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Use relative rpath to support relocatable binary packages
[gromacs.git] / include / gmx_math_x86_sse4_1_double.h
blob37a9cac29b2d8272e38bd9dfc6e53bb24cbcbfc5
1 /* -*- mode: c; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4; c-file-style: "stroustrup"; -*-
2 *
3 *
4 * This file is part of GROMACS.
5 * Copyright (c) 2012-
6 *
7 * Written by the Gromacs development team under coordination of
8 * David van der Spoel, Berk Hess, and Erik Lindahl.
9 *
10 * This library is free software; you can redistribute it and/or
11 * modify it under the terms of the GNU Lesser General Public License
12 * as published by the Free Software Foundation; either version 2
13 * of the License, or (at your option) any later version.
14 *
15 * To help us fund GROMACS development, we humbly ask that you cite
16 * the research papers on the package. Check out http://www.gromacs.org
17 *
18 * And Hey:
19 * Gnomes, ROck Monsters And Chili Sauce
20 */
21 #ifndef _gmx_math_x86_sse4_1_double_h_
22 #define _gmx_math_x86_sse4_1_double_h_
23
24 #include <stdio.h>
25 #include <math.h>
26
27 #include "gmx_x86_sse4.h"
28
29
30
31 #ifndef M_PI
32 # define M_PI 3.14159265358979323846264338327950288
33 #endif
34
35 /************************
36 * *
37 * Simple math routines *
38 * *
39 ************************/
40
41 /* 1.0/sqrt(x) */
42 static gmx_inline __m128d
43 gmx_mm_invsqrt_pd(__m128d x)
44 {
45 const __m128d half = _mm_set1_pd(0.5);
46 const __m128d three = _mm_set1_pd(3.0);
47
48 /* Lookup instruction only exists in single precision, convert back and forth... */
49 __m128d lu = _mm_cvtps_pd(_mm_rsqrt_ps( _mm_cvtpd_ps(x)));
50
51 lu = _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu,lu),x)),lu));
52 return _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu,lu),x)),lu));
53 }
54
55 /* 1.0/sqrt(x), done for a pair of arguments to improve throughput */
56 static void
57 gmx_mm_invsqrt_pair_pd(__m128d x1, __m128d x2, __m128d *invsqrt1, __m128d *invsqrt2)
58 {
59 const __m128d half = _mm_set1_pd(0.5);
60 const __m128d three = _mm_set1_pd(3.0);
61 const __m128 halff = _mm_set1_ps(0.5f);
62 const __m128 threef = _mm_set1_ps(3.0f);
63
64 __m128 xf,luf;
65 __m128d lu1,lu2;
66
67 /* Do first N-R step in float for 2x throughput */
68 xf = _mm_shuffle_ps(_mm_cvtpd_ps(x1),_mm_cvtpd_ps(x2),MM_SHUFFLE(1,0,1,0));
69 luf = _mm_rsqrt_ps(xf);
70 luf = _mm_mul_ps(halff,_mm_mul_ps(_mm_sub_ps(threef,_mm_mul_ps(_mm_mul_ps(luf,luf),xf)),luf));
71
72 lu2 = _mm_cvtps_pd(_mm_shuffle_ps(luf,luf,MM_SHUFFLE(3,2,3,2)));
73 lu1 = _mm_cvtps_pd(luf);
74
75 *invsqrt1 = _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu1,lu1),x1)),lu1));
76 *invsqrt2 = _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu2,lu2),x2)),lu2));
77 }
78
79 /* sqrt(x) - Do NOT use this (but rather invsqrt) if you actually need 1.0/sqrt(x) */
80 static gmx_inline __m128d
81 gmx_mm_sqrt_pd(__m128d x)
82 {
83 __m128d mask;
84 __m128d res;
85
86 mask = _mm_cmpeq_pd(x,_mm_setzero_pd());
87 res = _mm_andnot_pd(mask,gmx_mm_invsqrt_pd(x));
88
89 res = _mm_mul_pd(x,res);
90
91 return res;
92 }
93
94 /* 1.0/x */
95 static gmx_inline __m128d
96 gmx_mm_inv_pd(__m128d x)
97 {
98 const __m128d two = _mm_set1_pd(2.0);
99
100 /* Lookup instruction only exists in single precision, convert back and forth... */
101 __m128d lu = _mm_cvtps_pd(_mm_rcp_ps( _mm_cvtpd_ps(x)));
102
103 /* Perform two N-R steps for double precision */
104 lu = _mm_mul_pd(lu,_mm_sub_pd(two,_mm_mul_pd(x,lu)));
105 return _mm_mul_pd(lu,_mm_sub_pd(two,_mm_mul_pd(x,lu)));
106 }
107
108 static gmx_inline __m128d
109 gmx_mm_abs_pd(__m128d x)
110 {
111 const __m128d signmask = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF, 0xFFFFFFFF, 0x7FFFFFFF, 0xFFFFFFFF) );
112
113 return _mm_and_pd(x,signmask);
114 }
115
116
117 /*
118 * 2^x function.
119 *
120 * The 2^w term is calculated from a (6,0)-th order (no denominator) Minimax polynomia on the interval
121 * [-0.5,0.5].
122 *
123 * The approximation on [-0.5,0.5] is a rational Padé approximation, 1+2*P(x^2)/(Q(x^2)-P(x^2)),
124 * according to the same algorithm as used in the Cephes/netlib math routines.
125 */
126 static __m128d
127 gmx_mm_exp2_pd(__m128d x)
128 {
129 /* Lower bound: We do not allow numbers that would lead to an IEEE fp representation exponent smaller than -126. */
130 const __m128d arglimit = _mm_set1_pd(1022.0);
131 const __m128i expbase = _mm_set1_epi32(1023);
132
133 const __m128d P2 = _mm_set1_pd(2.30933477057345225087e-2);
134 const __m128d P1 = _mm_set1_pd(2.02020656693165307700e1);
135 const __m128d P0 = _mm_set1_pd(1.51390680115615096133e3);
136 /* Q2 == 1.0 */
137 const __m128d Q1 = _mm_set1_pd(2.33184211722314911771e2);
138 const __m128d Q0 = _mm_set1_pd(4.36821166879210612817e3);
139 const __m128d one = _mm_set1_pd(1.0);
140 const __m128d two = _mm_set1_pd(2.0);
141
142 __m128d valuemask;
143 __m128i iexppart;
144 __m128d fexppart;
145 __m128d intpart;
146 __m128d z,z2;
147 __m128d PolyP,PolyQ;
148
149 iexppart = _mm_cvtpd_epi32(x);
150 intpart = _mm_round_pd(x,_MM_FROUND_TO_NEAREST_INT);
151
152 /* The two lowest elements of iexppart now contains 32-bit numbers with a correctly biased exponent.
153 * To be able to shift it into the exponent for a double precision number we first need to
154 * shuffle so that the lower half contains the first element, and the upper half the second.
155 * This should really be done as a zero-extension, but since the next instructions will shift
156 * the registers left by 52 bits it doesn't matter what we put there - it will be shifted out.
157 * (thus we just use element 2 from iexppart).
158 */
159 iexppart = _mm_shuffle_epi32(iexppart,_MM_SHUFFLE(2,1,2,0));
160
161 /* Do the shift operation on the 64-bit registers */
162 iexppart = _mm_add_epi32(iexppart,expbase);
163 iexppart = _mm_slli_epi64(iexppart,52);
164
165 valuemask = _mm_cmpge_pd(arglimit,gmx_mm_abs_pd(x));
166 fexppart = _mm_and_pd(valuemask,gmx_mm_castsi128_pd(iexppart));
167
168 z = _mm_sub_pd(x,intpart);
169 z2 = _mm_mul_pd(z,z);
170
171 PolyP = _mm_mul_pd(P2,z2);
172 PolyP = _mm_add_pd(PolyP,P1);
173 PolyQ = _mm_add_pd(z2,Q1);
174 PolyP = _mm_mul_pd(PolyP,z2);
175 PolyQ = _mm_mul_pd(PolyQ,z2);
176 PolyP = _mm_add_pd(PolyP,P0);
177 PolyQ = _mm_add_pd(PolyQ,Q0);
178 PolyP = _mm_mul_pd(PolyP,z);
179
180 z = _mm_mul_pd(PolyP,gmx_mm_inv_pd(_mm_sub_pd(PolyQ,PolyP)));
181 z = _mm_add_pd(one,_mm_mul_pd(two,z));
182
183 z = _mm_mul_pd(z,fexppart);
184
185 return z;
186 }
187
188 /* Exponential function. This could be calculated from 2^x as Exp(x)=2^(y), where y=log2(e)*x,
189 * but there will then be a small rounding error since we lose some precision due to the
190 * multiplication. This will then be magnified a lot by the exponential.
191 *
192 * Instead, we calculate the fractional part directly as a Padé approximation of
193 * Exp(z) on [-0.5,0.5]. We use extended precision arithmetics to calculate the fraction
194 * remaining after 2^y, which avoids the precision-loss.
195 */
196 static __m128d
197 gmx_mm_exp_pd(__m128d exparg)
198 {
199 const __m128d argscale = _mm_set1_pd(1.4426950408889634073599);
200 /* Lower bound: We do not allow numbers that would lead to an IEEE fp representation exponent smaller than -126. */
201 const __m128d arglimit = _mm_set1_pd(1022.0);
202 const __m128i expbase = _mm_set1_epi32(1023);
203
204 const __m128d invargscale0 = _mm_set1_pd(6.93145751953125e-1);
205 const __m128d invargscale1 = _mm_set1_pd(1.42860682030941723212e-6);
206
207 const __m128d P2 = _mm_set1_pd(1.26177193074810590878e-4);
208 const __m128d P1 = _mm_set1_pd(3.02994407707441961300e-2);
209 /* P0 == 1.0 */
210 const __m128d Q3 = _mm_set1_pd(3.00198505138664455042E-6);
211 const __m128d Q2 = _mm_set1_pd(2.52448340349684104192E-3);
212 const __m128d Q1 = _mm_set1_pd(2.27265548208155028766E-1);
213 /* Q0 == 2.0 */
214 const __m128d one = _mm_set1_pd(1.0);
215 const __m128d two = _mm_set1_pd(2.0);
216
217 __m128d valuemask;
218 __m128i iexppart;
219 __m128d fexppart;
220 __m128d intpart;
221 __m128d x,z,z2;
222 __m128d PolyP,PolyQ;
223
224 x = _mm_mul_pd(exparg,argscale);
225
226 iexppart = _mm_cvtpd_epi32(x);
227 intpart = _mm_round_pd(x,_MM_FROUND_TO_NEAREST_INT);
228
229 /* The two lowest elements of iexppart now contains 32-bit numbers with a correctly biased exponent.
