1 /* 2 * Double-precision 2^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 Shift __exp_data.exp2_shift 15 #define T __exp_data.tab 16 #define C1 __exp_data.exp2_poly[0] 17 #define C2 __exp_data.exp2_poly[1] 18 #define C3 __exp_data.exp2_poly[2] 19 #define C4 __exp_data.exp2_poly[3] 20 #define C5 __exp_data.exp2_poly[4] 21 #define C6 __exp_data.exp2_poly[5] 22 23 /* Handle cases that may overflow or underflow when computing the result that 24 is scale*(1+TMP) without intermediate rounding. The bit representation of 25 scale is in SBITS, however it has a computed exponent that may have 26 overflown into the sign bit so that needs to be adjusted before using it as 27 a double. (int32_t)KI is the k used in the argument reduction and exponent 28 adjustment of scale, positive k here means the result may overflow and 29 negative k means the result may underflow. */ 30 static inline double 31 specialcase (double_t tmp, uint64_t sbits, uint64_t ki) 32 { 33 double_t scale, y; 34 35 if ((ki & 0x80000000) == 0) 36 { 37 /* k > 0, the exponent of scale might have overflowed by 1. */ 38 sbits -= 1ull << 52; 39 scale = asdouble (sbits); 40 y = 2 * (scale + scale * tmp); 41 return check_oflow (eval_as_double (y)); 42 } 43 /* k < 0, need special care in the subnormal range. */ 44 sbits += 1022ull << 52; 45 scale = asdouble (sbits); 46 y = scale + scale * tmp; 47 if (y < 1.0) 48 { 49 /* Round y to the right precision before scaling it into the subnormal 50 range to avoid double rounding that can cause 0.5+E/2 ulp error where 51 E is the worst-case ulp error outside the subnormal range. So this 52 is only useful if the goal is better than 1 ulp worst-case error. */ 53 double_t hi, lo; 54 lo = scale - y + scale * tmp; 55 hi = 1.0 + y; 56 lo = 1.0 - hi + y + lo; 57 y = eval_as_double (hi + lo) - 1.0; 58 /* Avoid -0.0 with downward rounding. */ 59 if (WANT_ROUNDING && y == 0.0) 60 y = 0.0; 61 /* The underflow exception needs to be signaled explicitly. */ 62 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); 63 } 64 y = 0x1p-1022 * y; 65 return check_uflow (eval_as_double (y)); 66 } 67 68 /* Top 12 bits of a double (sign and exponent bits). */ 69 static inline uint32_t 70 top12 (double x) 71 { 72 return asuint64 (x) >> 52; 73 } 74 75 double 76 exp2 (double x) 77 { 78 uint32_t abstop; 79 uint64_t ki, idx, top, sbits; 80 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 81 double_t kd, r, r2, scale, tail, tmp; 82 83 abstop = top12 (x) & 0x7ff; 84 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) 85 { 86 if (abstop - top12 (0x1p-54) >= 0x80000000) 87 /* Avoid spurious underflow for tiny x. */ 88 /* Note: 0 is common input. */ 89 return WANT_ROUNDING ? 1.0 + x : 1.0; 90 if (abstop >= top12 (1024.0)) 91 { 92 if (asuint64 (x) == asuint64 (-INFINITY)) 93 return 0.0; 94 if (abstop >= top12 (INFINITY)) 95 return 1.0 + x; 96 if (!(asuint64 (x) >> 63)) 97 return __math_oflow (0); 98 else if (asuint64 (x) >= asuint64 (-1075.0)) 99 return __math_uflow (0); 100 } 101 if (2 * asuint64 (x) > 2 * asuint64 (928.0)) 102 /* Large x is special cased below. */ 103 abstop = 0; 104 } 105 106 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */ 107 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */ 108 kd = eval_as_double (x + Shift); 109 ki = asuint64 (kd); /* k. */ 110 kd -= Shift; /* k/N for int k. */ 111 r = x - kd; 112 /* 2^(k/N) ~= scale * (1 + tail). */ 113 idx = 2 * (ki % N); 114 top = ki << (52 - EXP_TABLE_BITS); 115 tail = asdouble (T[idx]); 116 /* This is only a valid scale when -1023*N < k < 1024*N. */ 117 sbits = T[idx + 1] + top; 118 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */ 119 /* Evaluation is optimized assuming superscalar pipelined execution. */ 120 r2 = r * r; 121 /* Without fma the worst case error is 0.5/N ulp larger. */ 122 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */ 123 #if EXP2_POLY_ORDER == 4 124 tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4); 125 #elif EXP2_POLY_ORDER == 5 126 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 127 #elif EXP2_POLY_ORDER == 6 128 tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); 129 #endif 130 if (unlikely (abstop == 0)) 131 return specialcase (tmp, sbits, ki); 132 scale = asdouble (sbits); 133 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there 134 is no spurious underflow here even without fma. */ 135 return eval_as_double (scale + scale * tmp); 136 } 137 #if USE_GLIBC_ABI 138 strong_alias (exp2, __exp2_finite) 139 hidden_alias (exp2, __ieee754_exp2) 140 # if LDBL_MANT_DIG == 53 141 long double exp2l (long double x) { return exp2 (x); } 142 # endif 143 #endif 144