sysdeps/ieee754/dbl-64/e_exp.c

   1 /* Double-precision e^x function.
   2    Copyright (C) 2018-2019 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-svid-compat.h>
  24 #include <shlib-compat.h>
  25 #include <libm-alias-double.h>
  26 #include "math_config.h"
  27
  28 #define N (1 << EXP_TABLE_BITS)
  29 #define InvLn2N __exp_data.invln2N
  30 #define NegLn2hiN __exp_data.negln2hiN
  31 #define NegLn2loN __exp_data.negln2loN
  32 #define Shift __exp_data.shift
  33 #define T __exp_data.tab
  34 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
  35 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
  36 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
  37 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
  38
  39 /* Handle cases that may overflow or underflow when computing the result that
  40    is scale*(1+TMP) without intermediate rounding.  The bit representation of
  41    scale is in SBITS, however it has a computed exponent that may have
  42    overflown into the sign bit so that needs to be adjusted before using it as
  43    a double.  (int32_t)KI is the k used in the argument reduction and exponent
  44    adjustment of scale, positive k here means the result may overflow and
  45    negative k means the result may underflow.  */
  46 static inline double
  47 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
  48 {
  49   double_t scale, y;
  50
  51   if ((ki & 0x80000000) == 0)
  52     {
  53       /* k > 0, the exponent of scale might have overflowed by <= 460.  */
  54       sbits -= 1009ull << 52;
  55       scale = asdouble (sbits);
  56       y = 0x1p1009 * (scale + scale * tmp);
  57       return check_oflow (y);
  58     }
  59   /* k < 0, need special care in the subnormal range.  */
  60   sbits += 1022ull << 52;
  61   scale = asdouble (sbits);
  62   y = scale + scale * tmp;
  63   if (y < 1.0)
  64     {
  65       /* Round y to the right precision before scaling it into the subnormal
  66          range to avoid double rounding that can cause 0.5+E/2 ulp error where
  67          E is the worst-case ulp error outside the subnormal range.  So this
  68          is only useful if the goal is better than 1 ulp worst-case error.  */
  69       double_t hi, lo;
  70       lo = scale - y + scale * tmp;
  71       hi = 1.0 + y;
  72       lo = 1.0 - hi + y + lo;
  73       y = math_narrow_eval (hi + lo) - 1.0;
  74       /* Avoid -0.0 with downward rounding.  */
  75       if (WANT_ROUNDING && y == 0.0)
  76         y = 0.0;
  77       /* The underflow exception needs to be signaled explicitly.  */
  78       math_force_eval (math_opt_barrier (0x1p-1022) * 0x1p-1022);
  79     }
  80   y = 0x1p-1022 * y;
  81   return check_uflow (y);
  82 }
  83
  84 /* Top 12 bits of a double (sign and exponent bits).  */
  85 static inline uint32_t
  86 top12 (double x)
  87 {
  88   return asuint64 (x) >> 52;
  89 }
  90
  91 #ifndef SECTION
  92 # define SECTION
  93 #endif
  94
  95 double
  96 SECTION
  97 __exp (double x)
  98 {
  99   uint32_t abstop;
 100   uint64_t ki, idx, top, sbits;
 101   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
 102   double_t kd, z, r, r2, scale, tail, tmp;
 103
 104   abstop = top12 (x) & 0x7ff;
 105   if (__glibc_unlikely (abstop - top12 (0x1p-54)
 106                         >= top12 (512.0) - top12 (0x1p-54)))
 107     {
 108       if (abstop - top12 (0x1p-54) >= 0x80000000)
 109         /* Avoid spurious underflow for tiny x.  */
 110         /* Note: 0 is common input.  */
 111         return WANT_ROUNDING ? 1.0 + x : 1.0;
 112       if (abstop >= top12 (1024.0))
 113         {
 114           if (asuint64 (x) == asuint64 (-INFINITY))
 115             return 0.0;
 116           if (abstop >= top12 (INFINITY))
 117             return 1.0 + x;
 118           if (asuint64 (x) >> 63)
 119             return __math_uflow (0);
 120           else
 121             return __math_oflow (0);
 122         }
 123       /* Large x is special cased below.  */
 124       abstop = 0;
 125     }
 126
 127   /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
 128   /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
 129   z = InvLn2N * x;
 130 #if TOINT_INTRINSICS
 131   kd = roundtoint (z);
 132   ki = converttoint (z);
 133 #else
 134   /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
 135   kd = math_narrow_eval (z + Shift);
 136   ki = asuint64 (kd);
 137   kd -= Shift;
 138 #endif
 139   r = x + kd * NegLn2hiN + kd * NegLn2loN;
 140   /* 2^(k/N) ~= scale * (1 + tail).  */
 141   idx = 2 * (ki % N);
 142   top = ki << (52 - EXP_TABLE_BITS);
 143   tail = asdouble (T[idx]);
 144   /* This is only a valid scale when -1023*N < k < 1024*N.  */
 145   sbits = T[idx + 1] + top;
 146   /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
 147   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 148   r2 = r * r;
 149   /* Without fma the worst case error is 0.25/N ulp larger.  */
 150   /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
 151   tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 152   if (__glibc_unlikely (abstop == 0))
 153     return specialcase (tmp, sbits, ki);
 154   scale = asdouble (sbits);
 155   /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-739, so there
 156      is no spurious underflow here even without fma.  */
 157   return scale + scale * tmp;
 158 }
 159 #ifndef __exp
 160 hidden_def (__exp)
 161 strong_alias (__exp, __ieee754_exp)
 162 strong_alias (__exp, __exp_finite)
 163 # if LIBM_SVID_COMPAT
 164 versioned_symbol (libm, __exp, exp, GLIBC_2_29);
 165 libm_alias_double_other (__exp, exp)
 166 # else
 167 libm_alias_double (__exp, exp)
 168 # endif
 169 #endif