230 * To be able to shift it into the exponent for a double precision number we first need to
231 * shuffle so that the lower half contains the first element, and the upper half the second.
232 * This should really be done as a zero-extension, but since the next instructions will shift
233 * the registers left by 52 bits it doesn't matter what we put there - it will be shifted out.
234 * (thus we just use element 2 from iexppart).
235 */
236 iexppart = _mm_shuffle_epi32(iexppart,_MM_SHUFFLE(2,1,2,0));
237
238 /* Do the shift operation on the 64-bit registers */
239 iexppart = _mm_add_epi32(iexppart,expbase);
240 iexppart = _mm_slli_epi64(iexppart,52);
241
242 valuemask = _mm_cmpge_pd(arglimit,gmx_mm_abs_pd(x));
243 fexppart = _mm_and_pd(valuemask,gmx_mm_castsi128_pd(iexppart));
244
245 z = _mm_sub_pd(exparg,_mm_mul_pd(invargscale0,intpart));
246 z = _mm_sub_pd(z,_mm_mul_pd(invargscale1,intpart));
247
248 z2 = _mm_mul_pd(z,z);
249
250 PolyQ = _mm_mul_pd(Q3,z2);
251 PolyQ = _mm_add_pd(PolyQ,Q2);
252 PolyP = _mm_mul_pd(P2,z2);
253 PolyQ = _mm_mul_pd(PolyQ,z2);
254 PolyP = _mm_add_pd(PolyP,P1);
255 PolyQ = _mm_add_pd(PolyQ,Q1);
256 PolyP = _mm_mul_pd(PolyP,z2);
257 PolyQ = _mm_mul_pd(PolyQ,z2);
258 PolyP = _mm_add_pd(PolyP,one);
259 PolyQ = _mm_add_pd(PolyQ,two);
260
261 PolyP = _mm_mul_pd(PolyP,z);
262
263 z = _mm_mul_pd(PolyP,gmx_mm_inv_pd(_mm_sub_pd(PolyQ,PolyP)));
264 z = _mm_add_pd(one,_mm_mul_pd(two,z));
265
266 z = _mm_mul_pd(z,fexppart);
267
268 return z;
269 }
270
271
272
273 static __m128d
274 gmx_mm_log_pd(__m128d x)
275 {
276 /* Same algorithm as cephes library */
277 const __m128d expmask = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FF00000, 0x00000000, 0x7FF00000, 0x00000000) );
278
279 const __m128i expbase_m1 = _mm_set1_epi32(1023-1); /* We want non-IEEE format */
280
281 const __m128d half = _mm_set1_pd(0.5);
282 const __m128d one = _mm_set1_pd(1.0);
283 const __m128d two = _mm_set1_pd(2.0);
284 const __m128d invsq2 = _mm_set1_pd(1.0/sqrt(2.0));
285
286 const __m128d corr1 = _mm_set1_pd(-2.121944400546905827679e-4);
287 const __m128d corr2 = _mm_set1_pd(0.693359375);
288
289 const __m128d P5 = _mm_set1_pd(1.01875663804580931796e-4);
290 const __m128d P4 = _mm_set1_pd(4.97494994976747001425e-1);
291 const __m128d P3 = _mm_set1_pd(4.70579119878881725854e0);
292 const __m128d P2 = _mm_set1_pd(1.44989225341610930846e1);
293 const __m128d P1 = _mm_set1_pd(1.79368678507819816313e1);
294 const __m128d P0 = _mm_set1_pd(7.70838733755885391666e0);
295
296 const __m128d Q4 = _mm_set1_pd(1.12873587189167450590e1);
297 const __m128d Q3 = _mm_set1_pd(4.52279145837532221105e1);
298 const __m128d Q2 = _mm_set1_pd(8.29875266912776603211e1);
299 const __m128d Q1 = _mm_set1_pd(7.11544750618563894466e1);
300 const __m128d Q0 = _mm_set1_pd(2.31251620126765340583e1);
301
302 const __m128d R2 = _mm_set1_pd(-7.89580278884799154124e-1);
303 const __m128d R1 = _mm_set1_pd(1.63866645699558079767e1);
304 const __m128d R0 = _mm_set1_pd(-6.41409952958715622951e1);
305
306 const __m128d S2 = _mm_set1_pd(-3.56722798256324312549E1);
307 const __m128d S1 = _mm_set1_pd(3.12093766372244180303E2);
308 const __m128d S0 = _mm_set1_pd(-7.69691943550460008604E2);
309
310 __m128d fexp;
311 __m128i iexp;
312
313 __m128d mask1,mask2;
314 __m128d corr,t1,t2,q;
315 __m128d zA,yA,xA,zB,yB,xB,z;
316 __m128d polyR,polyS;
317 __m128d polyP1,polyP2,polyQ1,polyQ2;
318
319 /* Separate x into exponent and mantissa, with a mantissa in the range [0.5..1[ (not IEEE754 standard!) */
320 fexp = _mm_and_pd(x,expmask);
321 iexp = gmx_mm_castpd_si128(fexp);
322 iexp = _mm_srli_epi64(iexp,52);
323 iexp = _mm_sub_epi32(iexp,expbase_m1);
324 iexp = _mm_shuffle_epi32(iexp, _MM_SHUFFLE(1,1,2,0) );
325 fexp = _mm_cvtepi32_pd(iexp);
326
327 x = _mm_andnot_pd(expmask,x);
328 x = _mm_or_pd(x,one);
329 x = _mm_mul_pd(x,half);
330
331 mask1 = _mm_cmpgt_pd(gmx_mm_abs_pd(fexp),two);
332 mask2 = _mm_cmplt_pd(x,invsq2);
333
334 fexp = _mm_sub_pd(fexp,_mm_and_pd(mask2,one));
335
336 /* If mask1 is set ('A') */
337 zA = _mm_sub_pd(x,half);
338 t1 = _mm_blendv_pd( zA,x,mask2 );
339 zA = _mm_sub_pd(t1,half);
340 t2 = _mm_blendv_pd( x,zA,mask2 );
341 yA = _mm_mul_pd(half,_mm_add_pd(t2,one));
342
343 xA = _mm_mul_pd(zA,gmx_mm_inv_pd(yA));
344 zA = _mm_mul_pd(xA,xA);
345
346 /* EVALUATE POLY */
347 polyR = _mm_mul_pd(R2,zA);
348 polyR = _mm_add_pd(polyR,R1);
349 polyR = _mm_mul_pd(polyR,zA);
350 polyR = _mm_add_pd(polyR,R0);
351
352 polyS = _mm_add_pd(zA,S2);
353 polyS = _mm_mul_pd(polyS,zA);
354 polyS = _mm_add_pd(polyS,S1);
355 polyS = _mm_mul_pd(polyS,zA);
356 polyS = _mm_add_pd(polyS,S0);
357
358 q = _mm_mul_pd(polyR,gmx_mm_inv_pd(polyS));
359 zA = _mm_mul_pd(_mm_mul_pd(xA,zA),q);
360
361 zA = _mm_add_pd(zA,_mm_mul_pd(corr1,fexp));
362 zA = _mm_add_pd(zA,xA);
363 zA = _mm_add_pd(zA,_mm_mul_pd(corr2,fexp));
364
365 /* If mask1 is not set ('B') */
366 corr = _mm_and_pd(mask2,x);
367 xB = _mm_add_pd(x,corr);
368 xB = _mm_sub_pd(xB,one);
369 zB = _mm_mul_pd(xB,xB);
370
371 polyP1 = _mm_mul_pd(P5,zB);
372 polyP2 = _mm_mul_pd(P4,zB);
373 polyP1 = _mm_add_pd(polyP1,P3);
374 polyP2 = _mm_add_pd(polyP2,P2);
375 polyP1 = _mm_mul_pd(polyP1,zB);
376 polyP2 = _mm_mul_pd(polyP2,zB);
377 polyP1 = _mm_add_pd(polyP1,P1);
378 polyP2 = _mm_add_pd(polyP2,P0);
379 polyP1 = _mm_mul_pd(polyP1,xB);
380 polyP1 = _mm_add_pd(polyP1,polyP2);
381
382 polyQ2 = _mm_mul_pd(Q4,zB);
383 polyQ1 = _mm_add_pd(zB,Q3);
384 polyQ2 = _mm_add_pd(polyQ2,Q2);
385 polyQ1 = _mm_mul_pd(polyQ1,zB);
386 polyQ2 = _mm_mul_pd(polyQ2,zB);
387 polyQ1 = _mm_add_pd(polyQ1,Q1);
388 polyQ2 = _mm_add_pd(polyQ2,Q0);
389 polyQ1 = _mm_mul_pd(polyQ1,xB);
390 polyQ1 = _mm_add_pd(polyQ1,polyQ2);
391
392 fexp = _mm_and_pd(fexp,_mm_cmpneq_pd(fexp,_mm_setzero_pd()));
393
394 q = _mm_mul_pd(polyP1,gmx_mm_inv_pd(polyQ1));
395 yB = _mm_mul_pd(_mm_mul_pd(xB,zB),q);
396
397 yB = _mm_add_pd(yB,_mm_mul_pd(corr1,fexp));
398 yB = _mm_sub_pd(yB,_mm_mul_pd(half,zB));
399 zB = _mm_add_pd(xB,yB);
400 zB = _mm_add_pd(zB,_mm_mul_pd(corr2,fexp));
401
402 z = _mm_blendv_pd( zB,zA,mask1 );
403
404 return z;
405 }
406
407
408 static __m128d
409 gmx_mm_erf_pd(__m128d x)
410 {
411 /* Coefficients for minimax approximation of erf(x)=x*(CAoffset + P(x^2)/Q(x^2)) in range [-0.75,0.75] */
412 const __m128d CAP4 = _mm_set1_pd(-0.431780540597889301512e-4);
413 const __m128d CAP3 = _mm_set1_pd(-0.00578562306260059236059);
414 const __m128d CAP2 = _mm_set1_pd(-0.028593586920219752446);
415 const __m128d CAP1 = _mm_set1_pd(-0.315924962948621698209);
416 const __m128d CAP0 = _mm_set1_pd(0.14952975608477029151);
417
418 const __m128d CAQ5 = _mm_set1_pd(-0.374089300177174709737e-5);
