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