libc/src/math/generic/sincosf.cpp

   1 //===-- Single-precision sincos function ----------------------------------===//
   2 //
   3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
   4 // See https://llvm.org/LICENSE.txt for license information.
   5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
   6 //
   7 //===----------------------------------------------------------------------===//
   8
   9 #include "src/math/sincosf.h"
  10 #include "sincosf_utils.h"
  11 #include "src/__support/FPUtil/FEnvImpl.h"
  12 #include "src/__support/FPUtil/FPBits.h"
  13 #include "src/__support/FPUtil/multiply_add.h"
  14 #include "src/__support/common.h"
  15 #include "src/__support/macros/optimization.h"            // LIBC_UNLIKELY
  16 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
  17
  18 #include <errno.h>
  19
  20 namespace __llvm_libc {
  21
  22 // Exceptional values
  23 static constexpr int N_EXCEPTS = 6;
  24
  25 static constexpr uint32_t EXCEPT_INPUTS[N_EXCEPTS] = {
  26     0x46199998, // x = 0x1.33333p13   x
  27     0x55325019, // x = 0x1.64a032p43  x
  28     0x5922aa80, // x = 0x1.4555p51    x
  29     0x5f18b878, // x = 0x1.3170fp63   x
  30     0x6115cb11, // x = 0x1.2b9622p67  x
  31     0x7beef5ef, // x = 0x1.ddebdep120 x
  32 };
  33
  34 static constexpr uint32_t EXCEPT_OUTPUTS_SIN[N_EXCEPTS][4] = {
  35     {0xbeb1fa5d, 0, 1, 0}, // x = 0x1.33333p13, sin(x) = -0x1.63f4bap-2 (RZ)
  36     {0xbf171adf, 0, 1, 1}, // x = 0x1.64a032p43, sin(x) = -0x1.2e35bep-1 (RZ)
  37     {0xbf587521, 0, 1, 1}, // x = 0x1.4555p51, sin(x) = -0x1.b0ea42p-1 (RZ)
  38     {0x3dad60f6, 1, 0, 1}, // x = 0x1.3170fp63, sin(x) = 0x1.5ac1ecp-4 (RZ)
  39     {0xbe7cc1e0, 0, 1, 1}, // x = 0x1.2b9622p67, sin(x) = -0x1.f983cp-3 (RZ)
  40     {0xbf587d1b, 0, 1, 1}, // x = 0x1.ddebdep120, sin(x) = -0x1.b0fa36p-1 (RZ)
  41 };
  42
  43 static constexpr uint32_t EXCEPT_OUTPUTS_COS[N_EXCEPTS][4] = {
  44     {0xbf70090b, 0, 1, 0}, // x = 0x1.33333p13, cos(x) = -0x1.e01216p-1 (RZ)
  45     {0x3f4ea5d2, 1, 0, 0}, // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ)
  46     {0x3f08aebe, 1, 0, 1}, // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ)
  47     {0x3f7f14bb, 1, 0, 0}, // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ)
  48     {0x3f78142e, 1, 0, 1}, // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ)
  49     {0x3f08a21c, 1, 0, 0}, // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ)
  50 };
  51
  52 LLVM_LIBC_FUNCTION(void, sincosf, (float x, float *sinp, float *cosp)) {
  53   using FPBits = typename fputil::FPBits<float>;
  54   FPBits xbits(x);
  55
  56   uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
  57   double xd = static_cast<double>(x);
  58
  59   // Range reduction:
  60   // For |x| >= 2^-12, we perform range reduction as follows:
  61   // Find k and y such that:
  62   //   x = (k + y) * pi/32
  63   //   k is an integer
  64   //   |y| < 0.5
  65   // For small range (|x| < 2^45 when FMA instructions are available, 2^22
  66   // otherwise), this is done by performing:
  67   //   k = round(x * 32/pi)
  68   //   y = x * 32/pi - k
  69   // For large range, we will omit all the higher parts of 32/pi such that the
  70   // least significant bits of their full products with x are larger than 63,
  71   // since:
  72   //     sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x), and
  73   //     cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x).
  74   //
  75   // When FMA instructions are not available, we store the digits of 32/pi in
  76   // chunks of 28-bit precision.  This will make sure that the products:
  77   //   x * THIRTYTWO_OVER_PI_28[i] are all exact.
  78   // When FMA instructions are available, we simply store the digits of326/pi in
  79   // chunks of doubles (53-bit of precision).
  80   // So when multiplying by the largest values of single precision, the
  81   // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80.  By the
  82   // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
  83   // us more than 40 bits of accuracy. For the worst-case estimation of range
  84   // reduction, see for instances:
  85   //   Elementary Functions by J-M. Muller, Chapter 11,
  86   //   Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
  87   //   Chapter 10.2.
