| //===-- Single-precision sin 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/sinf.h" |
| #include "sincosf_utils.h" |
| #include "src/__support/FPUtil/BasicOperations.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/common.h" |
| |
| #include <errno.h> |
| |
| #if defined(LIBC_TARGET_HAS_FMA) |
| #include "range_reduction_fma.h" |
| // using namespace __llvm_libc::fma; |
| using __llvm_libc::fma::N_EXCEPTS; |
| using __llvm_libc::fma::SinfExcepts; |
| #else |
| #include "range_reduction.h" |
| // using namespace __llvm_libc::generic; |
| using __llvm_libc::generic::N_EXCEPTS; |
| using __llvm_libc::generic::SinfExcepts; |
| #endif |
| |
| namespace __llvm_libc { |
| |
| LLVM_LIBC_FUNCTION(float, sinf, (float x)) { |
| using FPBits = typename fputil::FPBits<float>; |
| FPBits xbits(x); |
| |
| uint32_t x_u = xbits.uintval(); |
| uint32_t x_abs = x_u & 0x7fff'ffffU; |
| double xd = static_cast<double>(x); |
| |
| // Range reduction: |
| // For |x| > pi/16, we perform range reduction as follows: |
| // Find k and y such that: |
| // x = (k + y) * pi/16 |
| // k is an integer |
| // |y| < 0.5 |
| // For small range (|x| < 2^46 when FMA instructions are available, 2^22 |
| // otherwise), this is done by performing: |
| // k = round(x * 16/pi) |
| // y = x * 16/pi - k |
| // For large range, we will omit all the higher parts of 16/pi such that the |
| // least significant bits of their full products with x are larger than 31, |
| // since sin((k + y + 32*i) * pi/16) = sin(x + i * 2pi) = sin(x). |
| // |
| // When FMA instructions are not available, we store the digits of 16/pi in |
| // chunks of 28-bit precision. This will make sure that the products: |
| // x * SIXTEEN_OVER_PI_28[i] are all exact. |
| // When FMA instructions are available, we simply store the digits of 16/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 of sum |
| // formula: |
| // sin(x) = sin((k + y)*pi/16) |
| // = sin(y*pi/16) * cos(k*pi/16) + cos(y*pi/16) * sin(k*pi/16) |
| // The values of sin(k*pi/16) and cos(k*pi/16) for k = 0..31 are precomputed |
| // and stored using a vector of 32 doubles. Sin(y*pi/16) and cos(y*pi/16) are |
| // computed using degree-7 and degree-8 minimax polynomials generated by |
| // Sollya respectively. |
| |
| // |x| <= pi/16 |
| if (unlikely(x_abs <= 0x3e49'0fdbU)) { |
| |
| // |x| < 0x1.d12ed2p-12f |
| if (unlikely(x_abs < 0x39e8'9769U)) { |
| if (unlikely(x_abs == 0U)) { |
| // For signed zeros. |
| return x; |
| } |
| // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x |
| // is: |
| // |sin(x) - x| / |sin(x)| < |x^3| / (6|x|) |
| // = x^2 / 6 |
| // < 2^-25 |
| // < epsilon(1)/2. |
| // So the correctly rounded values of sin(x) are: |
| // = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO, |
| // or (rounding mode = FE_UPWARD and x is |
| // negative), |
| // = x otherwise. |
| // To simplify the rounding decision and make it more efficient, we use |
| // fma(x, -2^-25, x) instead. |
| // An exhaustive test shows that this formula work correctly for all |
| // rounding modes up to |x| < 0x1.c555dep-11f. |
| // 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_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_HAS_FMA |
| } |
| |
| // |x| < pi/16. |
| double xsq = xd * xd; |
| |
| // Degree-9 polynomial approximation: |
| // sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9 |
| // = x (1 + a_3 x^2 + ... + a_9 x^8) |
| // = x * P(x^2) |
| // generated by Sollya with the following commands: |
| // > display = hexadecimal; |
| // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]); |
| double result = |
| fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7, |
| -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19); |
| return xd * result; |
| } |
| |
| using ExceptChecker = typename fputil::ExceptionChecker<float, N_EXCEPTS>; |
| { |
| float result; |
| if (ExceptChecker::check_odd_func(SinfExcepts, x_abs, xbits.get_sign(), |
| result)) |
| return result; |
| } |
| |
| if (unlikely(x_abs >= 0x7f80'0000U)) { |
| if (x_abs == 0x7f80'0000U) { |
| errno = EDOM; |
| fputil::set_except(FE_INVALID); |
| } |
| return x + |
| FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1)); |
| } |
| |
| // Combine the results with the sine of sum formula: |
| // sin(x) = sin((k + y)*pi/16) |
| // = sin(y*pi/16) * cos(k*pi/16) + cos(y*pi/16) * sin(k*pi/16) |
| // = sin_y * cos_k + (1 + cosm1_y) * sin_k |
| // = sin_y * cos_k + (cosm1_y * sin_k + sin_k) |
| double sin_k, cos_k, sin_y, cosm1_y; |
| |
| sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y); |
| |
| return fputil::multiply_add(sin_y, cos_k, |
| fputil::multiply_add(cosm1_y, sin_k, sin_k)); |
| } |
| |
| } // namespace __llvm_libc |