419 const __m128d CAQ4 = _mm_set1_pd(0.00015126584532155383535);
420 const __m128d CAQ3 = _mm_set1_pd(0.00536692680669480725423);
421 const __m128d CAQ2 = _mm_set1_pd(0.0668686825594046122636);
422 const __m128d CAQ1 = _mm_set1_pd(0.402604990869284362773);
423 /* CAQ0 == 1.0 */
424 const __m128d CAoffset = _mm_set1_pd(0.9788494110107421875);
425
426 /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)*x*(P(x-1)/Q(x-1)) in range [1.0,4.5] */
427 const __m128d CBP6 = _mm_set1_pd(2.49650423685462752497647637088e-10);
428 const __m128d CBP5 = _mm_set1_pd(0.00119770193298159629350136085658);
429 const __m128d CBP4 = _mm_set1_pd(0.0164944422378370965881008942733);
430 const __m128d CBP3 = _mm_set1_pd(0.0984581468691775932063932439252);
431 const __m128d CBP2 = _mm_set1_pd(0.317364595806937763843589437418);
432 const __m128d CBP1 = _mm_set1_pd(0.554167062641455850932670067075);
433 const __m128d CBP0 = _mm_set1_pd(0.427583576155807163756925301060);
434 const __m128d CBQ7 = _mm_set1_pd(0.00212288829699830145976198384930);
435 const __m128d CBQ6 = _mm_set1_pd(0.0334810979522685300554606393425);
436 const __m128d CBQ5 = _mm_set1_pd(0.2361713785181450957579508850717);
437 const __m128d CBQ4 = _mm_set1_pd(0.955364736493055670530981883072);
438 const __m128d CBQ3 = _mm_set1_pd(2.36815675631420037315349279199);
439 const __m128d CBQ2 = _mm_set1_pd(3.55261649184083035537184223542);
440 const __m128d CBQ1 = _mm_set1_pd(2.93501136050160872574376997993);
441 /* CBQ0 == 1.0 */
442
443 /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)/x*(P(1/x)/Q(1/x)) in range [4.5,inf] */
444 const __m128d CCP6 = _mm_set1_pd(-2.8175401114513378771);
445 const __m128d CCP5 = _mm_set1_pd(-3.22729451764143718517);
446 const __m128d CCP4 = _mm_set1_pd(-2.5518551727311523996);
447 const __m128d CCP3 = _mm_set1_pd(-0.687717681153649930619);
448 const __m128d CCP2 = _mm_set1_pd(-0.212652252872804219852);
449 const __m128d CCP1 = _mm_set1_pd(0.0175389834052493308818);
450 const __m128d CCP0 = _mm_set1_pd(0.00628057170626964891937);
451
452 const __m128d CCQ6 = _mm_set1_pd(5.48409182238641741584);
453 const __m128d CCQ5 = _mm_set1_pd(13.5064170191802889145);
454 const __m128d CCQ4 = _mm_set1_pd(22.9367376522880577224);
455 const __m128d CCQ3 = _mm_set1_pd(15.930646027911794143);
456 const __m128d CCQ2 = _mm_set1_pd(11.0567237927800161565);
457 const __m128d CCQ1 = _mm_set1_pd(2.79257750980575282228);
458 /* CCQ0 == 1.0 */
459 const __m128d CCoffset = _mm_set1_pd(0.5579090118408203125);
460
461 const __m128d one = _mm_set1_pd(1.0);
462 const __m128d two = _mm_set1_pd(2.0);
463
464 const __m128d signbit = gmx_mm_castsi128_pd( _mm_set_epi32(0x80000000,0x00000000,0x80000000,0x00000000) );
465
466 __m128d xabs,x2,x4,t,t2,w,w2;
467 __m128d PolyAP0,PolyAP1,PolyAQ0,PolyAQ1;
468 __m128d PolyBP0,PolyBP1,PolyBQ0,PolyBQ1;
469 __m128d PolyCP0,PolyCP1,PolyCQ0,PolyCQ1;
470 __m128d res_erf,res_erfcB,res_erfcC,res_erfc,res;
471 __m128d mask,expmx2;
472
473 /* Calculate erf() */
474 xabs = gmx_mm_abs_pd(x);
475 x2 = _mm_mul_pd(x,x);
476 x4 = _mm_mul_pd(x2,x2);
477
478 PolyAP0 = _mm_mul_pd(CAP4,x4);
479 PolyAP1 = _mm_mul_pd(CAP3,x4);
480 PolyAP0 = _mm_add_pd(PolyAP0,CAP2);
481 PolyAP1 = _mm_add_pd(PolyAP1,CAP1);
482 PolyAP0 = _mm_mul_pd(PolyAP0,x4);
483 PolyAP1 = _mm_mul_pd(PolyAP1,x2);
484 PolyAP0 = _mm_add_pd(PolyAP0,CAP0);
485 PolyAP0 = _mm_add_pd(PolyAP0,PolyAP1);
486
487 PolyAQ1 = _mm_mul_pd(CAQ5,x4);
488 PolyAQ0 = _mm_mul_pd(CAQ4,x4);
489 PolyAQ1 = _mm_add_pd(PolyAQ1,CAQ3);
490 PolyAQ0 = _mm_add_pd(PolyAQ0,CAQ2);
491 PolyAQ1 = _mm_mul_pd(PolyAQ1,x4);
492 PolyAQ0 = _mm_mul_pd(PolyAQ0,x4);
493 PolyAQ1 = _mm_add_pd(PolyAQ1,CAQ1);
494 PolyAQ0 = _mm_add_pd(PolyAQ0,one);
495 PolyAQ1 = _mm_mul_pd(PolyAQ1,x2);
496 PolyAQ0 = _mm_add_pd(PolyAQ0,PolyAQ1);
497
498 res_erf = _mm_mul_pd(PolyAP0,gmx_mm_inv_pd(PolyAQ0));
499 res_erf = _mm_add_pd(CAoffset,res_erf);
500 res_erf = _mm_mul_pd(x,res_erf);
501
502 /* Calculate erfc() in range [1,4.5] */
503 t = _mm_sub_pd(xabs,one);
504 t2 = _mm_mul_pd(t,t);
505
506 PolyBP0 = _mm_mul_pd(CBP6,t2);
507 PolyBP1 = _mm_mul_pd(CBP5,t2);
508 PolyBP0 = _mm_add_pd(PolyBP0,CBP4);
509 PolyBP1 = _mm_add_pd(PolyBP1,CBP3);
510 PolyBP0 = _mm_mul_pd(PolyBP0,t2);
511 PolyBP1 = _mm_mul_pd(PolyBP1,t2);
512 PolyBP0 = _mm_add_pd(PolyBP0,CBP2);
513 PolyBP1 = _mm_add_pd(PolyBP1,CBP1);
514 PolyBP0 = _mm_mul_pd(PolyBP0,t2);
515 PolyBP1 = _mm_mul_pd(PolyBP1,t);
516 PolyBP0 = _mm_add_pd(PolyBP0,CBP0);
517 PolyBP0 = _mm_add_pd(PolyBP0,PolyBP1);
518
519 PolyBQ1 = _mm_mul_pd(CBQ7,t2);
520 PolyBQ0 = _mm_mul_pd(CBQ6,t2);
521 PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ5);
522 PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ4);
523 PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
524 PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
525 PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ3);
526 PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ2);
527 PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
528 PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
529 PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ1);
530 PolyBQ0 = _mm_add_pd(PolyBQ0,one);
531 PolyBQ1 = _mm_mul_pd(PolyBQ1,t);
532 PolyBQ0 = _mm_add_pd(PolyBQ0,PolyBQ1);
533
534 res_erfcB = _mm_mul_pd(PolyBP0,gmx_mm_inv_pd(PolyBQ0));
535
536 res_erfcB = _mm_mul_pd(res_erfcB,xabs);
537
538 /* Calculate erfc() in range [4.5,inf] */
539 w = gmx_mm_inv_pd(xabs);
540 w2 = _mm_mul_pd(w,w);
541
542 PolyCP0 = _mm_mul_pd(CCP6,w2);
543 PolyCP1 = _mm_mul_pd(CCP5,w2);
544 PolyCP0 = _mm_add_pd(PolyCP0,CCP4);
545 PolyCP1 = _mm_add_pd(PolyCP1,CCP3);
546 PolyCP0 = _mm_mul_pd(PolyCP0,w2);
547 PolyCP1 = _mm_mul_pd(PolyCP1,w2);
548 PolyCP0 = _mm_add_pd(PolyCP0,CCP2);
549 PolyCP1 = _mm_add_pd(PolyCP1,CCP1);
550 PolyCP0 = _mm_mul_pd(PolyCP0,w2);
551 PolyCP1 = _mm_mul_pd(PolyCP1,w);
552 PolyCP0 = _mm_add_pd(PolyCP0,CCP0);
553 PolyCP0 = _mm_add_pd(PolyCP0,PolyCP1);
554
555 PolyCQ0 = _mm_mul_pd(CCQ6,w2);
556 PolyCQ1 = _mm_mul_pd(CCQ5,w2);
557 PolyCQ0 = _mm_add_pd(PolyCQ0,CCQ4);
558 PolyCQ1 = _mm_add_pd(PolyCQ1,CCQ3);
559 PolyCQ0 = _mm_mul_pd(PolyCQ0,w2);
560 PolyCQ1 = _mm_mul_pd(PolyCQ1,w2);
561 PolyCQ0 = _mm_add_pd(PolyCQ0,CCQ2);
562 PolyCQ1 = _mm_add_pd(PolyCQ1,CCQ1);
563 PolyCQ0 = _mm_mul_pd(PolyCQ0,w2);
564 PolyCQ1 = _mm_mul_pd(PolyCQ1,w);
565 PolyCQ0 = _mm_add_pd(PolyCQ0,one);
566 PolyCQ0 = _mm_add_pd(PolyCQ0,PolyCQ1);
567
568 expmx2 = gmx_mm_exp_pd( _mm_or_pd(signbit, x2) );
569
570 res_erfcC = _mm_mul_pd(PolyCP0,gmx_mm_inv_pd(PolyCQ0));
571 res_erfcC = _mm_add_pd(res_erfcC,CCoffset);
572 res_erfcC = _mm_mul_pd(res_erfcC,w);
573
574 mask = _mm_cmpgt_pd(xabs,_mm_set1_pd(4.5));
575 res_erfc = _mm_blendv_pd(res_erfcB,res_erfcC,mask);
