libc/src/math/generic/expf.cpp

   1 //===-- Single-precision e^x 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/expf.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/FPBits.h"
  14 #include "src/__support/FPUtil/PolyEval.h"
  15 #include "src/__support/FPUtil/multiply_add.h"
  16 #include "src/__support/FPUtil/nearest_integer.h"
  17 #include "src/__support/FPUtil/rounding_mode.h"
  18 #include "src/__support/common.h"
  19 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
  20
  21 #include <errno.h>
  22
  23 namespace __llvm_libc {
  24
  25 LLVM_LIBC_FUNCTION(float, expf, (float x)) {
  26   using FPBits = typename fputil::FPBits<float>;
  27   FPBits xbits(x);
  28
  29   uint32_t x_u = xbits.uintval();
  30   uint32_t x_abs = x_u & 0x7fff'ffffU;
  31
  32   // Exceptional values
  33   if (LIBC_UNLIKELY(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f
  34     return 0x1.108a58p-66f - x * 0x1.0p-95f;
  35   }
  36
  37   // When |x| >= 89, |x| < 2^-25, or x is nan
  38   if (LIBC_UNLIKELY(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) {
  39     // |x| < 2^-25
  40     if (xbits.get_unbiased_exponent() <= 101) {
  41       return 1.0f + x;
  42     }
  43
  44     // When x < log(2^-150) or nan
  45     if (xbits.uintval() >= 0xc2cf'f1b5U) {
  46       // exp(-Inf) = 0
  47       if (xbits.is_inf())
  48         return 0.0f;
  49       // exp(nan) = nan
  50       if (xbits.is_nan())
  51         return x;
  52       if (fputil::fenv_is_round_up())
  53         return static_cast<float>(FPBits(FPBits::MIN_SUBNORMAL));
  54       fputil::set_errno_if_required(ERANGE);
  55       fputil::raise_except_if_required(FE_UNDERFLOW);
  56       return 0.0f;
  57     }
  58     // x >= 89 or nan
  59     if (!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000)) {
  60       // x is finite
  61       if (xbits.uintval() < 0x7f80'0000U) {
  62         int rounding = fputil::quick_get_round();
  63         if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
  64           return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
  65
  66         fputil::set_errno_if_required(ERANGE);
  67         fputil::raise_except_if_required(FE_OVERFLOW);
  68       }
  69       // x is +inf or nan
  70       return x + static_cast<float>(FPBits::inf());
  71     }
  72   }
  73   // For -104 < x < 89, to compute exp(x), we perform the following range
  74   // reduction: find hi, mid, lo such that:
  75   //   x = hi + mid + lo, in which
  76   //     hi is an integer,
  77   //     mid * 2^7 is an integer
  78   //     -2^(-8) <= lo < 2^-8.
  79   // In particular,
  80   //   hi + mid = round(x * 2^7) * 2^(-7).
  81   // Then,
  82   //   exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo).
  83   // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2
  84   // respectively.  exp(lo) is computed using a degree-4 minimax polynomial
  85   // generated by Sollya.
  86
  87   // x_hi = (hi + mid) * 2^7 = round(x * 2^7).
  88   float kf = fputil::nearest_integer(x * 0x1.0p7f);
  89   // Subtract (hi + mid) from x to get lo.
  90   double xd = static_cast<double>(fputil::multiply_add(kf, -0x1.0p-7f, x));
  91   int x_hi = static_cast<int>(kf);
  92   x_hi += 104 << 7;
  93   // hi = x_hi >> 7
  94   double exp_hi = EXP_M1[x_hi >> 7];
  95   // mid * 2^7 = x_hi & 0x0000'007fU;
  96   double exp_mid = EXP_M2[x_hi & 0x7f];
  97   // Degree-4 minimax polynomial generated by Sollya with the following
  98   // commands:
  99   //   > display = hexadecimal;
 100   //   > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]);
 101   //   > Q;
 102   double exp_lo =
 103       fputil::polyeval(xd, 0x1p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1,
 104                        0x1.555566668e5e7p-3, 0x1.55555555ef243p-5);
 105   return static_cast<float>(exp_hi * exp_mid * exp_lo);
 106 }
 107
 108 } // namespace __llvm_libc