libc/src/math/generic/exp2f.cpp

   1 //===-- Single-precision 2^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/exp2f.h"
  10 #include "src/__support/FPUtil/FEnvImpl.h"
  11 #include "src/__support/FPUtil/FPBits.h"
  12 #include "src/__support/FPUtil/PolyEval.h"
  13 #include "src/__support/FPUtil/multiply_add.h"
  14 #include "src/__support/FPUtil/nearest_integer.h"
  15 #include "src/__support/common.h"
  16 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
  17
  18 #include <errno.h>
  19
  20 #include "explogxf.h"
  21
  22 namespace __llvm_libc {
  23
  24 constexpr uint32_t EXVAL1 = 0x3b42'9d37U;
  25 constexpr uint32_t EXVAL2 = 0xbcf3'a937U;
  26 constexpr uint32_t EXVAL_MASK = EXVAL1 & EXVAL2;
  27
  28 LLVM_LIBC_FUNCTION(float, exp2f, (float x)) {
  29   using FPBits = typename fputil::FPBits<float>;
  30   FPBits xbits(x);
  31
  32   uint32_t x_u = xbits.uintval();
  33   uint32_t x_abs = x_u & 0x7fff'ffffU;
  34
  35   // |x| < 2^-25
  36   if (LIBC_UNLIKELY(x_abs <= 0x3280'0000U)) {
  37     return 1.0f + x;
  38   }
  39
  40   // // When |x| >= 128, or x is nan
  41   if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U)) {
  42
  43     // x >= 128
  44     if (!xbits.get_sign()) {
  45       // x is finite
  46       if (x_u < 0x7f80'0000U) {
  47         int rounding = fputil::get_round();
  48         if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
  49           return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
  50
  51         fputil::set_errno_if_required(ERANGE);
  52         fputil::raise_except_if_required(FE_OVERFLOW);
  53       }
  54       // x is +inf or nan
  55       return x + FPBits::inf().get_val();
  56     }
  57     // x <= -150
  58     if (x_u >= 0xc316'0000U) {
  59       // exp(-Inf) = 0
  60       if (xbits.is_inf())
  61         return 0.0f;
  62       // exp(nan) = nan
  63       if (xbits.is_nan())
  64         return x;
  65       if (fputil::get_round() == FE_UPWARD)
  66         return FPBits(FPBits::MIN_SUBNORMAL).get_val();
  67       if (x != 0.0f) {
  68         fputil::set_errno_if_required(ERANGE);
  69         fputil::raise_except_if_required(FE_UNDERFLOW);
  70       }
  71       return 0.0f;
  72     }
  73   }
  74
  75   // Check exceptional values.
  76   if (LIBC_UNLIKELY((x_u & EXVAL_MASK) == EXVAL_MASK)) {
  77     if (LIBC_UNLIKELY(x_u == EXVAL1)) { // x = 0x1.853a6ep-9f
  78       if (fputil::get_round() == FE_TONEAREST)
  79         return 0x1.00870ap+0f;
  80     } else if (LIBC_UNLIKELY(x_u == EXVAL2)) { // x = -0x1.e7526ep-6f
  81       if (fputil::get_round() == FE_TONEAREST)
  82         return 0x1.f58d62p-1f;
  83     }
  84   }
  85
  86   // For -150 < x < 128, to compute 2^x, we perform the following range
  87   // reduction: find hi, mid, lo such that:
  88   //   x = hi + mid + lo, in which
  89   //     hi is an integer,
  90   //     0 <= mid * 2^5 < 32 is an integer
  91   //     -2^(-6) <= lo <= 2^-6.
  92   // In particular,
  93   //   hi + mid = round(x * 2^5) * 2^(-5).
  94   // Then,
  95   //   2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
  96   // 2^mid is stored in the lookup table of 32 elements.
  97   // 2^lo is computed using a degree-5 minimax polynomial
  98   // generated by Sollya.
  99   // We perform 2^hi * 2^mid by simply add hi to the exponent field
 100   // of 2^mid.
 101
 102   // kf = (hi + mid) * 2^5 = round(x * 2^5)
 103   float kf = fputil::nearest_integer(x * 32.0f);
 104   // dx = lo = x - (hi + mid) = x - kf * 2^(-5)
 105   double dx = fputil::multiply_add(-0x1.0p-5f, kf, x);
 106
 107   int k = static_cast<int>(kf);
 108   // hi = floor(kf * 2^(-4))
 109   // exp_hi = shift hi to the exponent field of double precision.
 110   int64_t exp_hi = static_cast<int64_t>(k >> ExpBase::MID_BITS)
 111                    << fputil::FloatProperties<double>::MANTISSA_WIDTH;
 112   // mh = 2^hi * 2^mid
 113   // mh_bits = bit field of mh
 114   int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp_hi;
 115   double mh = fputil::FPBits<double>(uint64_t(mh_bits)).get_val();
 116
 117   // Degree-5 polynomial approximating (2^x - 1)/x generating by Sollya with:
 118   // > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-1/32. 1/32]);
 119   constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
 120                                 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
 121                                 0x1.5d88091198529p-10};
 122   double dx_sq = dx * dx;
 123   double c1 = fputil::multiply_add(dx, COEFFS[0], 1.0);
 124   double c2 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
 125   double c3 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
 126   double p = fputil::multiply_add(dx_sq, c3, c2);
 127   // 2^x = 2^(hi + mid + lo)
 128   //     = 2^(hi + mid) * 2^lo
 129   //     ~ mh * (1 + lo * P(lo))
 130   //     = mh + (mh*lo) * P(lo)
 131   return static_cast<float>(fputil::multiply_add(p, dx_sq * mh, c1 * mh));
 132 }
 133
 134 } // namespace __llvm_libc