576
577 res_erfc = _mm_mul_pd(res_erfc,expmx2);
578
579 /* erfc(x<0) = 2-erfc(|x|) */
580 mask = _mm_cmplt_pd(x,_mm_setzero_pd());
581 res_erfc = _mm_blendv_pd(res_erfc,_mm_sub_pd(two,res_erfc),mask);
582
583 /* Select erf() or erfc() */
584 mask = _mm_cmplt_pd(xabs,one);
585 res = _mm_blendv_pd(_mm_sub_pd(one,res_erfc),res_erf,mask);
586
587 return res;
588 }
589
590
591 static __m128d
592 gmx_mm_erfc_pd(__m128d x)
593 {
594 /* Coefficients for minimax approximation of erf(x)=x*(CAoffset + P(x^2)/Q(x^2)) in range [-0.75,0.75] */
595 const __m128d CAP4 = _mm_set1_pd(-0.431780540597889301512e-4);
596 const __m128d CAP3 = _mm_set1_pd(-0.00578562306260059236059);
597 const __m128d CAP2 = _mm_set1_pd(-0.028593586920219752446);
598 const __m128d CAP1 = _mm_set1_pd(-0.315924962948621698209);
599 const __m128d CAP0 = _mm_set1_pd(0.14952975608477029151);
600
601 const __m128d CAQ5 = _mm_set1_pd(-0.374089300177174709737e-5);
602 const __m128d CAQ4 = _mm_set1_pd(0.00015126584532155383535);
603 const __m128d CAQ3 = _mm_set1_pd(0.00536692680669480725423);
604 const __m128d CAQ2 = _mm_set1_pd(0.0668686825594046122636);
605 const __m128d CAQ1 = _mm_set1_pd(0.402604990869284362773);
606 /* CAQ0 == 1.0 */
607 const __m128d CAoffset = _mm_set1_pd(0.9788494110107421875);
608
609 /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)*x*(P(x-1)/Q(x-1)) in range [1.0,4.5] */
610 const __m128d CBP6 = _mm_set1_pd(2.49650423685462752497647637088e-10);
611 const __m128d CBP5 = _mm_set1_pd(0.00119770193298159629350136085658);
612 const __m128d CBP4 = _mm_set1_pd(0.0164944422378370965881008942733);
613 const __m128d CBP3 = _mm_set1_pd(0.0984581468691775932063932439252);
614 const __m128d CBP2 = _mm_set1_pd(0.317364595806937763843589437418);
615 const __m128d CBP1 = _mm_set1_pd(0.554167062641455850932670067075);
616 const __m128d CBP0 = _mm_set1_pd(0.427583576155807163756925301060);
617 const __m128d CBQ7 = _mm_set1_pd(0.00212288829699830145976198384930);
618 const __m128d CBQ6 = _mm_set1_pd(0.0334810979522685300554606393425);
619 const __m128d CBQ5 = _mm_set1_pd(0.2361713785181450957579508850717);
620 const __m128d CBQ4 = _mm_set1_pd(0.955364736493055670530981883072);
621 const __m128d CBQ3 = _mm_set1_pd(2.36815675631420037315349279199);
622 const __m128d CBQ2 = _mm_set1_pd(3.55261649184083035537184223542);
623 const __m128d CBQ1 = _mm_set1_pd(2.93501136050160872574376997993);
624 /* CBQ0 == 1.0 */
625
626 /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)/x*(P(1/x)/Q(1/x)) in range [4.5,inf] */
627 const __m128d CCP6 = _mm_set1_pd(-2.8175401114513378771);
628 const __m128d CCP5 = _mm_set1_pd(-3.22729451764143718517);
629 const __m128d CCP4 = _mm_set1_pd(-2.5518551727311523996);
630 const __m128d CCP3 = _mm_set1_pd(-0.687717681153649930619);
631 const __m128d CCP2 = _mm_set1_pd(-0.212652252872804219852);
632 const __m128d CCP1 = _mm_set1_pd(0.0175389834052493308818);
633 const __m128d CCP0 = _mm_set1_pd(0.00628057170626964891937);
634
635 const __m128d CCQ6 = _mm_set1_pd(5.48409182238641741584);
636 const __m128d CCQ5 = _mm_set1_pd(13.5064170191802889145);
637 const __m128d CCQ4 = _mm_set1_pd(22.9367376522880577224);
638 const __m128d CCQ3 = _mm_set1_pd(15.930646027911794143);
639 const __m128d CCQ2 = _mm_set1_pd(11.0567237927800161565);
640 const __m128d CCQ1 = _mm_set1_pd(2.79257750980575282228);
641 /* CCQ0 == 1.0 */
642 const __m128d CCoffset = _mm_set1_pd(0.5579090118408203125);
643
644 const __m128d one = _mm_set1_pd(1.0);
645 const __m128d two = _mm_set1_pd(2.0);
646
647 const __m128d signbit = gmx_mm_castsi128_pd( _mm_set_epi32(0x80000000,0x00000000,0x80000000,0x00000000) );
648
649 __m128d xabs,x2,x4,t,t2,w,w2;
650 __m128d PolyAP0,PolyAP1,PolyAQ0,PolyAQ1;
651 __m128d PolyBP0,PolyBP1,PolyBQ0,PolyBQ1;
652 __m128d PolyCP0,PolyCP1,PolyCQ0,PolyCQ1;
653 __m128d res_erf,res_erfcB,res_erfcC,res_erfc,res;
654 __m128d mask,expmx2;
655
656 /* Calculate erf() */
657 xabs = gmx_mm_abs_pd(x);
658 x2 = _mm_mul_pd(x,x);
659 x4 = _mm_mul_pd(x2,x2);
660
661 PolyAP0 = _mm_mul_pd(CAP4,x4);
662 PolyAP1 = _mm_mul_pd(CAP3,x4);
663 PolyAP0 = _mm_add_pd(PolyAP0,CAP2);
664 PolyAP1 = _mm_add_pd(PolyAP1,CAP1);
665 PolyAP0 = _mm_mul_pd(PolyAP0,x4);
666 PolyAP1 = _mm_mul_pd(PolyAP1,x2);
667 PolyAP0 = _mm_add_pd(PolyAP0,CAP0);
668 PolyAP0 = _mm_add_pd(PolyAP0,PolyAP1);
669
670 PolyAQ1 = _mm_mul_pd(CAQ5,x4);
671 PolyAQ0 = _mm_mul_pd(CAQ4,x4);
672 PolyAQ1 = _mm_add_pd(PolyAQ1,CAQ3);
673 PolyAQ0 = _mm_add_pd(PolyAQ0,CAQ2);
674 PolyAQ1 = _mm_mul_pd(PolyAQ1,x4);
675 PolyAQ0 = _mm_mul_pd(PolyAQ0,x4);
676 PolyAQ1 = _mm_add_pd(PolyAQ1,CAQ1);
677 PolyAQ0 = _mm_add_pd(PolyAQ0,one);
678 PolyAQ1 = _mm_mul_pd(PolyAQ1,x2);
679 PolyAQ0 = _mm_add_pd(PolyAQ0,PolyAQ1);
680
681 res_erf = _mm_mul_pd(PolyAP0,gmx_mm_inv_pd(PolyAQ0));
682 res_erf = _mm_add_pd(CAoffset,res_erf);
683 res_erf = _mm_mul_pd(x,res_erf);
684
685 /* Calculate erfc() in range [1,4.5] */
686 t = _mm_sub_pd(xabs,one);
687 t2 = _mm_mul_pd(t,t);
688
689 PolyBP0 = _mm_mul_pd(CBP6,t2);
690 PolyBP1 = _mm_mul_pd(CBP5,t2);
691 PolyBP0 = _mm_add_pd(PolyBP0,CBP4);
692 PolyBP1 = _mm_add_pd(PolyBP1,CBP3);
693 PolyBP0 = _mm_mul_pd(PolyBP0,t2);
694 PolyBP1 = _mm_mul_pd(PolyBP1,t2);
695 PolyBP0 = _mm_add_pd(PolyBP0,CBP2);
696 PolyBP1 = _mm_add_pd(PolyBP1,CBP1);
697 PolyBP0 = _mm_mul_pd(PolyBP0,t2);
698 PolyBP1 = _mm_mul_pd(PolyBP1,t);
699 PolyBP0 = _mm_add_pd(PolyBP0,CBP0);
700 PolyBP0 = _mm_add_pd(PolyBP0,PolyBP1);
701
702 PolyBQ1 = _mm_mul_pd(CBQ7,t2);
703 PolyBQ0 = _mm_mul_pd(CBQ6,t2);
704 PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ5);
705 PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ4);
706 PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
707 PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
708 PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ3);
709 PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ2);
710 PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
711 PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
712 PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ1);
713 PolyBQ0 = _mm_add_pd(PolyBQ0,one);
714 PolyBQ1 = _mm_mul_pd(PolyBQ1,t);
715 PolyBQ0 = _mm_add_pd(PolyBQ0,PolyBQ1);
716
717 res_erfcB = _mm_mul_pd(PolyBP0,gmx_mm_inv_pd(PolyBQ0));
718
719 res_erfcB = _mm_mul_pd(res_erfcB,xabs);
720
721 /* Calculate erfc() in range [4.5,inf] */
722 w = gmx_mm_inv_pd(xabs);
723 w2 = _mm_mul_pd(w,w);
724
725 PolyCP0 = _mm_mul_pd(CCP6,w2);
726 PolyCP1 = _mm_mul_pd(CCP5,w2);
727 PolyCP0 = _mm_add_pd(PolyCP0,CCP4);
728 PolyCP1 = _mm_add_pd(PolyCP1,CCP3);
729 PolyCP0 = _mm_mul_pd(PolyCP0,w2);
730 PolyCP1 = _mm_mul_pd(PolyCP1,w2);
731 PolyCP0 = _mm_add_pd(PolyCP0,CCP2);
