src/math/exp2.c

   1 /*
   2  * Double-precision 2^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 Shift __exp_data.exp2_shift
  15 #define T __exp_data.tab
  16 #define C1 __exp_data.exp2_poly[0]
  17 #define C2 __exp_data.exp2_poly[1]
  18 #define C3 __exp_data.exp2_poly[2]
  19 #define C4 __exp_data.exp2_poly[3]
  20 #define C5 __exp_data.exp2_poly[4]
  21
  22 /* Handle cases that may overflow or underflow when computing the result that
  23    is scale*(1+TMP) without intermediate rounding.  The bit representation of
  24    scale is in SBITS, however it has a computed exponent that may have
  25    overflown into the sign bit so that needs to be adjusted before using it as
  26    a double.  (int32_t)KI is the k used in the argument reduction and exponent
  27    adjustment of scale, positive k here means the result may overflow and
  28    negative k means the result may underflow.  */
  29 static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
  30 {
  31         double_t scale, y;
  32
  33         if ((ki & 0x80000000) == 0) {
  34                 /* k > 0, the exponent of scale might have overflowed by 1.  */
  35                 sbits -= 1ull << 52;
  36                 scale = asdouble(sbits);
  37                 y = 2 * (scale + scale * tmp);
  38                 return eval_as_double(y);
  39         }
  40         /* k < 0, need special care in the subnormal range.  */
  41         sbits += 1022ull << 52;
  42         scale = asdouble(sbits);
  43         y = scale + scale * tmp;
  44         if (y < 1.0) {
  45                 /* Round y to the right precision before scaling it into the subnormal
  46                    range to avoid double rounding that can cause 0.5+E/2 ulp error where
  47                    E is the worst-case ulp error outside the subnormal range.  So this
  48                    is only useful if the goal is better than 1 ulp worst-case error.  */
  49                 double_t hi, lo;
  50                 lo = scale - y + scale * tmp;
  51                 hi = 1.0 + y;
  52                 lo = 1.0 - hi + y + lo;
  53                 y = eval_as_double(hi + lo) - 1.0;
  54                 /* Avoid -0.0 with downward rounding.  */
  55                 if (WANT_ROUNDING && y == 0.0)
  56                         y = 0.0;
  57                 /* The underflow exception needs to be signaled explicitly.  */
  58                 fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
  59         }
  60         y = 0x1p-1022 * y;
  61         return eval_as_double(y);
  62 }
  63
  64 /* Top 12 bits of a double (sign and exponent bits).  */
  65 static inline uint32_t top12(double x)
  66 {
  67         return asuint64(x) >> 52;
  68 }
  69
  70 double exp2(double x)
  71 {
  72         uint32_t abstop;
  73         uint64_t ki, idx, top, sbits;
  74         double_t kd, r, r2, scale, tail, tmp;
  75
  76         abstop = top12(x) & 0x7ff;
  77         if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
  78                 if (abstop - top12(0x1p-54) >= 0x80000000)
  79                         /* Avoid spurious underflow for tiny x.  */
  80                         /* Note: 0 is common input.  */
  81                         return WANT_ROUNDING ? 1.0 + x : 1.0;
  82                 if (abstop >= top12(1024.0)) {
  83                         if (asuint64(x) == asuint64(-INFINITY))
  84                                 return 0.0;
  85                         if (abstop >= top12(INFINITY))
  86                                 return 1.0 + x;
  87                         if (!(asuint64(x) >> 63))
  88                                 return __math_oflow(0);
  89                         else if (asuint64(x) >= asuint64(-1075.0))
  90                                 return __math_uflow(0);
  91                 }
  92                 if (2 * asuint64(x) > 2 * asuint64(928.0))
  93                         /* Large x is special cased below.  */
  94                         abstop = 0;
  95         }
  96
  97         /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)].  */
  98         /* x = k/N + r, with int k and r in [-1/2N, 1/2N].  */
  99         kd = eval_as_double(x + Shift);
 100         ki = asuint64(kd); /* k.  */
 101         kd -= Shift; /* k/N for int k.  */
 102         r = x - kd;
 103         /* 2^(k/N) ~= scale * (1 + tail).  */
 104         idx = 2 * (ki % N);
 105         top = ki << (52 - EXP_TABLE_BITS);
 106         tail = asdouble(T[idx]);
 107         /* This is only a valid scale when -1023*N < k < 1024*N.  */
 108         sbits = T[idx + 1] + top;
 109         /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1).  */
 110         /* Evaluation is optimized assuming superscalar pipelined execution.  */
 111         r2 = r * r;
 112         /* Without fma the worst case error is 0.5/N ulp larger.  */
 113         /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp.  */
 114         tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 115         if (predict_false(abstop == 0))
 116                 return specialcase(tmp, sbits, ki);
 117         scale = asdouble(sbits);
 118         /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
 119            is no spurious underflow here even without fma.  */
 120         return eval_as_double(scale + scale * tmp);
 121 }