1 /*
2 * Copyright (c) 2014,2015 Advanced Micro Devices, Inc.
3 *
4 * Permission is hereby granted, free of charge, to any person obtaining a copy
5 * of this software and associated documentation files (the "Software"), to deal
6 * in the Software without restriction, including without limitation the rights
7 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
8 * copies of the Software, and to permit persons to whom the Software is
9 * furnished to do so, subject to the following conditions:
10 *
11 * The above copyright notice and this permission notice shall be included in
12 * all copies or substantial portions of the Software.
13 *
14 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
15 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
16 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
17 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
18 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
19 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
20 * THE SOFTWARE.
21 */
22
23 #include "math.h"
24
25 /*
26 Algorithm:
27
28 Based on:
29 Ping-Tak Peter Tang
30 "Table-driven implementation of the logarithm function in IEEE
31 floating-point arithmetic"
32 ACM Transactions on Mathematical Software (TOMS)
33 Volume 16, Issue 4 (December 1990)
34
35
36 x very close to 1.0 is handled differently, for x everywhere else
37 a brief explanation is given below
38
39 x = (2^m)*A
40 x = (2^m)*(G+g) with (1 <= G < 2) and (g <= 2^(-8))
41 x = (2^m)*2*(G/2+g/2)
42 x = (2^m)*2*(F+f) with (0.5 <= F < 1) and (f <= 2^(-9))
43
44 Y = (2^(-1))*(2^(-m))*(2^m)*A
45 Now, range of Y is: 0.5 <= Y < 1
46
47 F = 0x80 + (first 7 mantissa bits) + (8th mantissa bit)
48 Now, range of F is: 128 <= F <= 256
49 F = F / 256
50 Now, range of F is: 0.5 <= F <= 1
51
52 f = -(Y-F), with (f <= 2^(-9))
53
54 log(x) = m*log(2) + log(2) + log(F-f)
55 log(x) = m*log(2) + log(2) + log(F) + log(1-(f/F))
56 log(x) = m*log(2) + log(2*F) + log(1-r)
57
58 r = (f/F), with (r <= 2^(-8))
59 r = f*(1/F) with (1/F) precomputed to avoid division
60
61 log(x) = m*log(2) + log(G) - poly
62
63 log(G) is precomputed
64 poly = (r + (r^2)/2 + (r^3)/3 + (r^4)/4) + (r^5)/5))
65
66 log(2) and log(G) need to be maintained in extra precision
67 to avoid losing precision in the calculations
68
69
70 For x close to 1.0, we employ the following technique to
71 ensure faster convergence.
72
73 log(x) = log((1+s)/(1-s)) = 2*s + (2/3)*s^3 + (2/5)*s^5 + (2/7)*s^7
74 x = ((1+s)/(1-s))
75 x = 1 + r
76 s = r/(2+r)
77
78 */
79
80 _CLC_OVERLOAD _CLC_DEF float
81 #if defined(COMPILING_LOG2)
log2(float x)82 log2(float x)
83 #elif defined(COMPILING_LOG10)
84 log10(float x)
85 #else
86 log(float x)
87 #endif
88 {
89
90 #if defined(COMPILING_LOG2)
91 const float LOG2E = 0x1.715476p+0f; // 1.4426950408889634
92 const float LOG2E_HEAD = 0x1.700000p+0f; // 1.4375
93 const float LOG2E_TAIL = 0x1.547652p-8f; // 0.00519504072
94 #elif defined(COMPILING_LOG10)
95 const float LOG10E = 0x1.bcb7b2p-2f; // 0.43429448190325182
96 const float LOG10E_HEAD = 0x1.bc0000p-2f; // 0.43359375
97 const float LOG10E_TAIL = 0x1.6f62a4p-11f; // 0.0007007319
98 const float LOG10_2_HEAD = 0x1.340000p-2f; // 0.30078125
99 const float LOG10_2_TAIL = 0x1.04d426p-12f; // 0.000248745637
100 #else
101 const float LOG2_HEAD = 0x1.62e000p-1f; // 0.693115234
