src/math/exp.c

   1 /*
   2  * Double-precision e^x function.
   3  *
   4  * Copyright (c) 2018, Arm Limited.
   5  * SPDX-License-Identifier: MIT
   6  */
   7
   8 #include <math.h>
   9 #include <stdint.h>
  10 #include "libm.h"
  11 #include "exp_data.h"
  12
  13 #define N (1 << EXP_TABLE_BITS)
  14 #define InvLn2N __exp_data.invln2N
  15 #define NegLn2hiN __exp_data.negln2hiN
  16 #define NegLn2loN __exp_data.negln2loN
  17 #define Shift __exp_data.shift
  18 #define T __exp_data.tab
  19 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
  20 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
  21 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
  22 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
  23
  24 /* Handle cases that may overflow or underflow when computing the result that
  25    is scale*(1+TMP) without intermediate rounding.  The bit representation of
  26    scale is in SBITS, however it has a computed exponent that may have
  27    overflown into the sign bit so that needs to be adjusted before using it as
  28    a double.  (int32_t)KI is the k used in the argument reduction and exponent
  29    adjustment of scale, positive k here means the result may overflow and
  30    negative k means the result may underflow.  */
  31 static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
  32 {
  33         double_t scale, y;
  34
  35         if ((ki & 0x80000000) == 0) {
  36                 /* k > 0, the exponent of scale might have overflowed by <= 460.  */
  37                 sbits -= 1009ull << 52;
  38                 scale = asdouble(sbits);
  39                 y = 0x1p1009 * (scale + scale * tmp);
  40                 return eval_as_double(y);
  41         }
  42         /* k < 0, need special care in the subnormal range.  */
  43         sbits += 1022ull << 52;
  44         scale = asdouble(sbits);
  45         y = scale + scale * tmp;
  46         if (y < 1.0) {
  47                 /* Round y to the right precision before scaling it into the subnormal
  48                  range to avoid double rounding that can cause 0.5+E/2 ulp error where
  49                  E is the worst-case ulp error outside the subnormal range.  So this
  50                  is only useful if the goal is better than 1 ulp worst-case error.  */
  51                 double_t hi, lo;
  52                 lo = scale - y + scale * tmp;
  53                 hi = 1.0 + y;
  54                 lo = 1.0 - hi + y + lo;
  55                 y = eval_as_double(hi + lo) - 1.0;
  56                 /* Avoid -0.0 with downward rounding.  */
  57                 if (WANT_ROUNDING && y == 0.0)
  58                         y = 0.0;
  59                 /* The underflow exception needs to be signaled explicitly.  */
  60                 fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
  61         }
  62         y = 0x1p-1022 * y;
  63         return eval_as_double(y);
  64 }
  65
  66 /* Top 12 bits of a double (sign and exponent bits).  */
  67 static inline uint32_t top12(double x)
  68 {
  69         return asuint64(x) >> 52;
  70 }
  71
  72 double exp(double x)
  73 {
  74         uint32_t abstop;
  75         uint64_t ki, idx, top, sbits;
  76         double_t kd, z, r, r2, scale, tail, tmp;
  77
  78         abstop = top12(x) & 0x7ff;
  79         if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
  80                 if (abstop - top12(0x1p-54) >= 0x80000000)
  81                         /* Avoid spurious underflow for tiny x.  */
  82                         /* Note: 0 is common input.  */
  83                         return WANT_ROUNDING ? 1.0 + x : 1.0;
  84                 if (abstop >= top12(1024.0)) {
  85                         if (asuint64(x) == asuint64(-INFINITY))
  86                                 return 0.0;
  87                         if (abstop >= top12(INFINITY))
  88                                 return 1.0 + x;
  89                         if (asuint64(x) >> 63)
  90                                 return __math_uflow(0);
  91                         else
  92                                 return __math_oflow(0);
  93                 }
  94                 /* Large x is special cased below.  */
  95                 abstop = 0;
  96         }
  97
  98         /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
  99         /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
 100         z = InvLn2N * x;
 101 #if TOINT_INTRINSICS
 102         kd = roundtoint(z);
 103         ki = converttoint(z);
 104 #elif EXP_USE_TOINT_NARROW
 105         /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
 106         kd = eval_as_double(z + Shift);
 107         ki = asuint64(kd) >> 16;
 108         kd = (double_t)(int32_t)ki;
 109 #else
 110         /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
 111         kd = eval_as_double(z + Shift);
 112         ki = asuint64(kd);
 113         kd -= Shift;
 114 #endif
 115         r = x + kd * NegLn2hiN + kd * NegLn2loN;
 116         /* 2^(k/N) ~= scale * (1 + tail).  */
 117         idx = 2 * (ki % N);
 118         top = ki << (52 - EXP_TABLE_BITS);
 119         tail = asdouble(T[idx]);
 120         /* This is only a valid scale when -1023*N < k < 1024*N.  */
 121         sbits = T[idx + 1] + top;
 122         /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
 123         /* Evaluation is optimized assuming superscalar pipelined execution.  */
 124         r2 = r * r;
 125         /* Without fma the worst case error is 0.25/N ulp larger.  */
 126         /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
 127         tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 128         if (predict_false(abstop == 0))
 129                 return specialcase(tmp, sbits, ki);
 130         scale = asdouble(sbits);
 131         /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
 132            is no spurious underflow here even without fma.  */
 133         return eval_as_double(scale + scale * tmp);
 134 }