libc/src/math/generic/sinf.cpp

   1 //===-- Single-precision sin 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/sinf.h"
  10 #include "sincosf_utils.h"
  11 #include "src/__support/FPUtil/BasicOperations.h"
  12 #include "src/__support/FPUtil/FEnvImpl.h"
  13 #include "src/__support/FPUtil/FPBits.h"
  14 #include "src/__support/FPUtil/PolyEval.h"
  15 #include "src/__support/FPUtil/multiply_add.h"
  16 #include "src/__support/common.h"
  17 #include "src/__support/macros/optimization.h"            // LIBC_UNLIKELY
  18 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
  19
  20 #include <errno.h>
  21
  22 #if defined(LIBC_TARGET_CPU_HAS_FMA)
  23 #include "range_reduction_fma.h"
  24 #else
  25 #include "range_reduction.h"
  26 #endif
  27
  28 namespace __llvm_libc {
  29
  30 LLVM_LIBC_FUNCTION(float, sinf, (float x)) {
  31   using FPBits = typename fputil::FPBits<float>;
  32   FPBits xbits(x);
  33
  34   uint32_t x_u = xbits.uintval();
  35   uint32_t x_abs = x_u & 0x7fff'ffffU;
  36   double xd = static_cast<double>(x);
  37
  38   // Range reduction:
  39   // For |x| > pi/32, we perform range reduction as follows:
  40   // Find k and y such that:
  41   //   x = (k + y) * pi/32
  42   //   k is an integer
  43   //   |y| < 0.5
  44   // For small range (|x| < 2^45 when FMA instructions are available, 2^22
  45   // otherwise), this is done by performing:
  46   //   k = round(x * 32/pi)
  47   //   y = x * 32/pi - k
  48   // For large range, we will omit all the higher parts of 32/pi such that the
  49   // least significant bits of their full products with x are larger than 63,
  50   // since sin((k + y + 64*i) * pi/32) = sin(x + i * 2pi) = sin(x).
  51   //
  52   // When FMA instructions are not available, we store the digits of 32/pi in
  53   // chunks of 28-bit precision.  This will make sure that the products:
  54   //   x * THIRTYTWO_OVER_PI_28[i] are all exact.
  55   // When FMA instructions are available, we simply store the digits of 32/pi in
  56   // chunks of doubles (53-bit of precision).
  57   // So when multiplying by the largest values of single precision, the
  58   // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80.  By the
  59   // worst-case analysis of range reduction, |y| >= 2^-38, so this should give
  60   // us more than 40 bits of accuracy. For the worst-case estimation of range
  61   // reduction, see for instances:
  62   //   Elementary Functions by J-M. Muller, Chapter 11,
  63   //   Handbook of Floating-Point Arithmetic by J-M. Muller et. al.,
  64   //   Chapter 10.2.
  65   //
  66   // Once k and y are computed, we then deduce the answer by the sine of sum
  67   // formula:
  68   //   sin(x) = sin((k + y)*pi/32)
  69   //          = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
  70   // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..31 are precomputed
  71   // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are
  72   // computed using degree-7 and degree-6 minimax polynomials generated by
  73   // Sollya respectively.
  74
  75   // |x| <= pi/16
  76   if (LIBC_UNLIKELY(x_abs <= 0x3e49'0fdbU)) {
  77
  78     // |x| < 0x1.d12ed2p-12f
  79     if (LIBC_UNLIKELY(x_abs < 0x39e8'9769U)) {
  80       if (LIBC_UNLIKELY(x_abs == 0U)) {
  81         // For signed zeros.
  82         return x;
  83       }
  84       // When |x| < 2^-12, the relative error of the approximation sin(x) ~ x
  85       // is:
  86       //   |sin(x) - x| / |sin(x)| < |x^3| / (6|x|)
  87       //                           = x^2 / 6
  88       //                           < 2^-25
  89       //                           < epsilon(1)/2.
  90       // So the correctly rounded values of sin(x) are:
  91       //   = x - sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
  92       //                        or (rounding mode = FE_UPWARD and x is
  93       //                        negative),
  94       //   = x otherwise.
  95       // To simplify the rounding decision and make it more efficient, we use
  96       //   fma(x, -2^-25, x) instead.
  97       // An exhaustive test shows that this formula work correctly for all
  98       // rounding modes up to |x| < 0x1.c555dep-11f.
  99       // Note: to use the formula x - 2^-25*x to decide the correct rounding, we
 100       // do need fma(x, -2^-25, x) to prevent underflow caused by -2^-25*x when
 101       // |x| < 2^-125. For targets without FMA instructions, we simply use
 102       // double for intermediate results as it is more efficient than using an
 103       // emulated version of FMA.
 104 #if defined(LIBC_TARGET_CPU_HAS_FMA)
 105       return fputil::multiply_add(x, -0x1.0p-25f, x);
 106 #else
 107       return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, xd));
 108 #endif // LIBC_TARGET_CPU_HAS_FMA
 109     }
 110
 111     // |x| < pi/16.
 112     double xsq = xd * xd;
 113
 114     // Degree-9 polynomial approximation:
 115     //   sin(x) ~ x + a_3 x^3 + a_5 x^5 + a_7 x^7 + a_9 x^9
 116     //          = x (1 + a_3 x^2 + ... + a_9 x^8)
 117     //          = x * P(x^2)
 118     // generated by Sollya with the following commands:
 119     // > display = hexadecimal;
 120     // > Q = fpminimax(sin(x)/x, [|0, 2, 4, 6, 8|], [|1, D...|], [0, pi/16]);
 121     double result =
 122         fputil::polyeval(xsq, 1.0, -0x1.55555555554c6p-3, 0x1.1111111085e65p-7,
 123                          -0x1.a019f70fb4d4fp-13, 0x1.718d179815e74p-19);
 124     return static_cast<float>(xd * result);
 125   }
 126
 127   if (LIBC_UNLIKELY(x_abs == 0x4619'9998U)) { // x = 0x1.33333p13
 128     float r = -0x1.63f4bap-2f;
 129     int rounding = fputil::get_round();
 130     bool sign = xbits.get_sign();
 131     if ((rounding == FE_DOWNWARD && !sign) || (rounding == FE_UPWARD && sign))
 132       r = -0x1.63f4bcp-2f;
 133     return xbits.get_sign() ? -r : r;
 134   }
 135
 136   if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
 137     if (x_abs == 0x7f80'0000U) {
 138       fputil::set_errno_if_required(EDOM);
 139       fputil::raise_except_if_required(FE_INVALID);
 140     }
 141     return x + FPBits::build_quiet_nan(0);
 142   }
 143
 144   // Combine the results with the sine of sum formula:
 145   //   sin(x) = sin((k + y)*pi/32)
 146   //          = sin(y*pi/32) * cos(k*pi/32) + cos(y*pi/32) * sin(k*pi/32)
 147   //          = sin_y * cos_k + (1 + cosm1_y) * sin_k
 148   //          = sin_y * cos_k + (cosm1_y * sin_k + sin_k)
 149   double sin_k, cos_k, sin_y, cosm1_y;
 150
 151   sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
 152
 153   return static_cast<float>(fputil::multiply_add(
 154       sin_y, cos_k, fputil::multiply_add(cosm1_y, sin_k, sin_k)));
 155 }
 156
 157 } // namespace __llvm_libc