  88   //
  89   // Once k and y are computed, we then deduce the answer by the sine and cosine
  90   // of sum formulas:
  91   //   sin(x) = sin((k + y)*pi/32)
  92   //          = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
  93   //   cos(x) = cos((k + y)*pi/32)
  94   //          = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
  95   // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed
  96   // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
  97   // computed using degree-7 and degree-6 minimax polynomials generated by
  98   // Sollya respectively.
  99
 100   // |x| < 0x1.0p-12f
 101   if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) {
 102     if (LIBC_UNLIKELY(x_abs == 0U)) {
 103       // For signed zeros.
 104       *sinp = x;
 105       *cosp = 1.0f;
 106       return;
 107     }
 108     // When |x| < 2^-12, the relative errors of the approximations
 109     //   sin(x) ~ x, cos(x) ~ 1
 110     // are:
 111     //   |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
 112     //                           = x^2 / 6
 113     //                           < 2^-25
 114     //                           < epsilon(1)/2.
 115     //   |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2.
 116     // So the correctly rounded values of sin(x) and cos(x) are:
 117     //   sin(x) = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
 118     //                        or (rounding mode = FE_UPWARD and x is
 119     //                        negative),
 120     //          = x otherwise.
 121     //   cos(x) = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD,
 122     //          = 1 otherwise.
 123     // To simplify the rounding decision and make it more efficient and to
 124     // prevent compiler to perform constant folding, we use
 125     //   sin(x) = fma(x, -2^-25, x),
 126     //   cos(x) = fma(x*0.5f, -x, 1)
 127     // instead.
 128     // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
 129     // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
 130     // |x| < 2^-125. For targets without FMA instructions, we simply use
 131     // double for intermediate results as it is more efficient than using an
 132     // emulated version of FMA.
 133 #if defined(LIBC_TARGET_CPU_HAS_FMA)
 134     *sinp = fputil::multiply_add(x, -0x1.0p-25f, x);
 135     *cosp = fputil::multiply_add(FPBits(x_abs).get_val(), -0x1.0p-25f, 1.0f);
 136 #else
 137     *sinp = static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
 138     *cosp = static_cast<float>(fputil::multiply_add(
 139         static_cast<double>(FPBits(x_abs).get_val()), -0x1.0p-25, 1.0));
 140 #endif // LIBC_TARGET_CPU_HAS_FMA
 141     return;
 142   }
 143
 144   // x is inf or nan.
 145   if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
 146     if (x_abs == 0x7f80'0000U) {
 147       fputil::set_errno_if_required(EDOM);
 148       fputil::raise_except_if_required(FE_INVALID);
 149     }
 150     *sinp =
 151         x + FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1));
 152     *cosp = *sinp;
 153     return;
 154   }
 155
 156   // Check exceptional values.
 157   for (int i = 0; i < N_EXCEPTS; ++i) {
 158     if (LIBC_UNLIKELY(x_abs == EXCEPT_INPUTS[i])) {
 159       uint32_t s = EXCEPT_OUTPUTS_SIN[i][0]; // FE_TOWARDZERO
 160       uint32_t c = EXCEPT_OUTPUTS_COS[i][0]; // FE_TOWARDZERO
 161       bool x_sign = x < 0;
 162       switch (fputil::get_round()) {
 163       case FE_UPWARD:
 164         s += x_sign ? EXCEPT_OUTPUTS_SIN[i][2] : EXCEPT_OUTPUTS_SIN[i][1];
 165         c += EXCEPT_OUTPUTS_COS[i][1];
 166         break;
 167       case FE_DOWNWARD:
 168         s += x_sign ? EXCEPT_OUTPUTS_SIN[i][1] : EXCEPT_OUTPUTS_SIN[i][2];
 169         c += EXCEPT_OUTPUTS_COS[i][2];
 170         break;
 171       case FE_TONEAREST:
 172         s += EXCEPT_OUTPUTS_SIN[i][3];
 173         c += EXCEPT_OUTPUTS_COS[i][3];
 174         break;
 175       }
 176       *sinp = x_sign ? -FPBits(s).get_val() : FPBits(s).get_val();
 177       *cosp = FPBits(c).get_val();
 178
 179       return;
 180     }
 181   }
 182
 183   // Combine the results with the sine and cosine of sum formulas:
 184   //   sin(x) = sin((k + y)*pi/32)
 185   //          = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
 186   //          = sin_y * cos_k + (1 + cosm1_y) * sin_k
 187   //          = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
 188   //   cos(x) = cos((k + y)*pi/32)
 189   //          = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32)
 190   //          = cosm1_y * cos_k + sin_y * sin_k
 191   //          = (cosm1_y * cos_k + cos_k) + sin_y * sin_k
 192   double sin_k, cos_k, sin_y, cosm1_y;
 193
 194   sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
 195
 196   *sinp = static_cast<float>(fputil::multiply_add(
 197       sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
 198   *cosp = static_cast<float>(fputil::multiply_add(
 199       sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k)));
 200 }
 201
 202 } // namespace __llvm_libc