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