libc/src/math/generic/expm1f.cpp

   1 //===-- Single-precision e^x - 1 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/expm1f.h"
  10 #include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
  11 #include "src/__support/FPUtil/BasicOperations.h"
  12 #include "src/__support/FPUtil/FEnvImpl.h"
  13 #include "src/__support/FPUtil/FMA.h"
  14 #include "src/__support/FPUtil/FPBits.h"
  15 #include "src/__support/FPUtil/PolyEval.h"
  16 #include "src/__support/FPUtil/multiply_add.h"
  17 #include "src/__support/FPUtil/nearest_integer.h"
  18 #include "src/__support/FPUtil/rounding_mode.h"
  19 #include "src/__support/common.h"
  20 #include "src/__support/macros/optimization.h"            // LIBC_UNLIKELY
  21 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
  22
  23 #include <errno.h>
  24
  25 namespace LIBC_NAMESPACE {
  26
  27 LLVM_LIBC_FUNCTION(float, expm1f, (float x)) {
  28   using FPBits = typename fputil::FPBits<float>;
  29   FPBits xbits(x);
  30
  31   uint32_t x_u = xbits.uintval();
  32   uint32_t x_abs = x_u & 0x7fff'ffffU;
  33
  34   // Exceptional value
  35   if (LIBC_UNLIKELY(x_u == 0x3e35'bec5U)) { // x = 0x1.6b7d8ap-3f
  36     int round_mode = fputil::quick_get_round();
  37     if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD)
  38       return 0x1.8dbe64p-3f;
  39     return 0x1.8dbe62p-3f;
  40   }
  41
  42 #if !defined(LIBC_TARGET_CPU_HAS_FMA)
  43   if (LIBC_UNLIKELY(x_u == 0xbdc1'c6cbU)) { // x = -0x1.838d96p-4f
  44     int round_mode = fputil::quick_get_round();
  45     if (round_mode == FE_TONEAREST || round_mode == FE_DOWNWARD)
  46       return -0x1.71c884p-4f;
  47     return -0x1.71c882p-4f;
  48   }
  49 #endif // LIBC_TARGET_CPU_HAS_FMA
  50
  51   // When |x| > 25*log(2), or nan
  52   if (LIBC_UNLIKELY(x_abs >= 0x418a'a123U)) {
  53     // x < log(2^-25)
  54     if (xbits.get_sign()) {
  55       // exp(-Inf) = 0
  56       if (xbits.is_inf())
  57         return -1.0f;
  58       // exp(nan) = nan
  59       if (xbits.is_nan())
  60         return x;
  61       int round_mode = fputil::quick_get_round();
  62       if (round_mode == FE_UPWARD || round_mode == FE_TOWARDZERO)
  63         return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f
  64       return -1.0f;
  65     } else {
  66       // x >= 89 or nan
  67       if (xbits.uintval() >= 0x42b2'0000) {
  68         if (xbits.uintval() < 0x7f80'0000U) {
  69           int rounding = fputil::quick_get_round();
  70           if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
  71             return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
  72
  73           fputil::set_errno_if_required(ERANGE);
  74           fputil::raise_except_if_required(FE_OVERFLOW);
  75         }
  76         return x + static_cast<float>(FPBits::inf());
  77       }
  78     }
  79   }
  80
  81   // |x| < 2^-4
  82   if (x_abs < 0x3d80'0000U) {
  83     // |x| < 2^-25
  84     if (x_abs < 0x3300'0000U) {
  85       // x = -0.0f
  86       if (LIBC_UNLIKELY(xbits.uintval() == 0x8000'0000U))
  87         return x;
  88         // When |x| < 2^-25, the relative error of the approximation e^x - 1 ~ x
  89         // is:
  90         //   |(e^x - 1) - x| / |e^x - 1| < |x^2| / |x|
  91         //                               = |x|
  92         //                               < 2^-25
  93         //                               < epsilon(1)/2.
