1 /* 2 * Double-precision 2^x function. 3 * 4 * Copyright (c) 2018, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include <math.h> 9 #include <stdint.h> 10 #include "libm.h" 11 #include "exp_data.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 22 /* Handle cases that may overflow or underflow when computing the result that 23 is scale*(1+TMP) without intermediate rounding. The bit representation of 24 scale is in SBITS, however it has a computed exponent that may have 25 overflown into the sign bit so that needs to be adjusted before using it as 26 a double. (int32_t)KI is the k used in the argument reduction and exponent 27 adjustment of scale, positive k here means the result may overflow and 28 negative k means the result may underflow. */ 29 static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki) 30 { 31 double_t scale, y; 32 33 if ((ki & 0x80000000) == 0) { 34 /* k > 0, the exponent of scale might have overflowed by 1. */ 35 sbits -= 1ull << 52; 36 scale = asdouble(sbits); 37 y = 2 * (scale + scale * tmp); 38 return eval_as_double(y); 39 } 40 /* k < 0, need special care in the subnormal range. */ 41 sbits += 1022ull << 52; 42 scale = asdouble(sbits); 43 y = scale + scale * tmp; 44 if (y < 1.0) { 45 /* Round y to the right precision before scaling it into the subnormal 46 range to avoid double rounding that can cause 0.5+E/2 ulp error where 47 E is the worst-case ulp error outside the subnormal range. So this 48 is only useful if the goal is better than 1 ulp worst-case error. */ 49 double_t hi, lo; 50 lo = scale - y + scale * tmp; 51 hi = 1.0 + y; 52 lo = 1.0 - hi + y + lo; 53 y = eval_as_double(hi + lo) - 1.0; 54 /* Avoid -0.0 with downward rounding. */ 55 if (WANT_ROUNDING && y == 0.0) 56 y = 0.0; 57 /* The underflow exception needs to be signaled explicitly. */ 58 fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022); 59 } 60 y = 0x1p-1022 * y; 61 return eval_as_double(y); 62 } 63 64 /* Top 12 bits of a double (sign and exponent bits). */ 65 static inline uint32_t top12(double x) 66 { 67 return asuint64(x) >> 52; 68 } 69 70 double exp2(double x) 71 { 72 uint32_t abstop; 73 uint64_t ki, idx, top, sbits; 74 double_t kd, r, r2, scale, tail, tmp; 75 76 abstop = top12(x) & 0x7ff; 77 if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) { 78 if (abstop - top12(0x1p-54) >= 0x80000000) 79 /* Avoid spurious underflow for tiny x. */ 80 /* Note: 0 is common input. */ 81 return WANT_ROUNDING ? 1.0 + x : 1.0; 82 if (abstop >= top12(1024.0)) { 83 if (asuint64(x) == asuint64(-INFINITY)) 84 return 0.0; 85 if (abstop >= top12(INFINITY)) 86 return 1.0 + x; 87 if (!(asuint64(x) >> 63)) 88 return __math_oflow(0); 89 else if (asuint64(x) >= asuint64(-1075.0)) 90 return __math_uflow(0); 91 } 92 if (2 * asuint64(x) > 2 * asuint64(928.0)) 93 /* Large x is special cased below. */ 94 abstop = 0; 95 } 96 97 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */ 98 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */ 99 kd = eval_as_double(x + Shift); 100 ki = asuint64(kd); /* k. */ 101 kd -= Shift; /* k/N for int k. */ 102 r = x - kd; 103 /* 2^(k/N) ~= scale * (1 + tail). */ 104 idx = 2 * (ki % N); 105 top = ki << (52 - EXP_TABLE_BITS); 106 tail = asdouble(T[idx]); 107 /* This is only a valid scale when -1023*N < k < 1024*N. */ 108 sbits = T[idx + 1] + top; 109 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */ 110 /* Evaluation is optimized assuming superscalar pipelined execution. */ 111 r2 = r * r; 112 /* Without fma the worst case error is 0.5/N ulp larger. */ 113 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */ 114 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); 115 if (predict_false(abstop == 0)) 116 return specialcase(tmp, sbits, ki); 117 scale = asdouble(sbits); 118 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there 119 is no spurious underflow here even without fma. */ 120 return eval_as_double(scale + scale * tmp); 121 } 122