1 /* 2 * Double-precision e^x function. 3 * 4 * Copyright (c) 2018-2024, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include <float.h> 9 #include <math.h> 10 #include <stdint.h> 11 #include "math_config.h" 12 #include "test_defs.h" 13 #include "test_sig.h" 14 15 #define N (1 << EXP_TABLE_BITS) 16 #define InvLn2N __exp_data.invln2N 17 #define NegLn2hiN __exp_data.negln2hiN 18 #define NegLn2loN __exp_data.negln2loN 19 #define Shift __exp_data.shift 20 #define T __exp_data.tab 21 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER] 22 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER] 23 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER] 24 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER] 25 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER] 26 27 /* Handle cases that may overflow or underflow when computing the result that 28 is scale*(1+TMP) without intermediate rounding. The bit representation of 29 scale is in SBITS, however it has a computed exponent that may have 30 overflown into the sign bit so that needs to be adjusted before using it as 31 a double. (int32_t)KI is the k used in the argument reduction and exponent 32 adjustment of scale, positive k here means the result may overflow and 33 negative k means the result may underflow. */ 34 static inline double 35 specialcase (double_t tmp, uint64_t sbits, uint64_t ki) 36 { 37 double_t scale, y; 38 39 if ((ki & 0x80000000) == 0) 40 { 41 /* k > 0, the exponent of scale might have overflowed by <= 460. */ 42 sbits -= 1009ull << 52; 43 scale = asdouble (sbits); 44 y = 0x1p1009 * (scale + scale * tmp); 45 return check_oflow (eval_as_double (y)); 46 } 47 /* k < 0, need special care in the subnormal range. */ 48 sbits += 1022ull << 52; 49 scale = asdouble (sbits); 50 y = scale + scale * tmp; 51 if (y < 1.0) 52 { 53 /* Round y to the right precision before scaling it into the subnormal 54 range to avoid double rounding that can cause 0.5+E/2 ulp error where 55 E is the worst-case ulp error outside the subnormal range. So this 56 is only useful if the goal is better than 1 ulp worst-case error. */ 57 double_t hi, lo; 58 lo = scale - y + scale * tmp; 59 hi = 1.0 + y; 60 lo = 1.0 - hi + y + lo; 61 y = eval_as_double (hi + lo) - 1.0; 62 /* Avoid -0.0 with downward rounding. */ 63 if (WANT_ROUNDING && y == 0.0) 64 y = 0.0; 65 /* The underflow exception needs to be signaled explicitly. */ 66 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); 67 } 68 y = 0x1p-1022 * y; 69 return check_uflow (eval_as_double (y)); 70 } 71 72 /* Top 12 bits of a double (sign and exponent bits). */ 73 static inline uint32_t 74 top12 (double x) 75 { 76 return asuint64 (x) >> 52; 77 } 78 79 /* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. 80 If hastail is 0 then xtail is assumed to be 0 too. */ 81 static inline double 82 exp_inline (double x, double xtail) 83 { 84 uint32_t abstop; 85 uint64_t ki, idx, top, sbits; 86 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 87 double_t kd, z, r, r2, scale, tail, tmp; 88 89 abstop = top12 (x) & 0x7ff; 90 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) 91 { 92 if (abstop - top12 (0x1p-54) >= 0x80000000) 93 /* Avoid spurious underflow for tiny x. */ 94 /* Note: 0 is common input. */ 95 return WANT_ROUNDING ? 1.0 + x : 1.0; 96 if (abstop >= top12 (1024.0)) 97 { 98 if (asuint64 (x) == asuint64 (-INFINITY)) 99 return 0.0; 100 if (abstop >= top12 (INFINITY)) 101 return 1.0 + x; 102 if (asuint64 (x) >> 63) 103 return __math_uflow (0); 104 else 105 return __math_oflow (0); 106 } 107 /* Large x is special cased below. */ 108 abstop = 0; 109 } 110 111 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ 112 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ 113 z = InvLn2N * x; 114 #if TOINT_INTRINSICS 115 kd = roundtoint (z); 116 ki = converttoint (z); 117 #elif EXP_USE_TOINT_NARROW 118 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ 119 kd = eval_as_double (z + Shift); 120 ki = asuint64 (kd) >> 16; 121 kd = (double_t) (int32_t) ki; 122 #else 123 /* z - kd is in [-1, 1] in non-nearest rounding modes. */ 124 kd = eval_as_double (z + Shift); 125 ki = asuint64 (kd); 126 kd -= Shift; 127 #endif 128 r = x + kd * NegLn2hiN + kd * NegLn2loN; 129 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ 130 if (!__builtin_constant_p (xtail) || xtail != 0.0) 131 r += xtail; 132 /* 2^(k/N) ~= scale * (1 + tail). */ 133 idx = 2 * (ki % N); 134 top = ki << (52 - EXP_TABLE_BITS); 135 tail = asdouble (T[idx]); 136 /* This is only a valid scale when -1023*N < k < 1024*N. */ 137 sbits = T[idx + 1] + top; 138 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ 139 /* Evaluation is optimized assuming superscalar pipelined execution. */ 140 r2 = r * r; 141 /* Without fma the worst case error is 0.25/N ulp larger. */ 142 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ 143 #if EXP_POLY_ORDER == 4 144 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4); 145 #elif EXP_POLY_ORDER == 5 146 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 147 #elif EXP_POLY_ORDER == 6 148 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); 149 #endif 150 if (unlikely (abstop == 0)) 151 return specialcase (tmp, sbits, ki); 152 scale = asdouble (sbits); 153 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there 154 is no spurious underflow here even without fma. */ 155 return eval_as_double (scale + scale * tmp); 156 } 157 158 double 159 exp (double x) 160 { 161 return exp_inline (x, 0); 162 } 163 164 #if USE_GLIBC_ABI 165 strong_alias (exp, __exp_finite) 166 hidden_alias (exp, __ieee754_exp) 167 # if LDBL_MANT_DIG == 53 168 long double expl (long double x) { return exp (x); } 169 # endif 170 #endif 171 172 TEST_SIG (S, D, 1, exp, -9.9, 9.9) 173 TEST_ULP (exp, 0.01) 174 TEST_ULP_NONNEAREST (exp, 0.5) 175 TEST_INTERVAL (exp, 0, 0xffff000000000000, 10000) 176 TEST_SYM_INTERVAL (exp, 0x1p-6, 0x1p6, 400000) 177 TEST_SYM_INTERVAL (exp, 633.3, 733.3, 10000) 178