleDouble cos_k = SIN_K_PI_OVER_128[(k + 64) & 255];
// After range reduction, k = round(x * 128 / pi) and y = x - k * (pi / 128).
// So k is an integer and -pi / 256 <= y <= pi / 256.
// Then cos(x) = cos((k * pi/128 + y)
// = cos(y) * cos(k*pi/128) - sin(y) * sin(k*pi/128)
DoubleDouble cos_k_cos_y = fputil::quick_mult<NO_FMA>(cos_y, cos_k);
DoubleDouble msin_k_sin_y = fputil::quick_mult<NO_FMA>(sin_y, msin_k);
DoubleDouble rr = fputil::exact_add<false>(cos_k_cos_y.hi, msin_k_sin_y.hi);
rr.lo += msin_k_sin_y.lo + cos_k_cos_y.lo;
#ifdef LIBC_MATH_COS_SKIP_ACCURATE_PASS
return rr.hi + rr.lo;
#else
// Accurate test and pass for correctly rounded implementation.
#ifdef LIBC_TARGET_CPU_HAS_FMA
constexpr double ERR = 0x1.0p-70;
#else
// TODO: Improve non-FMA fast pass accuracy.
constexpr double ERR = 0x1.0p-66;
#endif // LIBC_TARGET_CPU_HAS_FMA
double rlp = rr.lo + ERR;
double rlm = rr.lo - ERR;
double r_upper = rr.hi + rlp; // (rr.lo + ERR);
double r_lower = rr.hi + rlm; // (rr.lo - ERR);
// Ziv's rounding test.
if (LIBC_LIKELY(r_upper == r_lower))
return r_upper;
Float128 u_f128, sin_u, cos_u;
if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT))
u_f128 = generic::range_reduction_small_f128(x);
else
u_f128 = range_reduction_large.accurate();
generic::sincos_eval(u_f128, sin_u, cos_u);
auto get_sin_k = [](unsigned kk) -> Float128 {
unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);
Float128 ans = generic::SIN_K_PI_OVER_128_F128[idx];
if (kk & 128)
ans.sign = Sign::NEG;
return ans;
};
// -sin(k * pi/128) = sin((k + 128) * pi/128)
// cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
Float128 msin_k_f128 = get_sin_k(k + 128);
Float128 cos_k_f128 = get_sin_k(k + 64);
// cos(x) = cos((k * pi/128 + u)
// = cos(u) * cos(k*pi/128) - sin(u) * sin(k*pi/128)
Float128 r = fputil::quick_add(fputil::quick_mul(cos_k_f128, cos_u),
fputil::quick_mul(msin_k_f128, sin_u));
// TODO: Add assertion if Ziv's accuracy tests fail in debug mode.
// https://github.com/llvm/llvm-project/issues/96452.
return static_cast<double>(r);
#endif // !LIBC_MATH_COS_SKIP_ACCURATE_PASS
}
} // namespace LIBC_NAMESPACE_DECL
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