732 PolyCP1 = _mm_add_pd(PolyCP1,CCP1);
733 PolyCP0 = _mm_mul_pd(PolyCP0,w2);
734 PolyCP1 = _mm_mul_pd(PolyCP1,w);
735 PolyCP0 = _mm_add_pd(PolyCP0,CCP0);
736 PolyCP0 = _mm_add_pd(PolyCP0,PolyCP1);
737
738 PolyCQ0 = _mm_mul_pd(CCQ6,w2);
739 PolyCQ1 = _mm_mul_pd(CCQ5,w2);
740 PolyCQ0 = _mm_add_pd(PolyCQ0,CCQ4);
741 PolyCQ1 = _mm_add_pd(PolyCQ1,CCQ3);
742 PolyCQ0 = _mm_mul_pd(PolyCQ0,w2);
743 PolyCQ1 = _mm_mul_pd(PolyCQ1,w2);
744 PolyCQ0 = _mm_add_pd(PolyCQ0,CCQ2);
745 PolyCQ1 = _mm_add_pd(PolyCQ1,CCQ1);
746 PolyCQ0 = _mm_mul_pd(PolyCQ0,w2);
747 PolyCQ1 = _mm_mul_pd(PolyCQ1,w);
748 PolyCQ0 = _mm_add_pd(PolyCQ0,one);
749 PolyCQ0 = _mm_add_pd(PolyCQ0,PolyCQ1);
750
751 expmx2 = gmx_mm_exp_pd( _mm_or_pd(signbit, x2) );
752
753 res_erfcC = _mm_mul_pd(PolyCP0,gmx_mm_inv_pd(PolyCQ0));
754 res_erfcC = _mm_add_pd(res_erfcC,CCoffset);
755 res_erfcC = _mm_mul_pd(res_erfcC,w);
756
757 mask = _mm_cmpgt_pd(xabs,_mm_set1_pd(4.5));
758 res_erfc = _mm_blendv_pd(res_erfcB,res_erfcC,mask);
759
760 res_erfc = _mm_mul_pd(res_erfc,expmx2);
761
762 /* erfc(x<0) = 2-erfc(|x|) */
763 mask = _mm_cmplt_pd(x,_mm_setzero_pd());
764 res_erfc = _mm_blendv_pd(res_erfc,_mm_sub_pd(two,res_erfc),mask);
765
766 /* Select erf() or erfc() */
767 mask = _mm_cmplt_pd(xabs,one);
768 res = _mm_blendv_pd(res_erfc,_mm_sub_pd(one,res_erf),mask);
769
770 return res;
771 }
772
773
774 /* Calculate the force correction due to PME analytically.
775 *
776 * This routine is meant to enable analytical evaluation of the
777 * direct-space PME electrostatic force to avoid tables.
778 *
779 * The direct-space potential should be Erfc(beta*r)/r, but there
780 * are some problems evaluating that:
781 *
782 * First, the error function is difficult (read: expensive) to
783 * approxmiate accurately for intermediate to large arguments, and
784 * this happens already in ranges of beta*r that occur in simulations.
785 * Second, we now try to avoid calculating potentials in Gromacs but
786 * use forces directly.
787 *
788 * We can simply things slight by noting that the PME part is really
789 * a correction to the normal Coulomb force since Erfc(z)=1-Erf(z), i.e.
790 *
791 * V= 1/r - Erf(beta*r)/r
792 *
793 * The first term we already have from the inverse square root, so
794 * that we can leave out of this routine.
795 *
796 * For pme tolerances of 1e-3 to 1e-8 and cutoffs of 0.5nm to 1.8nm,
797 * the argument beta*r will be in the range 0.15 to ~4. Use your
798 * favorite plotting program to realize how well-behaved Erf(z)/z is
799 * in this range!
800 *
801 * We approximate f(z)=erf(z)/z with a rational minimax polynomial.
802 * However, it turns out it is more efficient to approximate f(z)/z and
803 * then only use even powers. This is another minor optimization, since
804 * we actually WANT f(z)/z, because it is going to be multiplied by
805 * the vector between the two atoms to get the vectorial force. The
806 * fastest flops are the ones we can avoid calculating!
807 *
808 * So, here's how it should be used:
809 *
810 * 1. Calculate r^2.
811 * 2. Multiply by beta^2, so you get z^2=beta^2*r^2.
812 * 3. Evaluate this routine with z^2 as the argument.
813 * 4. The return value is the expression:
814 *
815 *
816 * 2*exp(-z^2) erf(z)
817 * ------------ - --------
818 * sqrt(Pi)*z^2 z^3
819 *
820 * 5. Multiply the entire expression by beta^3. This will get you
821 *
822 * beta^3*2*exp(-z^2) beta^3*erf(z)
823 * ------------------ - ---------------
824 * sqrt(Pi)*z^2 z^3
825 *
826 * or, switching back to r (z=r*beta):
827 *
828 * 2*beta*exp(-r^2*beta^2) erf(r*beta)
829 * ----------------------- - -----------
830 * sqrt(Pi)*r^2 r^3
831 *
832 *
833 * With a bit of math exercise you should be able to confirm that
834 * this is exactly D[Erf[beta*r]/r,r] divided by r another time.
835 *
836 * 6. Add the result to 1/r^3, multiply by the product of the charges,
837 * and you have your force (divided by r). A final multiplication
838 * with the vector connecting the two particles and you have your
839 * vectorial force to add to the particles.
840 *
841 */
842 __m128d
843 gmx_mm_pmecorrF_pd(__m128d z2)
844 {
845 const __m128d FN10 = _mm_set1_pd(-8.0072854618360083154e-14);
846 const __m128d FN9 = _mm_set1_pd(1.1859116242260148027e-11);
847 const __m128d FN8 = _mm_set1_pd(-8.1490406329798423616e-10);
848 const __m128d FN7 = _mm_set1_pd(3.4404793543907847655e-8);
849 const __m128d FN6 = _mm_set1_pd(-9.9471420832602741006e-7);
850 const __m128d FN5 = _mm_set1_pd(0.000020740315999115847456);
851 const __m128d FN4 = _mm_set1_pd(-0.00031991745139313364005);
852 const __m128d FN3 = _mm_set1_pd(0.0035074449373659008203);
853 const __m128d FN2 = _mm_set1_pd(-0.031750380176100813405);
854 const __m128d FN1 = _mm_set1_pd(0.13884101728898463426);
855 const __m128d FN0 = _mm_set1_pd(-0.75225277815249618847);
856
857 const __m128d FD5 = _mm_set1_pd(0.000016009278224355026701);
858 const __m128d FD4 = _mm_set1_pd(0.00051055686934806966046);
859 const __m128d FD3 = _mm_set1_pd(0.0081803507497974289008);
860 const __m128d FD2 = _mm_set1_pd(0.077181146026670287235);
861 const __m128d FD1 = _mm_set1_pd(0.41543303143712535988);
862 const __m128d FD0 = _mm_set1_pd(1.0);
863
864 __m128d z4;
865 __m128d polyFN0,polyFN1,polyFD0,polyFD1;
866
867 z4 = _mm_mul_pd(z2,z2);
868
869 polyFD1 = _mm_mul_pd(FD5,z4);
870 polyFD0 = _mm_mul_pd(FD4,z4);
871 polyFD1 = _mm_add_pd(polyFD1,FD3);
872 polyFD0 = _mm_add_pd(polyFD0,FD2);
873 polyFD1 = _mm_mul_pd(polyFD1,z4);
874 polyFD0 = _mm_mul_pd(polyFD0,z4);
875 polyFD1 = _mm_add_pd(polyFD1,FD1);
876 polyFD0 = _mm_add_pd(polyFD0,FD0);
877 polyFD1 = _mm_mul_pd(polyFD1,z2);
878 polyFD0 = _mm_add_pd(polyFD0,polyFD1);
879
880 polyFD0 = gmx_mm_inv_pd(polyFD0);
881
882 polyFN0 = _mm_mul_pd(FN10,z4);
883 polyFN1 = _mm_mul_pd(FN9,z4);
884 polyFN0 = _mm_add_pd(polyFN0,FN8);
885 polyFN1 = _mm_add_pd(polyFN1,FN7);
886 polyFN0 = _mm_mul_pd(polyFN0,z4);
887 polyFN1 = _mm_mul_pd(polyFN1,z4);
888 polyFN0 = _mm_add_pd(polyFN0,FN6);
889 polyFN1 = _mm_add_pd(polyFN1,FN5);
890 polyFN0 = _mm_mul_pd(polyFN0,z4);
891 polyFN1 = _mm_mul_pd(polyFN1,z4);
892 polyFN0 = _mm_add_pd(polyFN0,FN4);
893 polyFN1 = _mm_add_pd(polyFN1,FN3);
894 polyFN0 = _mm_mul_pd(polyFN0,z4);
895 polyFN1 = _mm_mul_pd(polyFN1,z4);
896 polyFN0 = _mm_add_pd(polyFN0,FN2);
897 polyFN1 = _mm_add_pd(polyFN1,FN1);
898 polyFN0 = _mm_mul_pd(polyFN0,z4);
899 polyFN1 = _mm_mul_pd(polyFN1,z2);
900 polyFN0 = _mm_add_pd(polyFN0,FN0);
901 polyFN0 = _mm_add_pd(polyFN0,polyFN1);
902
903 return _mm_mul_pd(polyFN0,polyFD0);
904 }
905
906
907
908
909 /* Calculate the potential correction due to PME analytically.
910 *
911 * See gmx_mm256_pmecorrF_ps() for details about the approximation.
