1 /* 2 * Double-precision 2^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 Shift __exp_data.exp2_shift 17 #define T __exp_data.tab 18 #define C1 __exp_data.exp2_poly[0] 19 #define C2 __exp_data.exp2_poly[1] 20 #define C3 __exp_data.exp2_poly[2] 21 #define C4 __exp_data.exp2_poly[3] 22 #define C5 __exp_data.exp2_poly[4] 23 #define C6 __exp_data.exp2_poly[5] 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 1. */ 40 sbits -= 1ull << 52; 41 scale = asdouble (sbits); 42 y = 2 * (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 double 78 exp2 (double x) 79 { 80 uint32_t abstop; 81 uint64_t ki, idx, top, sbits; 82 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 83 double_t kd, r, r2, scale, tail, tmp; 84 85 abstop = top12 (x) & 0x7ff; 86 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) 87 { 88 if (abstop - top12 (0x1p-54) >= 0x80000000) 89 /* Avoid spurious underflow for tiny x. */ 90 /* Note: 0 is common input. */ 91 return WANT_ROUNDING ? 1.0 + x : 1.0; 92 if (abstop >= top12 (1024.0)) 93 { 94 if (asuint64 (x) == asuint64 (-INFINITY)) 95 return 0.0; 96 if (abstop >= top12 (INFINITY)) 97 return 1.0 + x; 98 if (!(asuint64 (x) >> 63)) 99 return __math_oflow (0); 100 else if (asuint64 (x) >= asuint64 (-1075.0)) 101 return __math_uflow (0); 102 } 103 if (2 * asuint64 (x) > 2 * asuint64 (928.0)) 104 /* Large x is special cased below. */ 105 abstop = 0; 106 } 107 108 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */ 109 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */ 110 kd = eval_as_double (x + Shift); 111 ki = asuint64 (kd); /* k. */ 112 kd -= Shift; /* k/N for int k. */ 113 r = x - kd; 114 /* 2^(k/N) ~= scale * (1 + tail). */ 115 idx = 2 * (ki % N); 116 top = ki << (52 - EXP_TABLE_BITS); 117 tail = asdouble (T[idx]); 118 /* This is only a valid scale when -1023*N < k < 1024*N. */ 119 sbits = T[idx + 1] + top; 120 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */ 121 /* Evaluation is optimized assuming superscalar pipelined execution. */ 122 r2 = r * r; 123 /* Without fma the worst case error is 0.5/N ulp larger. */ 124 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */ 125 #if EXP2_POLY_ORDER == 4 126 tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4); 127 #elif EXP2_POLY_ORDER == 5 128 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 129 #elif EXP2_POLY_ORDER == 6 130 tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); 131 #endif 132 if (unlikely (abstop == 0)) 133 return specialcase (tmp, sbits, ki); 134 scale = asdouble (sbits); 135 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there 136 is no spurious underflow here even without fma. */ 137 return eval_as_double (scale + scale * tmp); 138 } 139 #if USE_GLIBC_ABI 140 strong_alias (exp2, __exp2_finite) 141 hidden_alias (exp2, __ieee754_exp2) 142 # if LDBL_MANT_DIG == 53 143 long double exp2l (long double x) { return exp2 (x); } 144 # endif 145 #endif 146 147 TEST_SIG (S, D, 1, exp2, -9.9, 9.9) 148 TEST_ULP (exp2, 0.01) 149 TEST_ULP_NONNEAREST (exp2, 0.5) 150 TEST_INTERVAL (exp2, 0, 0xffff000000000000, 10000) 151 TEST_SYM_INTERVAL (exp2, 0x1p-6, 0x1p6, 40000) 152 TEST_SYM_INTERVAL (exp2, 633.3, 733.3, 10000) 153