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.  */
29static 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).  */
65static inline uint32_t top12(double x)
66{
67	return asuint64(x) >> 52;
68}
69
70double 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