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36 /* -------------------------------------------------------------- */
37 /* PROLOG END TAG zYx */
38 #ifdef __SPU__
39
40 #ifndef _LGAMMAD2_H_
41 #define _LGAMMAD2_H_ 1
42
43 #include <spu_intrinsics.h>
44 #include "divd2.h"
45 #include "recipd2.h"
46 #include "logd2.h"
47 #include "sind2.h"
48 #include "truncd2.h"
49
50
51 /*
52 * FUNCTION
53 * vector double _lgammad2(vector double x) - Natural Log of Gamma Function
54 *
55 * DESCRIPTION
56 * _lgammad2 calculates the natural logarithm of the absolute value of the gamma
57 * function for the corresponding elements of the input vector.
58 *
59 * C99 Special Cases:
60 * lgamma(0) returns +infinite
61 * lgamma(1) returns +0
62 * lgamma(2) returns +0
63 * lgamma(negative integer) returns +infinite
64 * lgamma(+infinite) returns +infinite
65 * lgamma(-infinite) returns +infinite
66 *
67 * Other Cases:
68 * lgamma(Nan) returns Nan
69 * lgamma(Denorm) treated as lgamma(0) and returns +infinite
70 *
71 */
72
73 #define PI 3.1415926535897932384626433832795028841971693993751058209749445923078164
74 #define HALFLOG2PI 9.1893853320467274178032973640561763986139747363778341281715154048276570E-1
75
76 #define EULER_MASCHERONI 0.5772156649015328606065
77
78 /*
79 * Zeta constants for Maclaurin approx. near zero
80 */
81 #define ZETA_02_DIV_02 8.2246703342411321823620758332301E-1
82 #define ZETA_03_DIV_03 -4.0068563438653142846657938717048E-1
83 #define ZETA_04_DIV_04 2.7058080842778454787900092413529E-1
84 #define ZETA_05_DIV_05 -2.0738555102867398526627309729141E-1
85 #define ZETA_06_DIV_06 1.6955717699740818995241965496515E-1
86
87 /*
88 * More Maclaurin coefficients
89 */
90 /*
91 #define ZETA_07_DIV_07 -1.4404989676884611811997107854997E-1
92 #define ZETA_08_DIV_08 1.2550966952474304242233565481358E-1
93 #define ZETA_09_DIV_09 -1.1133426586956469049087252991471E-1
94 #define ZETA_10_DIV_10 1.0009945751278180853371459589003E-1
95 #define ZETA_11_DIV_11 -9.0954017145829042232609298411497E-2
96 #define ZETA_12_DIV_12 8.3353840546109004024886499837312E-2
97 #define ZETA_13_DIV_13 -7.6932516411352191472827064348181E-2
98 #define ZETA_14_DIV_14 7.1432946295361336059232753221795E-2
99 #define ZETA_15_DIV_15 -6.6668705882420468032903448567376E-2
100 #define ZETA_16_DIV_16 6.2500955141213040741983285717977E-2
101 #define ZETA_17_DIV_17 -5.8823978658684582338957270605504E-2
102 #define ZETA_18_DIV_18 5.5555767627403611102214247869146E-2
103 #define ZETA_19_DIV_19 -5.2631679379616660733627666155673E-2
104 #define ZETA_20_DIV_20 5.0000047698101693639805657601934E-2
105 */
106
107 /*
108 * Coefficients for Stirling's Series for Lgamma()
109 */
110 #define STIRLING_01 8.3333333333333333333333333333333333333333333333333333333333333333333333E-2
111 #define STIRLING_02 -2.7777777777777777777777777777777777777777777777777777777777777777777778E-3
112 #define STIRLING_03 7.9365079365079365079365079365079365079365079365079365079365079365079365E-4
113 #define STIRLING_04 -5.9523809523809523809523809523809523809523809523809523809523809523809524E-4
114 #define STIRLING_05 8.4175084175084175084175084175084175084175084175084175084175084175084175E-4
115 #define STIRLING_06 -1.9175269175269175269175269175269175269175269175269175269175269175269175E-3
116 #define STIRLING_07 6.4102564102564102564102564102564102564102564102564102564102564102564103E-3
117 #define STIRLING_08 -2.9550653594771241830065359477124183006535947712418300653594771241830065E-2
118 #define STIRLING_09 1.7964437236883057316493849001588939669435025472177174963552672531000704E-1
119 #define STIRLING_10 -1.3924322169059011164274322169059011164274322169059011164274322169059011E0
120 #define STIRLING_11 1.3402864044168391994478951000690131124913733609385783298826777087646653E1