102 const float LOG2_TAIL = 0x1.0bfbe8p-15f; // 0.0000319461833
103 #endif
104
105 uint xi = as_uint(x);
106 uint ax = xi & EXSIGNBIT_SP32;
107
108 // Calculations for |x-1| < 2^-4
109 float r = x - 1.0f;
110 int near1 = fabs(r) < 0x1.0p-4f;
111 float u2 = MATH_DIVIDE(r, 2.0f + r);
112 float corr = u2 * r;
113 float u = u2 + u2;
114 float v = u * u;
115 float znear1, z1, z2;
116
117 // 2/(5 * 2^5), 2/(3 * 2^3)
118 z2 = mad(u, mad(v, 0x1.99999ap-7f, 0x1.555556p-4f)*v, -corr);
119
120 #if defined(COMPILING_LOG2)
121 z1 = as_float(as_int(r) & 0xffff0000);
122 z2 = z2 + (r - z1);
123 znear1 = mad(z1, LOG2E_HEAD, mad(z2, LOG2E_HEAD, mad(z1, LOG2E_TAIL, z2*LOG2E_TAIL)));
124 #elif defined(COMPILING_LOG10)
125 z1 = as_float(as_int(r) & 0xffff0000);
126 z2 = z2 + (r - z1);
127 znear1 = mad(z1, LOG10E_HEAD, mad(z2, LOG10E_HEAD, mad(z1, LOG10E_TAIL, z2*LOG10E_TAIL)));
128 #else
129 znear1 = z2 + r;
130 #endif
131
132 // Calculations for x not near 1
133 int m = (int)(xi >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;
134
135 // Normalize subnormal
136 uint xis = as_uint(as_float(xi | 0x3f800000) - 1.0f);
137 int ms = (int)(xis >> EXPSHIFTBITS_SP32) - 253;
138 int c = m == -127;
139 m = c ? ms : m;
140 uint xin = c ? xis : xi;
141
142 float mf = (float)m;
143 uint indx = (xin & 0x007f0000) + ((xin & 0x00008000) << 1);
144
145 // F - Y
146 float f = as_float(0x3f000000 | indx) - as_float(0x3f000000 | (xin & MANTBITS_SP32));
147
148 indx = indx >> 16;
149 r = f * USE_TABLE(log_inv_tbl, indx);
150
151 // 1/3, 1/2
152 float poly = mad(mad(r, 0x1.555556p-2f, 0.5f), r*r, r);
153
154 #if defined(COMPILING_LOG2)
155 float2 tv = USE_TABLE(log2_tbl, indx);
156 z1 = tv.s0 + mf;
157 z2 = mad(poly, -LOG2E, tv.s1);
158 #elif defined(COMPILING_LOG10)
159 float2 tv = USE_TABLE(log10_tbl, indx);
160 z1 = mad(mf, LOG10_2_HEAD, tv.s0);
161 z2 = mad(poly, -LOG10E, mf*LOG10_2_TAIL) + tv.s1;
162 #else
163 float2 tv = USE_TABLE(log_tbl, indx);
164 z1 = mad(mf, LOG2_HEAD, tv.s0);
165 z2 = mad(mf, LOG2_TAIL, -poly) + tv.s1;
166 #endif
167
168 float z = z1 + z2;
169 z = near1 ? znear1 : z;
170
171 // Corner cases
172 z = ax >= PINFBITPATT_SP32 ? x : z;
173 z = xi != ax ? as_float(QNANBITPATT_SP32) : z;
174 z = ax == 0 ? as_float(NINFBITPATT_SP32) : z;
175
176 return z;
177 }
178
179 #ifdef cl_khr_fp64
180
181 _CLC_OVERLOAD _CLC_DEF double
182 #if defined(COMPILING_LOG2)
log2(double x)183 log2(double x)
184 #elif defined(COMPILING_LOG10)
185 log10(double x)
186 #else
187 log(double x)
188 #endif
189 {
190
191 #ifndef COMPILING_LOG2
192 // log2_lead and log2_tail sum to an extra-precise version of ln(2)
193 const double log2_lead = 6.93147122859954833984e-01; /* 0x3fe62e42e0000000 */
194 const double log2_tail = 5.76999904754328540596e-08; /* 0x3e6efa39ef35793c */
195 #endif
196
197 #if defined(COMPILING_LOG10)
198 // log10e_lead and log10e_tail sum to an extra-precision version of log10(e) (19 bits in lead)
199 const double log10e_lead = 4.34293746948242187500e-01; /* 0x3fdbcb7800000000 */
200 const double log10e_tail = 7.3495500964015109100644e-7; /* 0x3ea8a93728719535 */
201 #elif defined(COMPILING_LOG2)
202 // log2e_lead and log2e_tail sum to an extra-precision version of log2(e) (19 bits in lead)
203 const double log2e_lead = 1.44269180297851562500E+00; /* 0x3FF7154400000000 */
