1 /* 2 * Double-precision e^x function. 3 * 4 * Copyright (c) 2018-2019, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include <float.h> 9 #include <math.h> 10 #include <stdint.h> 11 #include "math_config.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 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER] 24 25 /* Handle cases that may overflow or underflow when computing the result that 26 is scale*(1+TMP) without intermediate rounding. The bit representation of 27 scale is in SBITS, however it has a computed exponent that may have 28 overflown into the sign bit so that needs to be adjusted before using it as 29 a double. (int32_t)KI is the k used in the argument reduction and exponent 30 adjustment of scale, positive k here means the result may overflow and 31 negative k means the result may underflow. */ 32 static inline double 33 specialcase (double_t tmp, uint64_t sbits, uint64_t ki) 34 { 35 double_t scale, y; 36 37 if ((ki & 0x80000000) == 0) 38 { 39 /* k > 0, the exponent of scale might have overflowed by <= 460. */ 40 sbits -= 1009ull << 52; 41 scale = asdouble (sbits); 42 y = 0x1p1009 * (scale + scale * tmp); 43 return check_oflow (eval_as_double (y)); 44 } 45 /* k < 0, need special care in the subnormal range. */ 46 sbits += 1022ull << 52; 47 scale = asdouble (sbits); 48 y = scale + scale * tmp; 49 if (y < 1.0) 50 { 51 /* Round y to the right precision before scaling it into the subnormal 52 range to avoid double rounding that can cause 0.5+E/2 ulp error where 53 E is the worst-case ulp error outside the subnormal range. So this 54 is only useful if the goal is better than 1 ulp worst-case error. */ 55 double_t hi, lo; 56 lo = scale - y + scale * tmp; 57 hi = 1.0 + y; 58 lo = 1.0 - hi + y + lo; 59 y = eval_as_double (hi + lo) - 1.0; 60 /* Avoid -0.0 with downward rounding. */ 61 if (WANT_ROUNDING && y == 0.0) 62 y = 0.0; 63 /* The underflow exception needs to be signaled explicitly. */ 64 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); 65 } 66 y = 0x1p-1022 * y; 67 return check_uflow (eval_as_double (y)); 68 } 69 70 /* Top 12 bits of a double (sign and exponent bits). */ 71 static inline uint32_t 72 top12 (double x) 73 { 74 return asuint64 (x) >> 52; 75 } 76 77 /* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. 78 If hastail is 0 then xtail is assumed to be 0 too. */ 79 static inline double 80 exp_inline (double x, double xtail, int hastail) 81 { 82 uint32_t abstop; 83 uint64_t ki, idx, top, sbits; 84 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 85 double_t kd, z, r, r2, scale, tail, tmp; 86 87 abstop = top12 (x) & 0x7ff; 88 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) 89 { 90 if (abstop - top12 (0x1p-54) >= 0x80000000) 91 /* Avoid spurious underflow for tiny x. */ 92 /* Note: 0 is common input. */ 93 return WANT_ROUNDING ? 1.0 + x : 1.0; 94 if (abstop >= top12 (1024.0)) 95 { 96 if (asuint64 (x) == asuint64 (-INFINITY)) 97 return 0.0; 98 if (abstop >= top12 (INFINITY)) 99 return 1.0 + x; 100 if (asuint64 (x) >> 63) 101 return __math_uflow (0); 102 else 103 return __math_oflow (0); 104 } 105 /* Large x is special cased below. */ 106 abstop = 0; 107 } 108 109 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ 110 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ 111 z = InvLn2N * x; 112 #if TOINT_INTRINSICS 113 kd = roundtoint (z); 114 ki = converttoint (z); 115 #elif EXP_USE_TOINT_NARROW 116 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ 117 kd = eval_as_double (z + Shift); 118 ki = asuint64 (kd) >> 16; 119 kd = (double_t) (int32_t) ki; 120 #else 121 /* z - kd is in [-1, 1] in non-nearest rounding modes. */ 122 kd = eval_as_double (z + Shift); 123 ki = asuint64 (kd); 124 kd -= Shift; 125 #endif 126 r = x + kd * NegLn2hiN + kd * NegLn2loN; 127 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ 128 if (hastail) 129 r += xtail; 130 /* 2^(k/N) ~= scale * (1 + tail). */ 131 idx = 2 * (ki % N); 132 top = ki << (52 - EXP_TABLE_BITS); 133 tail = asdouble (T[idx]); 134 /* This is only a valid scale when -1023*N < k < 1024*N. */ 135 sbits = T[idx + 1] + top; 136 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ 137 /* Evaluation is optimized assuming superscalar pipelined execution. */ 138 r2 = r * r; 139 /* Without fma the worst case error is 0.25/N ulp larger. */ 140 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ 141 #if EXP_POLY_ORDER == 4 142 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4); 143 #elif EXP_POLY_ORDER == 5 144 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 145 #elif EXP_POLY_ORDER == 6 146 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); 147 #endif 148 if (unlikely (abstop == 0)) 149 return specialcase (tmp, sbits, ki); 150 scale = asdouble (sbits); 151 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there 152 is no spurious underflow here even without fma. */ 153 return eval_as_double (scale + scale * tmp); 154 } 155 156 double 157 exp (double x) 158 { 159 return exp_inline (x, 0, 0); 160 } 161 162 /* May be useful for implementing pow where more than double 163 precision input is needed. */ 164 double 165 __exp_dd (double x, double xtail) 166 { 167 return exp_inline (x, xtail, 1); 168 } 169 #if USE_GLIBC_ABI 170 strong_alias (exp, __exp_finite) 171 hidden_alias (exp, __ieee754_exp) 172 hidden_alias (__exp_dd, __exp1) 173 # if LDBL_MANT_DIG == 53 174 long double expl (long double x) { return exp (x); } 175 # endif 176 #endif 177