sysdeps/ieee754/dbl-64/e_exp.c

   1 /* Double-precision e^x function.
   2    Copyright (C) 2018 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4
   5    The GNU C Library is free software; you can redistribute it and/or
   6    modify it under the terms of the GNU Lesser General Public
   7    License as published by the Free Software Foundation; either
   8    version 2.1 of the License, or (at your option) any later version.
   9
  10    The GNU C Library is distributed in the hope that it will be useful,
  11    but WITHOUT ANY WARRANTY; without even the implied warranty of
  12    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  13    Lesser General Public License for more details.
  14
  15    You should have received a copy of the GNU Lesser General Public
  16    License along with the GNU C Library; if not, see
  17    <http://www.gnu.org/licenses/>.  */
  18
  19 #include <math.h>
  20 #include <stdint.h>
  21 #include <math-barriers.h>
  22 #include <math-narrow-eval.h>
  23 #include "math_config.h"
  24
  25 #define N (1 << EXP_TABLE_BITS)
  26 #define InvLn2N __exp_data.invln2N
  27 #define NegLn2hiN __exp_data.negln2hiN
  28 #define NegLn2loN __exp_data.negln2loN
  29 #define Shift __exp_data.shift
  30 #define T __exp_data.tab
  31 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
  32 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
  33 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
  34 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
  35
  36 /* Handle cases that may overflow or underflow when computing the result that
  37    is scale*(1+TMP) without intermediate rounding.  The bit representation of
  38    scale is in SBITS, however it has a computed exponent that may have
  39    overflown into the sign bit so that needs to be adjusted before using it as
  40    a double.  (int32_t)KI is the k used in the argument reduction and exponent
  41    adjustment of scale, positive k here means the result may overflow and
  42    negative k means the result may underflow.  */
  43 static inline double
  44 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
  45 {
  46   double_t scale, y;
  47
  48   if ((ki & 0x80000000) == 0)
  49     {
  50       /* k > 0, the exponent of scale might have overflowed by <= 460.  */
  51       sbits -= 1009ull << 52;
  52       scale = asdouble (sbits);
  53       y = 0x1p1009 * (scale + scale * tmp);
  54       return check_oflow (y);
  55     }
  56   /* k < 0, need special care in the subnormal range.  */
  57   sbits += 1022ull << 52;
  58   scale = asdouble (sbits);
  59   y = scale + scale * tmp;
  60   if (y < 1.0)
  61     {
  62       /* Round y to the right precision before scaling it into the subnormal
  63          range to avoid double rounding that can cause 0.5+E/2 ulp error where
  64          E is the worst-case ulp error outside the subnormal range.  So this
  65          is only useful if the goal is better than 1 ulp worst-case error.  */
  66       double_t hi, lo;
  67       lo = scale - y + scale * tmp;
  68       hi = 1.0 + y;
  69       lo = 1.0 - hi + y + lo;
  70       y = math_narrow_eval (hi + lo) - 1.0;
  71       /* Avoid -0.0 with downward rounding.  */
  72       if (WANT_ROUNDING && y == 0.0)
  73         y = 0.0;
  74       /* The underflow exception needs to be signaled explicitly.  */
  75       math_force_eval (math_opt_barrier (0x1p-1022) * 0x1p-1022);
  76     }
  77   y = 0x1p-1022 * y;
  78   return check_uflow (y);
  79 }
  80
  81 /* Top 12 bits of a double (sign and exponent bits).  */
  82 static inline uint32_t
  83 top12 (double x)
  84 {
  85   return asuint64 (x) >> 52;
  86 }
  87
  88 #ifndef SECTION
  89 # define SECTION
  90 #endif
  91
  92 double
  93 SECTION
  94 __ieee754_exp (double x)
  95 {
  96   uint32_t abstop;
  97   uint64_t ki, idx, top, sbits;
  98   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  99   double_t kd, z, r, r2, scale, tail, tmp;
 100
 101   abstop = top12 (x) & 0x7ff;
 102   if (__glibc_unlikely (abstop - top12 (0x1p-54)
 103                         >= top12 (512.0) - top12 (0x1p-54)))
 104     {
 105       if (abstop - top12 (0x1p-54) >= 0x80000000)
 106         /* Avoid spurious underflow for tiny x.  */
 107         /* Note: 0 is common input.  */
 108         return WANT_ROUNDING ? 1.0 + x : 1.0;
 109       if (abstop >= top12 (1024.0))
 110         {
 111           if (asuint64 (x) == asuint64 (-INFINITY))
 112             return 0.0;
 113           if (abstop >= top12 (INFINITY))
 114             return 1.0 + x;
 115           if (asuint64 (x) >> 63)
 116             return __math_uflow (0);
 117           else
 118             return __math_oflow (0);
 119         }
 120       /* Large x is special cased below.  */
 121       abstop = 0;
 122     }
 123
 124   /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
 125   /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
 126   z = InvLn2N * x;
 127 #if TOINT_INTRINSICS
 128   kd = roundtoint (z);
 129   ki = converttoint (z);
 130 #else
 131   /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
 132   kd = math_narrow_eval (z + Shift);
 133   ki = asuint64 (kd);
 134   kd -= Shift;
 135 #endif
 136   r = x + kd * NegLn2hiN + kd * NegLn2loN;
 137   /* 2^(k/N) ~= scale * (1 + tail).  */
 138   idx = 2 * (ki % N);
 139   top = ki << (52 - EXP_TABLE_BITS);
 140   tail = asdouble (T[idx]);
 141   /* This is only a valid scale when -1023*N < k < 1024*N.  */
 142   sbits = T[idx + 1] + top;
 143   /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
 144   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 145   r2 = r * r;
 146   /* Without fma the worst case error is 0.25/N ulp larger.  */
 147   /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
 148   tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 149   if (__glibc_unlikely (abstop == 0))
 150     return specialcase (tmp, sbits, ki);
 151   scale = asdouble (sbits);
 152   /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-739, so there
 153      is no spurious underflow here even without fma.  */
 154   return scale + scale * tmp;
 155 }
 156 #ifndef __ieee754_exp
 157 strong_alias (__ieee754_exp, __exp_finite)
 158 #endif