1/*
2 * Copyright (c) 2017-2018, Arm Limited.
3 * SPDX-License-Identifier: MIT
4 */
5
6#include <math.h>
7#include <stdint.h>
8#include "libm.h"
9#include "exp2f_data.h"
10#include "powf_data.h"
11
12/*
13POWF_LOG2_POLY_ORDER = 5
14EXP2F_TABLE_BITS = 5
15
16ULP error: 0.82 (~ 0.5 + relerr*2^24)
17relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
18relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
19relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
20*/
21
22#define N (1 << POWF_LOG2_TABLE_BITS)
23#define T __powf_log2_data.tab
24#define A __powf_log2_data.poly
25#define OFF 0x3f330000
26
27/* Subnormal input is normalized so ix has negative biased exponent.
28   Output is multiplied by N (POWF_SCALE) if TOINT_INTRINICS is set.  */
29static inline double_t log2_inline(uint32_t ix)
30{
31	double_t z, r, r2, r4, p, q, y, y0, invc, logc;
32	uint32_t iz, top, tmp;
33	int k, i;
34
35	/* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
36	   The range is split into N subintervals.
37	   The ith subinterval contains z and c is near its center.  */
38	tmp = ix - OFF;
39	i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
40	top = tmp & 0xff800000;
41	iz = ix - top;
42	k = (int32_t)top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
43	invc = T[i].invc;
44	logc = T[i].logc;
45	z = (double_t)asfloat(iz);
46
47	/* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
48	r = z * invc - 1;
49	y0 = logc + (double_t)k;
50
51	/* Pipelined polynomial evaluation to approximate log1p(r)/ln2.  */
52	r2 = r * r;
53	y = A[0] * r + A[1];
54	p = A[2] * r + A[3];
55	r4 = r2 * r2;
56	q = A[4] * r + y0;
57	q = p * r2 + q;
58	y = y * r4 + q;
59	return y;
60}
61
62#undef N
63#undef T
64#define N (1 << EXP2F_TABLE_BITS)
65#define T __exp2f_data.tab
66#define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
67
68/* The output of log2 and thus the input of exp2 is either scaled by N
69   (in case of fast toint intrinsics) or not.  The unscaled xd must be
70   in [-1021,1023], sign_bias sets the sign of the result.  */
71static inline float exp2_inline(double_t xd, uint32_t sign_bias)
72{
73	uint64_t ki, ski, t;
74	double_t kd, z, r, r2, y, s;
75
76#if TOINT_INTRINSICS
77#define C __exp2f_data.poly_scaled
78	/* N*x = k + r with r in [-1/2, 1/2] */
79	kd = roundtoint(xd); /* k */
80	ki = converttoint(xd);
81#else
82#define C __exp2f_data.poly
83#define SHIFT __exp2f_data.shift_scaled
84	/* x = k/N + r with r in [-1/(2N), 1/(2N)] */
85	kd = eval_as_double(xd + SHIFT);
86	ki = asuint64(kd);
87	kd -= SHIFT; /* k/N */
88#endif
89	r = xd - kd;
90
91	/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
92	t = T[ki % N];
93	ski = ki + sign_bias;
94	t += ski << (52 - EXP2F_TABLE_BITS);
95	s = asdouble(t);
96	z = C[0] * r + C[1];
97	r2 = r * r;
98	y = C[2] * r + 1;
99	y = z * r2 + y;
100	y = y * s;
101	return eval_as_float(y);
102}
103
104/* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
105   the bit representation of a non-zero finite floating-point value.  */
106static inline int checkint(uint32_t iy)
107{
108	int e = iy >> 23 & 0xff;
109	if (e < 0x7f)
110		return 0;
111	if (e > 0x7f + 23)
112		return 2;
113	if (iy & ((1 << (0x7f + 23 - e)) - 1))
114		return 0;
115	if (iy & (1 << (0x7f + 23 - e)))
116		return 1;
117	return 2;
118}
119
120static inline int zeroinfnan(uint32_t ix)
121{
122	return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
123}
124
125float powf(float x, float y)
126{
127	uint32_t sign_bias = 0;
128	uint32_t ix, iy;
129
130	ix = asuint(x);
131	iy = asuint(y);
132	if (predict_false(ix - 0x00800000 >= 0x7f800000 - 0x00800000 ||
133			  zeroinfnan(iy))) {
134		/* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan).  */
135		if (predict_false(zeroinfnan(iy))) {
136			if (2 * iy == 0)
137				return issignalingf_inline(x) ? x + y : 1.0f;
138			if (ix == 0x3f800000)
139				return issignalingf_inline(y) ? x + y : 1.0f;
140			if (2 * ix > 2u * 0x7f800000 ||
141			    2 * iy > 2u * 0x7f800000)
142				return x + y;
143			if (2 * ix == 2 * 0x3f800000)
144				return 1.0f;
145			if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
146				return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
147			return y * y;
148		}
149		if (predict_false(zeroinfnan(ix))) {
150			float_t x2 = x * x;
151			if (ix & 0x80000000 && checkint(iy) == 1)
152				x2 = -x2;
153			/* Without the barrier some versions of clang hoist the 1/x2 and
154			   thus division by zero exception can be signaled spuriously.  */
155			return iy & 0x80000000 ? fp_barrierf(1 / x2) : x2;
156		}
157		/* x and y are non-zero finite.  */
158		if (ix & 0x80000000) {
159			/* Finite x < 0.  */
160			int yint = checkint(iy);
161			if (yint == 0)
162				return __math_invalidf(x);
163			if (yint == 1)
164				sign_bias = SIGN_BIAS;
165			ix &= 0x7fffffff;
166		}
167		if (ix < 0x00800000) {
168			/* Normalize subnormal x so exponent becomes negative.  */
169			ix = asuint(x * 0x1p23f);
170			ix &= 0x7fffffff;
171			ix -= 23 << 23;
172		}
173	}
174	double_t logx = log2_inline(ix);
175	double_t ylogx = y * logx; /* cannot overflow, y is single prec.  */
176	if (predict_false((asuint64(ylogx) >> 47 & 0xffff) >=
177			  asuint64(126.0 * POWF_SCALE) >> 47)) {
178		/* |y*log(x)| >= 126.  */
179		if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
180			return __math_oflowf(sign_bias);
181		if (ylogx <= -150.0 * POWF_SCALE)
182			return __math_uflowf(sign_bias);
183	}
184	return exp2_inline(ylogx, sign_bias);
185}
186