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