  94         // So the correctly rounded values of expm1(x) are:
  95         //   = x + eps(x) if rounding mode = FE_UPWARD,
  96         //                   or (rounding mode = FE_TOWARDZERO and x is
  97         //                   negative),
  98         //   = x otherwise.
  99         // To simplify the rounding decision and make it more efficient, we use
 100         //   fma(x, x, x) ~ x + x^2 instead.
 101         // Note: to use the formula x + x^2 to decide the correct rounding, we
 102         // do need fma(x, x, x) to prevent underflow caused by x*x when |x| <
 103         // 2^-76. For targets without FMA instructions, we simply use double for
 104         // intermediate results as it is more efficient than using an emulated
 105         // version of FMA.
 106 #if defined(LIBC_TARGET_CPU_HAS_FMA)
 107       return fputil::fma(x, x, x);
 108 #else
 109       double xd = x;
 110       return static_cast<float>(fputil::multiply_add(xd, xd, xd));
 111 #endif // LIBC_TARGET_CPU_HAS_FMA
 112     }
 113
 114     constexpr double COEFFS[] = {0x1p-1,
 115                                  0x1.55555555557ddp-3,
 116                                  0x1.55555555552fap-5,
 117                                  0x1.111110fcd58b7p-7,
 118                                  0x1.6c16c1717660bp-10,
 119                                  0x1.a0241f0006d62p-13,
 120                                  0x1.a01e3f8d3c06p-16};
 121
 122     // 2^-25 <= |x| < 2^-4
 123     double xd = static_cast<double>(x);
 124     double xsq = xd * xd;
 125     // Degree-8 minimax polynomial generated by Sollya with:
 126     // > display = hexadecimal;
 127     // > P = fpminimax((expm1(x) - x)/x^2, 6, [|D...|], [-2^-4, 2^-4]);
 128
 129     double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);
 130     double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);
 131     double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);
 132
 133     double r = fputil::polyeval(xsq, c0, c1, c2, COEFFS[6]);
 134     return static_cast<float>(fputil::multiply_add(r, xsq, xd));
 135   }
 136
 137   // For -18 < x < 89, to compute expm1(x), we perform the following range
 138   // reduction: find hi, mid, lo such that:
 139   //   x = hi + mid + lo, in which
 140   //     hi is an integer,
 141   //     mid * 2^7 is an integer
 142   //     -2^(-8) <= lo < 2^-8.
 143   // In particular,
 144   //   hi + mid = round(x * 2^7) * 2^(-7).
 145   // Then,
 146   //   expm1(x) = exp(hi + mid + lo) - 1 = exp(hi) * exp(mid) * exp(lo) - 1.
 147   // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2
 148   // respectively.  exp(lo) is computed using a degree-4 minimax polynomial
 149   // generated by Sollya.
 150
 151   // x_hi = hi + mid.
 152   float kf = fputil::nearest_integer(x * 0x1.0p7f);
 153   int x_hi = static_cast<int>(kf);
 154   // Subtract (hi + mid) from x to get lo.
 155   double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x));
 156   x_hi += 104 << 7;
 157   // hi = x_hi >> 7
 158   double exp_hi = EXP_M1[x_hi >> 7];
 159   // lo = x_hi & 0x0000'007fU;
 160   double exp_mid = EXP_M2[x_hi & 0x7f];
 161   double exp_hi_mid = exp_hi * exp_mid;
 162   // Degree-4 minimax polynomial generated by Sollya with the following
 163   // commands:
 164   //   > display = hexadecimal;
 165   //   > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]);
 166   //   > Q;
 167   double exp_lo =
 168       fputil::polyeval(xd, 0x1.0p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1,
 169                        0x1.555566668e5e7p-3, 0x1.55555555ef243p-5);
 170   return static_cast<float>(fputil::multiply_add(exp_hi_mid, exp_lo, -1.0));
 171 }
 172
 173 } // namespace LIBC_NAMESPACE