912 *
913 * This routine calculates Erf(z)/z, although you should provide z^2
914 * as the input argument.
915 *
916 * Here's how it should be used:
917 *
918 * 1. Calculate r^2.
919 * 2. Multiply by beta^2, so you get z^2=beta^2*r^2.
920 * 3. Evaluate this routine with z^2 as the argument.
921 * 4. The return value is the expression:
922 *
923 *
924 * erf(z)
925 * --------
926 * z
927 *
928 * 5. Multiply the entire expression by beta and switching back to r (z=r*beta):
929 *
930 * erf(r*beta)
931 * -----------
932 * r
933 *
934 * 6. Add the result to 1/r, multiply by the product of the charges,
935 * and you have your potential.
936 *
937 */
938 __m128d
939 gmx_mm_pmecorrV_pd(__m256d z2)
940 {
941 const __m128d VN9 = _mm_set1_pd(-9.3723776169321855475e-13);
942 const __m128d VN8 = _mm_set1_pd(1.2280156762674215741e-10);
943 const __m128d VN7 = _mm_set1_pd(-7.3562157912251309487e-9);
944 const __m128d VN6 = _mm_set1_pd(2.6215886208032517509e-7);
945 const __m128d VN5 = _mm_set1_pd(-4.9532491651265819499e-6);
946 const __m128d VN4 = _mm_set1_pd(0.00025907400778966060389);
947 const __m128d VN3 = _mm_set1_pd(0.0010585044856156469792);
948 const __m128d VN2 = _mm_set1_pd(0.045247661136833092885);
949 const __m128d VN1 = _mm_set1_pd(0.11643931522926034421);
950 const __m128d VN0 = _mm_set1_pd(1.1283791671726767970);
951
952 const __m128d VD5 = _mm_set1_pd(0.000021784709867336150342);
953 const __m128d VD4 = _mm_set1_pd(0.00064293662010911388448);
954 const __m128d VD3 = _mm_set1_pd(0.0096311444822588683504);
955 const __m128d VD2 = _mm_set1_pd(0.085608012351550627051);
956 const __m128d VD1 = _mm_set1_pd(0.43652499166614811084);
957 const __m128d VD0 = _mm_set1_pd(1.0);
958
959 __m128d z4;
960 __m128d polyVN0,polyVN1,polyVD0,polyVD1;
961
962 z4 = _mm_mul_pd(z2,z2);
963
964 polyVD1 = _mm_mul_pd(VD5,z4);
965 polyVD0 = _mm_mul_pd(VD4,z4);
966 polyVD1 = _mm_add_pd(polyVD1,VD3);
967 polyVD0 = _mm_add_pd(polyVD0,VD2);
968 polyVD1 = _mm_mul_pd(polyVD1,z4);
969 polyVD0 = _mm_mul_pd(polyVD0,z4);
970 polyVD1 = _mm_add_pd(polyVD1,VD1);
971 polyVD0 = _mm_add_pd(polyVD0,VD0);
972 polyVD1 = _mm_mul_pd(polyVD1,z2);
973 polyVD0 = _mm_add_pd(polyVD0,polyVD1);
974
975 polyVD0 = gmx_mm_inv_pd(polyVD0);
976
977 polyVN1 = _mm_mul_pd(VN9,z4);
978 polyVN0 = _mm_mul_pd(VN8,z4);
979 polyVN1 = _mm_add_pd(polyVN1,VN7);
980 polyVN0 = _mm_add_pd(polyVN0,VN6);
981 polyVN1 = _mm_mul_pd(polyVN1,z4);
982 polyVN0 = _mm_mul_pd(polyVN0,z4);
983 polyVN1 = _mm_add_pd(polyVN1,VN5);
984 polyVN0 = _mm_add_pd(polyVN0,VN4);
985 polyVN1 = _mm_mul_pd(polyVN1,z4);
986 polyVN0 = _mm_mul_pd(polyVN0,z4);
987 polyVN1 = _mm_add_pd(polyVN1,VN3);
988 polyVN0 = _mm_add_pd(polyVN0,VN2);
989 polyVN1 = _mm_mul_pd(polyVN1,z4);
990 polyVN0 = _mm_mul_pd(polyVN0,z4);
991 polyVN1 = _mm_add_pd(polyVN1,VN1);
992 polyVN0 = _mm_add_pd(polyVN0,VN0);
993 polyVN1 = _mm_mul_pd(polyVN1,z2);
994 polyVN0 = _mm_add_pd(polyVN0,polyVN1);
995
996 return _mm_mul_pd(polyVN0,polyVD0);
997 }
998
999
1000 static int
1001 gmx_mm_sincos_pd(__m128d x,
1002 __m128d *sinval,
1003 __m128d *cosval)
1004 {
1005 #ifdef _MSC_VER
1006 __declspec(align(16))
1007 const double sintable[34] =
1008 {
1009 1.00000000000000000e+00 , 0.00000000000000000e+00 ,
1010 9.95184726672196929e-01 , 9.80171403295606036e-02 ,
1011 9.80785280403230431e-01 , 1.95090322016128248e-01 ,
1012 9.56940335732208824e-01 , 2.90284677254462331e-01 ,
1013 9.23879532511286738e-01 , 3.82683432365089782e-01 ,
1014 8.81921264348355050e-01 , 4.71396736825997642e-01 ,
1015 8.31469612302545236e-01 , 5.55570233019602178e-01 ,
1016 7.73010453362736993e-01 , 6.34393284163645488e-01 ,
1017 7.07106781186547573e-01 , 7.07106781186547462e-01 ,
1018 6.34393284163645599e-01 , 7.73010453362736882e-01 ,
1019 5.55570233019602289e-01 , 8.31469612302545125e-01 ,
1020 4.71396736825997809e-01 , 8.81921264348354939e-01 ,
1021 3.82683432365089837e-01 , 9.23879532511286738e-01 ,
1022 2.90284677254462276e-01 , 9.56940335732208935e-01 ,
1023 1.95090322016128304e-01 , 9.80785280403230431e-01 ,
1024 9.80171403295607702e-02 , 9.95184726672196818e-01 ,
1025 0.0 , 1.00000000000000000e+00
1026 };
1027 #else
1028 const __m128d sintable[17] =
1029 {
1030 _mm_set_pd( 0.0 , 1.0 ),
1031 _mm_set_pd( sin( 1.0 * (M_PI/2.0) / 16.0) , cos( 1.0 * (M_PI/2.0) / 16.0) ),
1032 _mm_set_pd( sin( 2.0 * (M_PI/2.0) / 16.0) , cos( 2.0 * (M_PI/2.0) / 16.0) ),
1033 _mm_set_pd( sin( 3.0 * (M_PI/2.0) / 16.0) , cos( 3.0 * (M_PI/2.0) / 16.0) ),
1034 _mm_set_pd( sin( 4.0 * (M_PI/2.0) / 16.0) , cos( 4.0 * (M_PI/2.0) / 16.0) ),
1035 _mm_set_pd( sin( 5.0 * (M_PI/2.0) / 16.0) , cos( 5.0 * (M_PI/2.0) / 16.0) ),
1036 _mm_set_pd( sin( 6.0 * (M_PI/2.0) / 16.0) , cos( 6.0 * (M_PI/2.0) / 16.0) ),
1037 _mm_set_pd( sin( 7.0 * (M_PI/2.0) / 16.0) , cos( 7.0 * (M_PI/2.0) / 16.0) ),
1038 _mm_set_pd( sin( 8.0 * (M_PI/2.0) / 16.0) , cos( 8.0 * (M_PI/2.0) / 16.0) ),
1039 _mm_set_pd( sin( 9.0 * (M_PI/2.0) / 16.0) , cos( 9.0 * (M_PI/2.0) / 16.0) ),
1040 _mm_set_pd( sin( 10.0 * (M_PI/2.0) / 16.0) , cos( 10.0 * (M_PI/2.0) / 16.0) ),
1041 _mm_set_pd( sin( 11.0 * (M_PI/2.0) / 16.0) , cos( 11.0 * (M_PI/2.0) / 16.0) ),
1042 _mm_set_pd( sin( 12.0 * (M_PI/2.0) / 16.0) , cos( 12.0 * (M_PI/2.0) / 16.0) ),
1043 _mm_set_pd( sin( 13.0 * (M_PI/2.0) / 16.0) , cos( 13.0 * (M_PI/2.0) / 16.0) ),
1044 _mm_set_pd( sin( 14.0 * (M_PI/2.0) / 16.0) , cos( 14.0 * (M_PI/2.0) / 16.0) ),
1045 _mm_set_pd( sin( 15.0 * (M_PI/2.0) / 16.0) , cos( 15.0 * (M_PI/2.0) / 16.0) ),
1046 _mm_set_pd( 1.0 , 0.0 )
1047 };
1048 #endif
1049
1050 const __m128d signmask = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1051 const __m128i signbit_epi32 = _mm_set1_epi32(0x80000000);
1052
1053 const __m128d tabscale = _mm_set1_pd(32.0/M_PI);
1054 const __m128d invtabscale0 = _mm_set1_pd(9.81747508049011230469e-02);
1055 const __m128d invtabscale1 = _mm_set1_pd(1.96197799156550576057e-08);
1056 const __m128i ione = _mm_set1_epi32(1);
1057 const __m128i i32 = _mm_set1_epi32(32);
1058 const __m128i i16 = _mm_set1_epi32(16);
1059 const __m128i tabmask = _mm_set1_epi32(0x3F);
1060 const __m128d sinP7 = _mm_set1_pd(-1.0/5040.0);
1061 const __m128d sinP5 = _mm_set1_pd(1.0/120.0);
1062 const __m128d sinP3 = _mm_set1_pd(-1.0/6.0);
1063 const __m128d sinP1 = _mm_set1_pd(1.0);
1064
1065 const __m128d cosP6 = _mm_set1_pd(-1.0/720.0);
1066 const __m128d cosP4 = _mm_set1_pd(1.0/24.0);
1067 const __m128d cosP2 = _mm_set1_pd(-1.0/2.0);
1068 const __m128d cosP0 = _mm_set1_pd(1.0);
1069
1070 __m128d scalex;
