libc/AOR_v20.02/math/exp2.c

   1 /*
   2  * Double-precision 2^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 Shift __exp_data.exp2_shift
  16 #define T __exp_data.tab
  17 #define C1 __exp_data.exp2_poly[0]
  18 #define C2 __exp_data.exp2_poly[1]
  19 #define C3 __exp_data.exp2_poly[2]
  20 #define C4 __exp_data.exp2_poly[3]
  21 #define C5 __exp_data.exp2_poly[4]
  22 #define C6 __exp_data.exp2_poly[5]
  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
  32 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
  33 {
  34   double_t scale, y;
  35
  36   if ((ki & 0x80000000) == 0)
  37     {
  38       /* k > 0, the exponent of scale might have overflowed by 1.  */
  39       sbits -= 1ull << 52;
  40       scale = asdouble (sbits);
  41       y = 2 * (scale + scale * tmp);
  42       return check_oflow (eval_as_double (y));
  43     }
  44   /* k < 0, need special care in the subnormal range.  */
  45   sbits += 1022ull << 52;
  46   scale = asdouble (sbits);
  47   y = scale + scale * tmp;
  48   if (y < 1.0)
  49     {
  50       /* Round y to the right precision before scaling it into the subnormal
  51          range to avoid double rounding that can cause 0.5+E/2 ulp error where
  52          E is the worst-case ulp error outside the subnormal range.  So this
  53          is only useful if the goal is better than 1 ulp worst-case error.  */
  54       double_t hi, lo;
  55       lo = scale - y + scale * tmp;
  56       hi = 1.0 + y;
  57       lo = 1.0 - hi + y + lo;
  58       y = eval_as_double (hi + lo) - 1.0;
  59       /* Avoid -0.0 with downward rounding.  */
  60       if (WANT_ROUNDING && y == 0.0)
  61         y = 0.0;
  62       /* The underflow exception needs to be signaled explicitly.  */
  63       force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
  64     }
  65   y = 0x1p-1022 * y;
  66   return check_uflow (eval_as_double (y));
  67 }
  68
  69 /* Top 12 bits of a double (sign and exponent bits).  */
  70 static inline uint32_t
  71 top12 (double x)
  72 {
  73   return asuint64 (x) >> 52;
  74 }
  75
  76 double
  77 exp2 (double x)
  78 {
  79   uint32_t abstop;
  80   uint64_t ki, idx, top, sbits;
  81   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  82   double_t kd, r, r2, scale, tail, tmp;
  83
  84   abstop = top12 (x) & 0x7ff;
  85   if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
  86     {
  87       if (abstop - top12 (0x1p-54) >= 0x80000000)
  88         /* Avoid spurious underflow for tiny x.  */
  89         /* Note: 0 is common input.  */
  90         return WANT_ROUNDING ? 1.0 + x : 1.0;
  91       if (abstop >= top12 (1024.0))
  92         {
  93           if (asuint64 (x) == asuint64 (-INFINITY))
  94             return 0.0;
  95           if (abstop >= top12 (INFINITY))
  96             return 1.0 + x;
  97           if (!(asuint64 (x) >> 63))
  98             return __math_oflow (0);
  99           else if (asuint64 (x) >= asuint64 (-1075.0))
 100             return __math_uflow (0);
 101         }
 102       if (2 * asuint64 (x) > 2 * asuint64 (928.0))
 103         /* Large x is special cased below.  */
 104         abstop = 0;
 105     }
 106
 107   /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)].  */
 108   /* x = k/N + r, with int k and r in [-1/2N, 1/2N].  */
 109   kd = eval_as_double (x + Shift);
 110   ki = asuint64 (kd); /* k.  */
 111   kd -= Shift; /* k/N for int k.  */
 112   r = x - kd;
 113   /* 2^(k/N) ~= scale * (1 + tail).  */
 114   idx = 2 * (ki % N);
 115   top = ki << (52 - EXP_TABLE_BITS);
 116   tail = asdouble (T[idx]);
 117   /* This is only a valid scale when -1023*N < k < 1024*N.  */
 118   sbits = T[idx + 1] + top;
 119   /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1).  */
 120   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 121   r2 = r * r;
 122   /* Without fma the worst case error is 0.5/N ulp larger.  */
 123   /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp.  */
 124 #if EXP2_POLY_ORDER == 4
 125   tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);
 126 #elif EXP2_POLY_ORDER == 5
 127   tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
 128 #elif EXP2_POLY_ORDER == 6
 129   tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
 130 #endif
 131   if (unlikely (abstop == 0))
 132     return specialcase (tmp, sbits, ki);
 133   scale = asdouble (sbits);
 134   /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
 135      is no spurious underflow here even without fma.  */
 136   return eval_as_double (scale + scale * tmp);
 137 }
 138 #if USE_GLIBC_ABI
 139 strong_alias (exp2, __exp2_finite)
 140 hidden_alias (exp2, __ieee754_exp2)
 141 # if LDBL_MANT_DIG == 53
 142 long double exp2l (long double x) { return exp2 (x); }
 143 # endif
 144 #endif