204 const double log2e_tail = 3.23791044778235969970E-06; /* 0x3ECB295C17F0BBBE */
205 #endif
206
207 // log_thresh1 = 9.39412117004394531250e-1 = 0x3fee0faa00000000
208 // log_thresh2 = 1.06449508666992187500 = 0x3ff1082c00000000
209 const double log_thresh1 = 0x1.e0faap-1;
210 const double log_thresh2 = 0x1.1082cp+0;
211
212 int is_near = x >= log_thresh1 & x <= log_thresh2;
213
214 // Near 1 code
215 double r = x - 1.0;
216 double u = r / (2.0 + r);
217 double correction = r * u;
218 u = u + u;
219 double v = u * u;
220 double r1 = r;
221
222 const double ca_1 = 8.33333333333317923934e-02; /* 0x3fb55555555554e6 */
223 const double ca_2 = 1.25000000037717509602e-02; /* 0x3f89999999bac6d4 */
224 const double ca_3 = 2.23213998791944806202e-03; /* 0x3f62492307f1519f */
225 const double ca_4 = 4.34887777707614552256e-04; /* 0x3f3c8034c85dfff0 */
226
227 double r2 = fma(u*v, fma(v, fma(v, fma(v, ca_4, ca_3), ca_2), ca_1), -correction);
228
229 #if defined(COMPILING_LOG10)
230 r = r1;
231 r1 = as_double(as_ulong(r1) & 0xffffffff00000000);
232 r2 = r2 + (r - r1);
233 double ret_near = fma(log10e_lead, r1, fma(log10e_lead, r2, fma(log10e_tail, r1, log10e_tail * r2)));
234 #elif defined(COMPILING_LOG2)
235 r = r1;
236 r1 = as_double(as_ulong(r1) & 0xffffffff00000000);
237 r2 = r2 + (r - r1);
238 double ret_near = fma(log2e_lead, r1, fma(log2e_lead, r2, fma(log2e_tail, r1, log2e_tail*r2)));
239 #else
240 double ret_near = r1 + r2;
241 #endif
242
243 // This is the far from 1 code
244
245 // Deal with subnormal
246 ulong ux = as_ulong(x);
247 ulong uxs = as_ulong(as_double(0x03d0000000000000UL | ux) - 0x1.0p-962);
248 int c = ux < IMPBIT_DP64;
249 ux = c ? uxs : ux;
250 int expadjust = c ? 60 : 0;
251
252 int xexp = ((as_int2(ux).hi >> 20) & 0x7ff) - EXPBIAS_DP64 - expadjust;
253 double f = as_double(HALFEXPBITS_DP64 | (ux & MANTBITS_DP64));
254 int index = as_int2(ux).hi >> 13;
255 index = ((0x80 | (index & 0x7e)) >> 1) + (index & 0x1);
256
257 double2 tv = USE_TABLE(ln_tbl, index - 64);
258 double z1 = tv.s0;
259 double q = tv.s1;
260
261 double f1 = index * 0x1.0p-7;
262 double f2 = f - f1;
263 u = f2 / fma(f2, 0.5, f1);
264 v = u * u;
265
266 const double cb_1 = 8.33333333333333593622e-02; /* 0x3fb5555555555557 */
267 const double cb_2 = 1.24999999978138668903e-02; /* 0x3f89999999865ede */
268 const double cb_3 = 2.23219810758559851206e-03; /* 0x3f6249423bd94741 */
269
270 double poly = v * fma(v, fma(v, cb_3, cb_2), cb_1);
271 double z2 = q + fma(u, poly, u);
272
273 double dxexp = (double)xexp;
274 #if defined (COMPILING_LOG10)
275 // Add xexp * log(2) to z1,z2 to get log(x)
276 r1 = fma(dxexp, log2_lead, z1);
277 r2 = fma(dxexp, log2_tail, z2);
278 double ret_far = fma(log10e_lead, r1, fma(log10e_lead, r2, fma(log10e_tail, r1, log10e_tail*r2)));
279 #elif defined(COMPILING_LOG2)
280 r1 = fma(log2e_lead, z1, dxexp);
281 r2 = fma(log2e_lead, z2, fma(log2e_tail, z1, log2e_tail*z2));
282 double ret_far = r1 + r2;
283 #else
284 r1 = fma(dxexp, log2_lead, z1);
285 r2 = fma(dxexp, log2_tail, z2);
286 double ret_far = r1 + r2;
287 #endif
288
289 double ret = is_near ? ret_near : ret_far;
290
291 ret = isinf(x) ? as_double(PINFBITPATT_DP64) : ret;
292 ret = isnan(x) | (x < 0.0) ? as_double(QNANBITPATT_DP64) : ret;
293 ret = x == 0.0 ? as_double(NINFBITPATT_DP64) : ret;
294 return ret;
295 }
296
297 #endif // cl_khr_fp64
298