1071 __m128i tabidx,corridx;
1072 __m128d xabs,z,z2,polySin,polyCos;
1073 __m128d xpoint;
1074 __m128d ypoint0,ypoint1;
1075
1076 __m128d sinpoint,cospoint;
1077 __m128d xsign,ssign,csign;
1078 __m128i imask,sswapsign,cswapsign;
1079 __m128d minusone;
1080
1081 xsign = _mm_andnot_pd(signmask,x);
1082 xabs = _mm_and_pd(x,signmask);
1083
1084 scalex = _mm_mul_pd(tabscale,xabs);
1085 tabidx = _mm_cvtpd_epi32(scalex);
1086
1087 xpoint = _mm_round_pd(scalex,_MM_FROUND_TO_NEAREST_INT);
1088
1089 /* Extended precision arithmetics */
1090 z = _mm_sub_pd(xabs,_mm_mul_pd(invtabscale0,xpoint));
1091 z = _mm_sub_pd(z,_mm_mul_pd(invtabscale1,xpoint));
1092
1093 /* Range reduction to 0..2*Pi */
1094 tabidx = _mm_and_si128(tabidx,tabmask);
1095
1096 /* tabidx is now in range [0,..,64] */
1097 imask = _mm_cmpgt_epi32(tabidx,i32);
1098 sswapsign = imask;
1099 cswapsign = imask;
1100 corridx = _mm_and_si128(imask,i32);
1101 tabidx = _mm_sub_epi32(tabidx,corridx);
1102
1103 /* tabidx is now in range [0..32] */
1104 imask = _mm_cmpgt_epi32(tabidx,i16);
1105 cswapsign = _mm_xor_si128(cswapsign,imask);
1106 corridx = _mm_sub_epi32(i32,tabidx);
1107 tabidx = _mm_blendv_epi8(tabidx,corridx,imask);
1108 /* tabidx is now in range [0..16] */
1109 ssign = _mm_cvtepi32_pd( _mm_or_si128( sswapsign , ione ) );
1110 csign = _mm_cvtepi32_pd( _mm_or_si128( cswapsign , ione ) );
1111
1112 #ifdef _MSC_VER
1113 ypoint0 = _mm_load_pd(sintable + 2*_mm_extract_epi32(tabidx,0));
1114 ypoint1 = _mm_load_pd(sintable + 2*_mm_extract_epi32(tabidx,1));
1115 #else
1116 ypoint0 = sintable[_mm_extract_epi32(tabidx,0)];
1117 ypoint1 = sintable[_mm_extract_epi32(tabidx,1)];
1118 #endif
1119 sinpoint = _mm_unpackhi_pd(ypoint0,ypoint1);
1120 cospoint = _mm_unpacklo_pd(ypoint0,ypoint1);
1121
1122 sinpoint = _mm_mul_pd(sinpoint,ssign);
1123 cospoint = _mm_mul_pd(cospoint,csign);
1124
1125 z2 = _mm_mul_pd(z,z);
1126
1127 polySin = _mm_mul_pd(sinP7,z2);
1128 polySin = _mm_add_pd(polySin,sinP5);
1129 polySin = _mm_mul_pd(polySin,z2);
1130 polySin = _mm_add_pd(polySin,sinP3);
1131 polySin = _mm_mul_pd(polySin,z2);
1132 polySin = _mm_add_pd(polySin,sinP1);
1133 polySin = _mm_mul_pd(polySin,z);
1134
1135 polyCos = _mm_mul_pd(cosP6,z2);
1136 polyCos = _mm_add_pd(polyCos,cosP4);
1137 polyCos = _mm_mul_pd(polyCos,z2);
1138 polyCos = _mm_add_pd(polyCos,cosP2);
1139 polyCos = _mm_mul_pd(polyCos,z2);
1140 polyCos = _mm_add_pd(polyCos,cosP0);
1141
1142 *sinval = _mm_xor_pd(_mm_add_pd( _mm_mul_pd(sinpoint,polyCos) , _mm_mul_pd(cospoint,polySin) ),xsign);
1143 *cosval = _mm_sub_pd( _mm_mul_pd(cospoint,polyCos) , _mm_mul_pd(sinpoint,polySin) );
1144
1145 return 0;
1146 }
1147
1148 /*
1149 * IMPORTANT: Do NOT call both sin & cos if you need both results, since each of them
1150 * will then call the sincos() routine and waste a factor 2 in performance!
1151 */
1152 static __m128d
1153 gmx_mm_sin_pd(__m128d x)
1154 {
1155 __m128d s,c;
1156 gmx_mm_sincos_pd(x,&s,&c);
1157 return s;
1158 }
1159
1160 /*
1161 * IMPORTANT: Do NOT call both sin & cos if you need both results, since each of them
1162 * will then call the sincos() routine and waste a factor 2 in performance!
1163 */
1164 static __m128d
1165 gmx_mm_cos_pd(__m128d x)
1166 {
1167 __m128d s,c;
1168 gmx_mm_sincos_pd(x,&s,&c);
1169 return c;
1170 }
1171
1172
1173
1174 static __m128d
1175 gmx_mm_tan_pd(__m128d x)
1176 {
1177 __m128d sinval,cosval;
1178 __m128d tanval;
1179
1180 gmx_mm_sincos_pd(x,&sinval,&cosval);
1181
1182 tanval = _mm_mul_pd(sinval,gmx_mm_inv_pd(cosval));
1183
1184 return tanval;
1185 }
1186
1187
1188
1189 static __m128d
1190 gmx_mm_asin_pd(__m128d x)
1191 {
1192 /* Same algorithm as cephes library */
1193 const __m128d signmask = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1194 const __m128d limit1 = _mm_set1_pd(0.625);
1195 const __m128d limit2 = _mm_set1_pd(1e-8);
1196 const __m128d one = _mm_set1_pd(1.0);
1197 const __m128d halfpi = _mm_set1_pd(M_PI/2.0);
1198 const __m128d quarterpi = _mm_set1_pd(M_PI/4.0);
1199 const __m128d morebits = _mm_set1_pd(6.123233995736765886130e-17);
1200
1201 const __m128d P5 = _mm_set1_pd(4.253011369004428248960e-3);
1202 const __m128d P4 = _mm_set1_pd(-6.019598008014123785661e-1);
1203 const __m128d P3 = _mm_set1_pd(5.444622390564711410273e0);
1204 const __m128d P2 = _mm_set1_pd(-1.626247967210700244449e1);
1205 const __m128d P1 = _mm_set1_pd(1.956261983317594739197e1);
1206 const __m128d P0 = _mm_set1_pd(-8.198089802484824371615e0);
1207
1208 const __m128d Q4 = _mm_set1_pd(-1.474091372988853791896e1);
1209 const __m128d Q3 = _mm_set1_pd(7.049610280856842141659e1);
1210 const __m128d Q2 = _mm_set1_pd(-1.471791292232726029859e2);
1211 const __m128d Q1 = _mm_set1_pd(1.395105614657485689735e2);
1212 const __m128d Q0 = _mm_set1_pd(-4.918853881490881290097e1);
1213
1214 const __m128d R4 = _mm_set1_pd(2.967721961301243206100e-3);
1215 const __m128d R3 = _mm_set1_pd(-5.634242780008963776856e-1);
1216 const __m128d R2 = _mm_set1_pd(6.968710824104713396794e0);
1217 const __m128d R1 = _mm_set1_pd(-2.556901049652824852289e1);
1218 const __m128d R0 = _mm_set1_pd(2.853665548261061424989e1);
1219
1220 const __m128d S3 = _mm_set1_pd(-2.194779531642920639778e1);
1221 const __m128d S2 = _mm_set1_pd(1.470656354026814941758e2);
1222 const __m128d S1 = _mm_set1_pd(-3.838770957603691357202e2);
1223 const __m128d S0 = _mm_set1_pd(3.424398657913078477438e2);
1224
1225 __m128d sign;
1226 __m128d mask;
1227 __m128d xabs;
1228 __m128d zz,ww,z,q,w,y,zz2,ww2;
1229 __m128d PA,PB;
1230 __m128d QA,QB;
1231 __m128d RA,RB;
1232 __m128d SA,SB;
1233 __m128d nom,denom;
1234
1235 sign = _mm_andnot_pd(signmask,x);
1236 xabs = _mm_and_pd(x,signmask);
1237
1238 mask = _mm_cmpgt_pd(xabs,limit1);
1239
1240 zz = _mm_sub_pd(one,xabs);
1241 ww = _mm_mul_pd(xabs,xabs);
1242 zz2 = _mm_mul_pd(zz,zz);
1243 ww2 = _mm_mul_pd(ww,ww);
1244
1245 /* R */
1246 RA = _mm_mul_pd(R4,zz2);
1247 RB = _mm_mul_pd(R3,zz2);
1248 RA = _mm_add_pd(RA,R2);
1249 RB = _mm_add_pd(RB,R1);
1250 RA = _mm_mul_pd(RA,zz2);
1251 RB = _mm_mul_pd(RB,zz);
1252 RA = _mm_add_pd(RA,R0);
1253 RA = _mm_add_pd(RA,RB);
1254
1255 /* S, SA = zz2 */
1256 SB = _mm_mul_pd(S3,zz2);
1257 SA = _mm_add_pd(zz2,S2);
1258 SB = _mm_add_pd(SB,S1);
1259 SA = _mm_mul_pd(SA,zz2);
1260 SB = _mm_mul_pd(SB,zz);
1261 SA = _mm_add_pd(SA,S0);
1262 SA = _mm_add_pd(SA,SB);
1263
1264 /* P */
1265 PA = _mm_mul_pd(P5,ww2);
1266 PB = _mm_mul_pd(P4,ww2);
1267 PA = _mm_add_pd(PA,P3);