121 #define STIRLING_12 -1.5684828462600201730636513245208897382810426288687158252375643679991506E2
122 #define STIRLING_13 2.1931033333333333333333333333333333333333333333333333333333333333333333E3
123 #define STIRLING_14 -3.6108771253724989357173265219242230736483610046828437633035334184759472E4
124 #define STIRLING_15 6.9147226885131306710839525077567346755333407168779805042318946657100161E5
125 /*
126 * More Stirling's coefficients
127 */
128 /*
129 #define STIRLING_16 -1.5238221539407416192283364958886780518659076533839342188488298545224541E7
130 #define STIRLING_17 3.8290075139141414141414141414141414141414141414141414141414141414141414E8
131 #define STIRLING_18 -1.0882266035784391089015149165525105374729434879810819660443720594096534E10
132 #define STIRLING_19 3.4732028376500225225225225225225225225225225225225225225225225225225225E11
133 #define STIRLING_20 -1.2369602142269274454251710349271324881080978641954251710349271324881081E13
134 #define STIRLING_21 4.8878806479307933507581516251802290210847053890567382180703629532735764E14
135 */
136
137
_lgammad2(vector double x)138 static __inline vector double _lgammad2(vector double x)
139 {
140 vec_uchar16 dup_even = ((vec_uchar16) { 0,1,2,3, 0,1,2,3, 8, 9,10,11, 8, 9,10,11 });
141 vec_uchar16 dup_odd = ((vec_uchar16) { 4,5,6,7, 4,5,6,7, 12,13,14,15, 12,13,14,15 });
142 vec_uchar16 swap_word = ((vec_uchar16) { 4,5,6,7, 0,1,2,3, 12,13,14,15, 8, 9,10,11 });
143 vec_double2 infinited = (vec_double2)spu_splats(0x7FF0000000000000ull);
144 vec_double2 zerod = spu_splats(0.0);
145 vec_double2 oned = spu_splats(1.0);
146 vec_double2 twod = spu_splats(2.0);
147 vec_double2 pi = spu_splats(PI);
148 vec_double2 sign_maskd = spu_splats(-0.0);
149
150 /* This is where we switch from near zero approx. */
151 vec_float4 zero_switch = spu_splats(0.001f);
152 vec_float4 shift_switch = spu_splats(6.0f);
153
154 vec_float4 xf;
155 vec_double2 inv_x, inv_xsqu;
156 vec_double2 xtrunc, xstirling;
157 vec_double2 sum, xabs;
158 vec_uint4 xhigh, xlow, xthigh, xtlow;
159 vec_uint4 x1, isnaninf, isnposint, iszero, isint, isneg, isshifted, is1, is2;
160 vec_double2 result, stresult, shresult, mresult, nresult;
161
162
163 /* Force Denorms to 0 */
164 x = spu_add(x, zerod);
165
166 xabs = spu_andc(x, sign_maskd);
167 xf = spu_roundtf(xabs);
168 xf = spu_shuffle(xf, xf, dup_even);
169
170
171 /*
172 * For 0 < x <= 0.001.
173 * Approximation Near Zero
174 *
175 * Use Maclaurin Expansion of lgamma()
176 *
177 * lgamma(z) = -ln(z) - z * EulerMascheroni + Sum[(-1)^n * z^n * Zeta(n)/n]
178 */
179 mresult = spu_madd(xabs, spu_splats(ZETA_06_DIV_06), spu_splats(ZETA_05_DIV_05));
180 mresult = spu_madd(xabs, mresult, spu_splats(ZETA_04_DIV_04));
181 mresult = spu_madd(xabs, mresult, spu_splats(ZETA_03_DIV_03));
182 mresult = spu_madd(xabs, mresult, spu_splats(ZETA_02_DIV_02));
183 mresult = spu_mul(xabs, spu_mul(xabs, mresult));
184 mresult = spu_sub(mresult, spu_add(_logd2(xabs), spu_mul(xabs, spu_splats(EULER_MASCHERONI))));
185
186
187 /*
188 * For 0.001 < x <= 6.0, we are going to push value
189 * out to an area where Stirling's approximation is
190 * accurate. Let's use a constant of 6.
191 *
192 * Use the recurrence relation:
193 * lgamma(x + 1) = ln(x) + lgamma(x)
194 *
195 * Note that we shift x here, before Stirling's calculation,
196 * then after Stirling's, we adjust the result.
197 *
198 */
199
200 isshifted = spu_cmpgt(shift_switch, xf);
201 xstirling = spu_sel(xabs, spu_add(xabs, spu_splats(6.0)), (vec_ullong2)isshifted);
202 inv_x = _recipd2(xstirling);
203 inv_xsqu = spu_mul(inv_x, inv_x);
204
205 /*
206 * For 6.0 < x < infinite
207 *
208 * Use Stirling's Series.
209 *
210 * 1 1 1 1 1
211 * lgamma(x) = --- ln (2*pi) + (z - ---) ln(x) - x + --- - ----- + ------ ...
