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