1268 PB = _mm_add_pd(PB,P2);
1269 PA = _mm_mul_pd(PA,ww2);
1270 PB = _mm_mul_pd(PB,ww2);
1271 PA = _mm_add_pd(PA,P1);
1272 PB = _mm_add_pd(PB,P0);
1273 PA = _mm_mul_pd(PA,ww);
1274 PA = _mm_add_pd(PA,PB);
1275
1276 /* Q, QA = ww2 */
1277 QB = _mm_mul_pd(Q4,ww2);
1278 QA = _mm_add_pd(ww2,Q3);
1279 QB = _mm_add_pd(QB,Q2);
1280 QA = _mm_mul_pd(QA,ww2);
1281 QB = _mm_mul_pd(QB,ww2);
1282 QA = _mm_add_pd(QA,Q1);
1283 QB = _mm_add_pd(QB,Q0);
1284 QA = _mm_mul_pd(QA,ww);
1285 QA = _mm_add_pd(QA,QB);
1286
1287 RA = _mm_mul_pd(RA,zz);
1288 PA = _mm_mul_pd(PA,ww);
1289
1290 nom = _mm_blendv_pd( PA,RA,mask );
1291 denom = _mm_blendv_pd( QA,SA,mask );
1292
1293 q = _mm_mul_pd( nom, gmx_mm_inv_pd(denom) );
1294
1295 zz = _mm_add_pd(zz,zz);
1296 zz = gmx_mm_sqrt_pd(zz);
1297 z = _mm_sub_pd(quarterpi,zz);
1298 zz = _mm_mul_pd(zz,q);
1299 zz = _mm_sub_pd(zz,morebits);
1300 z = _mm_sub_pd(z,zz);
1301 z = _mm_add_pd(z,quarterpi);
1302
1303 w = _mm_mul_pd(xabs,q);
1304 w = _mm_add_pd(w,xabs);
1305
1306 z = _mm_blendv_pd( w,z,mask );
1307
1308 mask = _mm_cmpgt_pd(xabs,limit2);
1309 z = _mm_blendv_pd( xabs,z,mask );
1310
1311 z = _mm_xor_pd(z,sign);
1312
1313 return z;
1314 }
1315
1316
1317 static __m128d
1318 gmx_mm_acos_pd(__m128d x)
1319 {
1320 const __m128d signmask = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1321 const __m128d one = _mm_set1_pd(1.0);
1322 const __m128d half = _mm_set1_pd(0.5);
1323 const __m128d pi = _mm_set1_pd(M_PI);
1324 const __m128d quarterpi0 = _mm_set1_pd(7.85398163397448309616e-1);
1325 const __m128d quarterpi1 = _mm_set1_pd(6.123233995736765886130e-17);
1326
1327
1328 __m128d mask1;
1329
1330 __m128d z,z1,z2;
1331
1332 mask1 = _mm_cmpgt_pd(x,half);
1333 z1 = _mm_mul_pd(half,_mm_sub_pd(one,x));
1334 z1 = gmx_mm_sqrt_pd(z1);
1335 z = _mm_blendv_pd( x,z1,mask1 );
1336
1337 z = gmx_mm_asin_pd(z);
1338
1339 z1 = _mm_add_pd(z,z);
1340
1341 z2 = _mm_sub_pd(quarterpi0,z);
1342 z2 = _mm_add_pd(z2,quarterpi1);
1343 z2 = _mm_add_pd(z2,quarterpi0);
1344
1345 z = _mm_blendv_pd(z2,z1,mask1);
1346
1347 return z;
1348 }
1349
1350 static __m128d
1351 gmx_mm_atan_pd(__m128d x)
1352 {
1353 /* Same algorithm as cephes library */
1354 const __m128d signmask = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1355 const __m128d limit1 = _mm_set1_pd(0.66);
1356 const __m128d limit2 = _mm_set1_pd(2.41421356237309504880);
1357 const __m128d quarterpi = _mm_set1_pd(M_PI/4.0);
1358 const __m128d halfpi = _mm_set1_pd(M_PI/2.0);
1359 const __m128d mone = _mm_set1_pd(-1.0);
1360 const __m128d morebits1 = _mm_set1_pd(0.5*6.123233995736765886130E-17);
1361 const __m128d morebits2 = _mm_set1_pd(6.123233995736765886130E-17);
1362
1363 const __m128d P4 = _mm_set1_pd(-8.750608600031904122785E-1);
1364 const __m128d P3 = _mm_set1_pd(-1.615753718733365076637E1);
1365 const __m128d P2 = _mm_set1_pd(-7.500855792314704667340E1);
1366 const __m128d P1 = _mm_set1_pd(-1.228866684490136173410E2);
1367 const __m128d P0 = _mm_set1_pd(-6.485021904942025371773E1);
1368
1369 const __m128d Q4 = _mm_set1_pd(2.485846490142306297962E1);
1370 const __m128d Q3 = _mm_set1_pd(1.650270098316988542046E2);
1371 const __m128d Q2 = _mm_set1_pd(4.328810604912902668951E2);
1372 const __m128d Q1 = _mm_set1_pd(4.853903996359136964868E2);
1373 const __m128d Q0 = _mm_set1_pd(1.945506571482613964425E2);
1374
1375 __m128d sign;
1376 __m128d mask1,mask2;
1377 __m128d y,t1,t2;
1378 __m128d z,z2;
1379 __m128d P_A,P_B,Q_A,Q_B;
1380
1381 sign = _mm_andnot_pd(signmask,x);
1382 x = _mm_and_pd(x,signmask);
1383
1384 mask1 = _mm_cmpgt_pd(x,limit1);
1385 mask2 = _mm_cmpgt_pd(x,limit2);
1386
1387 t1 = _mm_mul_pd(_mm_add_pd(x,mone),gmx_mm_inv_pd(_mm_sub_pd(x,mone)));
1388 t2 = _mm_mul_pd(mone,gmx_mm_inv_pd(x));
1389
1390 y = _mm_and_pd(mask1,quarterpi);
1391 y = _mm_or_pd( _mm_and_pd(mask2,halfpi) , _mm_andnot_pd(mask2,y) );
1392
1393 x = _mm_or_pd( _mm_and_pd(mask1,t1) , _mm_andnot_pd(mask1,x) );
1394 x = _mm_or_pd( _mm_and_pd(mask2,t2) , _mm_andnot_pd(mask2,x) );
1395
1396 z = _mm_mul_pd(x,x);
1397 z2 = _mm_mul_pd(z,z);
1398
1399 P_A = _mm_mul_pd(P4,z2);
1400 P_B = _mm_mul_pd(P3,z2);
1401 P_A = _mm_add_pd(P_A,P2);
1402 P_B = _mm_add_pd(P_B,P1);
1403 P_A = _mm_mul_pd(P_A,z2);
1404 P_B = _mm_mul_pd(P_B,z);
1405 P_A = _mm_add_pd(P_A,P0);
1406 P_A = _mm_add_pd(P_A,P_B);
1407
1408 /* Q_A = z2 */
1409 Q_B = _mm_mul_pd(Q4,z2);
1410 Q_A = _mm_add_pd(z2,Q3);
1411 Q_B = _mm_add_pd(Q_B,Q2);
1412 Q_A = _mm_mul_pd(Q_A,z2);
1413 Q_B = _mm_mul_pd(Q_B,z2);
1414 Q_A = _mm_add_pd(Q_A,Q1);
1415 Q_B = _mm_add_pd(Q_B,Q0);
1416 Q_A = _mm_mul_pd(Q_A,z);
1417 Q_A = _mm_add_pd(Q_A,Q_B);
1418
1419 z = _mm_mul_pd(z,P_A);
1420 z = _mm_mul_pd(z,gmx_mm_inv_pd(Q_A));
1421 z = _mm_mul_pd(z,x);
1422 z = _mm_add_pd(z,x);
1423
1424 t1 = _mm_and_pd(mask1,morebits1);
1425 t1 = _mm_or_pd( _mm_and_pd(mask2,morebits2) , _mm_andnot_pd(mask2,t1) );
1426
1427 z = _mm_add_pd(z,t1);
1428 y = _mm_add_pd(y,z);
1429
1430 y = _mm_xor_pd(y,sign);
1431
1432 return y;
1433 }
1434
1435
1436 static __m128d
1437 gmx_mm_atan2_pd(__m128d y, __m128d x)
1438 {
1439 const __m128d pi = _mm_set1_pd(M_PI);
1440 const __m128d minuspi = _mm_set1_pd(-M_PI);
1441 const __m128d halfpi = _mm_set1_pd(M_PI/2.0);
1442 const __m128d minushalfpi = _mm_set1_pd(-M_PI/2.0);
1443
1444 __m128d z,z1,z3,z4;
1445 __m128d w;
1446 __m128d maskx_lt,maskx_eq;
1447 __m128d masky_lt,masky_eq;
1448 __m128d mask1,mask2,mask3,mask4,maskall;
1449
1450 maskx_lt = _mm_cmplt_pd(x,_mm_setzero_pd());
1451 masky_lt = _mm_cmplt_pd(y,_mm_setzero_pd());
1452 maskx_eq = _mm_cmpeq_pd(x,_mm_setzero_pd());
1453 masky_eq = _mm_cmpeq_pd(y,_mm_setzero_pd());
1454
1455 z = _mm_mul_pd(y,gmx_mm_inv_pd(x));
1456 z = gmx_mm_atan_pd(z);
1457
1458 mask1 = _mm_and_pd(maskx_eq,masky_lt);
1459 mask2 = _mm_andnot_pd(maskx_lt,masky_eq);
1460 mask3 = _mm_andnot_pd( _mm_or_pd(masky_lt,masky_eq) , maskx_eq);
1461 mask4 = _mm_and_pd(masky_eq,maskx_lt);
1462
1463 maskall = _mm_or_pd( _mm_or_pd(mask1,mask2), _mm_or_pd(mask3,mask4) );
1464
1465 z = _mm_andnot_pd(maskall,z);
1466 z1 = _mm_and_pd(mask1,minushalfpi);
1467 z3 = _mm_and_pd(mask3,halfpi);
1468 z4 = _mm_and_pd(mask4,pi);
1469
1470 z = _mm_or_pd( _mm_or_pd(z,z1), _mm_or_pd(z3,z4) );
1471
1472 w = _mm_blendv_pd(pi,minuspi,masky_lt);
1473 w = _mm_and_pd(w,maskx_lt);
1474
1475 w = _mm_andnot_pd(maskall,w);
1476
1477 z = _mm_add_pd(z,w);
1478
1479 return z;
1480 }
1481
1482 #endif /*_gmx_math_x86_sse4_1_double_h_ */
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