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 /*
13 POWF_LOG2_POLY_ORDER = 5
14 EXP2F_TABLE_BITS = 5
15
16 ULP error: 0.82 (~ 0.5 + relerr*2^24)
17 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
18 relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
19 relerr_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. */
log2_inline(uint32_t ix)29 static 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. */
exp2_inline(double_t xd,uint32_t sign_bias)71 static 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. */
checkint(uint32_t iy)106 static 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
zeroinfnan(uint32_t ix)120 static inline int zeroinfnan(uint32_t ix)
121 {
122 return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
123 }
124
powf(float x,float y)125 float 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