| //===-- 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/common.h" |
| |
| #include <errno.h> |
| |
| namespace __llvm_libc { |
| |
| // Exceptional values |
| static constexpr int N_EXCEPTS = 10; |
| |
| static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = { |
| 0x3b5637f5, // x = 0x1.ac6feap-9 |
| 0x3fa7832a, // x = 0x1.4f0654p0 |
| 0x46199998, // x = 0x1.33333p13 |
| 0x55325019, // x = 0x1.64a032p43 |
| 0x55cafb2a, // x = 0x1.95f654p44 |
| 0x5922aa80, // x = 0x1.4555p51 |
| 0x5aa4542c, // x = 0x1.48a858p54 |
| 0x5f18b878, // x = 0x1.3170fp63 |
| 0x6115cb11, // x = 0x1.2b9622p67 |
| 0x7beef5ef, // x = 0x1.ddebdep120 |
| }; |
| |
| static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = { |
| {0x3b5637dc, 1, 0, 0}, // x = 0x1.ac6feap-9, sin(x) = 0x1.ac6fb8p-9 (RZ) |
| {0x3f7741b5, 1, 0, 1}, // x = 0x1.4f0654p0, sin(x) = 0x1.ee836ap-1 (RZ) |
| {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) |
| {0xbf7e7a16, 0, 1, 1}, // x = 0x1.95f654p44, sin(x) = -0x1.fcf42cp-1 (RZ) |
| {0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ) |
| {0x3f5f5646, 1, 0, 0}, // x = 0x1.48a858p54, sin(x) = 0x1.beac8cp-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] = { |
| {0x3f7fffa6, 1, 0, 0}, // x = 0x1.ac6feap-9, cos(x) = 0x1.ffff4cp-1 (RZ) |
| {0x3e84aabf, 1, 0, 1}, // x = 0x1.4f0654p0, cos(x) = 0x1.09557ep-2 (RZ) |
| {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) |
| {0x3ddf11f3, 1, 0, 1}, // x = 0x1.95f654p44, cos(x) = 0x1.be23e6p-4 (RZ) |
| {0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ) |
| {0x3efa40a4, 1, 0, 0}, // x = 0x1.48a858p54, cos(x) = 0x1.f48148p-2 (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) |
| }; |
| |
| // Fast sincosf implementation. Worst-case ULP is 0.5607, maximum relative |
| // error is 0.5303 * 2^-23. A single-step range reduction is used for |
| // small values. Large inputs have their range reduced using fast integer |
| // arithmetic. |
| 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| > 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), and |
| // cos((k + y + 32*i) * pi/16) = cos(x + i * 2pi) = cos(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 and cosine |
| // of sum formulas: |
| // sin(x) = sin((k + y)*pi/16) |
| // = sin(y*pi/16) * cos(k*pi/16) + cos(y*pi/16) * sin(k*pi/16) |
| // cos(x) = cos((k + y)*pi/16) |
| // = cos(y*pi/16) * cos(k*pi/16) - sin(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| < 0x1.0p-12f |
| if (unlikely(x_abs < 0x3980'0000U)) { |
| if (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_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_HAS_FMA |
| return; |
| } |
| |
| // x is inf or nan. |
| if (unlikely(x_abs >= 0x7f80'0000U)) { |
| if (x_abs == 0x7f80'0000U) { |
| errno = EDOM; |
| fputil::set_except(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 (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::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/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) |
| // cos(x) = cos((k + y)*pi/16) |
| // = cos(y*pi/16) * cos(k*pi/16) - sin(y*pi/16) * sin(k*pi/16) |
| // = 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 = fputil::multiply_add(sin_y, cos_k, |
| fputil::multiply_add(cosm1_y, sin_k, sin_k)); |
| *cosp = fputil::multiply_add(sin_y, -sin_k, |
| fputil::multiply_add(cosm1_y, cos_k, cos_k)); |
| } |
| |
| } // namespace __llvm_libc |