212 * 2 2 12x 360x^3 1260x^5
213 *
214 * Taking 10 terms of the sum gives good results for x > 6.0
215 *
216 */
217 sum = spu_madd(inv_xsqu, spu_splats(STIRLING_15), spu_splats(STIRLING_14));
218 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_13));
219 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_12));
220 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_11));
221 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_10));
222 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_09));
223 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_08));
224 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_07));
225 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_06));
226 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_05));
227 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_04));
228 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_03));
229 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_02));
230 sum = spu_madd(sum, inv_xsqu, spu_splats(STIRLING_01));
231 sum = spu_mul(sum, inv_x);
232
233 stresult = spu_madd(spu_sub(xstirling, spu_splats(0.5)), _logd2(xstirling), spu_splats(HALFLOG2PI));
234 stresult = spu_sub(stresult, xstirling);
235 stresult = spu_add(stresult, sum);
236
237 /*
238 * Adjust result if we shifted x into Stirling range.
239 *
240 * lgamma(x) = lgamma(x + n) - ln(x(x+1)(x+2)...(x+n-1)
241 *
242 */
243 shresult = spu_mul(xabs, spu_add(xabs, spu_splats(1.0)));
244 shresult = spu_mul(shresult, spu_add(xabs, spu_splats(2.0)));
245 shresult = spu_mul(shresult, spu_add(xabs, spu_splats(3.0)));
246 shresult = spu_mul(shresult, spu_add(xabs, spu_splats(4.0)));
247 shresult = spu_mul(shresult, spu_add(xabs, spu_splats(5.0)));
248 shresult = _logd2(shresult);
249 shresult = spu_sub(stresult, shresult);
250 stresult = spu_sel(stresult, shresult, (vec_ullong2)isshifted);
251
252
253 /*
254 * Select either Maclaurin or Stirling result before Negative X calc.
255 */
256 xf = spu_shuffle(xf, xf, dup_even);
257 vec_uint4 useStirlings = spu_cmpgt(xf, zero_switch);
258 result = spu_sel(mresult, stresult, (vec_ullong2)useStirlings);
259
260
261 /*
262 * Approximation for Negative X
263 *
264 * Use reflection relation
265 *
266 * gamma(x) * gamma(-x) = -pi/(x sin(pi x))
267 *
268 * lgamma(x) = log(pi/(-x sin(pi x))) - lgamma(-x)
269 *
270 */
271 nresult = spu_mul(x, _sind2(spu_mul(x, pi)));
272 nresult = spu_andc(nresult, sign_maskd);
273 nresult = _logd2(_divd2(pi, nresult));
274 nresult = spu_sub(nresult, result);
275
276
277 /*
278 * Select between the negative or positive x approximations.
279 */
280 isneg = (vec_uint4)spu_shuffle(x, x, dup_even);
281 isneg = spu_rlmaska(isneg, -32);
282 result = spu_sel(result, nresult, (vec_ullong2)isneg);
283
284
285 /*
286 * Finally, special cases/errors.
287 */
288 xhigh = (vec_uint4)spu_shuffle(xabs, xabs, dup_even);
289 xlow = (vec_uint4)spu_shuffle(xabs, xabs, dup_odd);
290
291 /* x = zero, return infinite */
292 x1 = spu_or(xhigh, xlow);
293 iszero = spu_cmpeq(x1, 0);
294
295 /* x = negative integer, return infinite */
296 xtrunc = _truncd2(xabs);
297 xthigh = (vec_uint4)spu_shuffle(xtrunc, xtrunc, dup_even);
298 xtlow = (vec_uint4)spu_shuffle(xtrunc, xtrunc, dup_odd);
299 isint = spu_and(spu_cmpeq(xthigh, xhigh), spu_cmpeq(xtlow, xlow));
300 isnposint = spu_or(spu_and(isint, isneg), iszero);
301 result = spu_sel(result, infinited, (vec_ullong2)isnposint);
302
303 /* x = 1.0 or 2.0, return 0.0 */
304 is1 = spu_cmpeq((vec_uint4)x, (vec_uint4)oned);
305 is1 = spu_and(is1, spu_shuffle(is1, is1, swap_word));
306 is2 = spu_cmpeq((vec_uint4)x, (vec_uint4)twod);
307 is2 = spu_and(is2, spu_shuffle(is2, is2, swap_word));
308 result = spu_sel(result, zerod, (vec_ullong2)spu_or(is1,is2));
309
310 /* x = +/- infinite or nan, return |x| */
311 isnaninf = spu_cmpgt(xhigh, 0x7FEFFFFF);
312 result = spu_sel(result, xabs, (vec_ullong2)isnaninf);
313
314 return result;
315 }
316
317 #endif /* _LGAMMAD2_H_ */
318 #endif /* __SPU__ */
319
