libc/AOR_v20.02/math/exp.c

   1 /*
   2  * Double-precision e^x function.
   3  *
   4  * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
   5  * See https://llvm.org/LICENSE.txt for license information.
   6  * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
   7  */
   8
   9 #include <float.h>
  10 #include <math.h>
  11 #include <stdint.h>
  12 #include "math_config.h"
  13
  14 #define N (1 << EXP_TABLE_BITS)
  15 #define InvLn2N __exp_data.invln2N
  16 #define NegLn2hiN __exp_data.negln2hiN
  17 #define NegLn2loN __exp_data.negln2loN
  18 #define Shift __exp_data.shift
  19 #define T __exp_data.tab
  20 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
  21 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
  22 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
  23 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
  24 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
  25
  26 /* Handle cases that may overflow or underflow when computing the result that
  27    is scale*(1+TMP) without intermediate rounding.  The bit representation of
  28    scale is in SBITS, however it has a computed exponent that may have
  29    overflown into the sign bit so that needs to be adjusted before using it as
  30    a double.  (int32_t)KI is the k used in the argument reduction and exponent
  31    adjustment of scale, positive k here means the result may overflow and
  32    negative k means the result may underflow.  */
  33 static inline double
  34 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
  35 {
  36   double_t scale, y;
  37
  38   if ((ki & 0x80000000) == 0)
  39     {
  40       /* k > 0, the exponent of scale might have overflowed by <= 460.  */
  41       sbits -= 1009ull << 52;
  42       scale = asdouble (sbits);
  43       y = 0x1p1009 * (scale + scale * tmp);
  44       return check_oflow (eval_as_double (y));
  45     }
  46   /* k < 0, need special care in the subnormal range.  */
  47   sbits += 1022ull << 52;
  48   scale = asdouble (sbits);
  49   y = scale + scale * tmp;
  50   if (y < 1.0)
  51     {
  52       /* Round y to the right precision before scaling it into the subnormal
  53          range to avoid double rounding that can cause 0.5+E/2 ulp error where
  54          E is the worst-case ulp error outside the subnormal range.  So this
  55          is only useful if the goal is better than 1 ulp worst-case error.  */
  56       double_t hi, lo;
  57       lo = scale - y + scale * tmp;
  58       hi = 1.0 + y;
  59       lo = 1.0 - hi + y + lo;
  60       y = eval_as_double (hi + lo) - 1.0;
  61       /* Avoid -0.0 with downward rounding.  */
  62       if (WANT_ROUNDING && y == 0.0)
  63         y = 0.0;
  64       /* The underflow exception needs to be signaled explicitly.  */
  65       force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
  66     }
  67   y = 0x1p-1022 * y;
  68   return check_uflow (eval_as_double (y));
  69 }
  70
  71 /* Top 12 bits of a double (sign and exponent bits).  */
  72 static inline uint32_t
  73 top12 (double x)
  74 {
  75   return asuint64 (x) >> 52;
  76 }
  77
  78 /* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
  79    If hastail is 0 then xtail is assumed to be 0 too.  */
  80 static inline double
  81 exp_inline (double x, double xtail, int hastail)
  82 {
  83   uint32_t abstop;
  84   uint64_t ki, idx, top, sbits;
  85   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  86   double_t kd, z, r, r2, scale, tail, tmp;
  87
  88   abstop = top12 (x) & 0x7ff;
  89   if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
  90     {
  91       if (abstop - top12 (0x1p-54) >= 0x80000000)
  92         /* Avoid spurious underflow for tiny x.  */
  93         /* Note: 0 is common input.  */
  94         return WANT_ROUNDING ? 1.0 + x : 1.0;
  95       if (abstop >= top12 (1024.0))
  96         {
  97           if (asuint64 (x) == asuint64 (-INFINITY))
  98             return 0.0;
  99           if (abstop >= top12 (INFINITY))
 100             return 1.0 + x;
 101           if (asuint64 (x) >> 63)
 102             return __math_uflow (0);
 103           else
 104             return __math_oflow (0);
 105         }
 106       /* Large x is special cased below.  */
 107       abstop = 0;
 108     }
 109
 110   /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
 111   /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
 112   z = InvLn2N * x;
 113 #if TOINT_INTRINSICS
 114   kd = roundtoint (z);
 115   ki = converttoint (z);
 116 #elif EXP_USE_TOINT_NARROW
 117   /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
 118   kd = eval_as_double (z + Shift);
 119   ki = asuint64 (kd) >> 16;
 120   kd = (double_t) (int32_t) ki;
 121 #else
 122   /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
 123   kd = eval_as_double (z + Shift);
 124   ki = asuint64 (kd);
 125   kd -= Shift;
 126 #endif
 127   r = x + kd * NegLn2hiN + kd * NegLn2loN;
 128   /* The code assumes 2^-200 < |xtail| < 2^-8/N.  */
 129   if (hastail)
 130     r += xtail;
 131   /* 2^(k/N) ~= scale * (1 + tail).  */
 132   idx = 2 * (ki % N);
 133   top = ki << (52 - EXP_TABLE_BITS);
 134   tail = asdouble (T[idx]);
 135   /* This is only a valid scale when -1023*N < k < 1024*N.  */
 136   sbits = T[idx + 1] + top;
 137   /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
 138   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 139   r2 = r * r;
 140   /* Without fma the worst case error is 0.25/N ulp larger.  */
 141   /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
 142 #if EXP_POLY_ORDER == 4
 143   tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
 144 #elif EXP_POLY_ORDER == 5
 145   tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 146 #elif EXP_POLY_ORDER == 6
 147   tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
 148 #endif
 149   if (unlikely (abstop == 0))
 150     return specialcase (tmp, sbits, ki);
 151   scale = asdouble (sbits);
 152   /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
 153      is no spurious underflow here even without fma.  */
 154   return eval_as_double (scale + scale * tmp);
 155 }
 156
 157 double
 158 exp (double x)
 159 {
 160   return exp_inline (x, 0, 0);
 161 }
 162
 163 /* May be useful for implementing pow where more than double
 164    precision input is needed.  */
 165 double
 166 __exp_dd (double x, double xtail)
 167 {
 168   return exp_inline (x, xtail, 1);
 169 }
 170 #if USE_GLIBC_ABI
 171 strong_alias (exp, __exp_finite)
 172 hidden_alias (exp, __ieee754_exp)
 173 hidden_alias (__exp_dd, __exp1)
 174 # if LDBL_MANT_DIG == 53
 175 long double expl (long double x) { return exp (x); }
 176 # endif
 177 #endif