include/gmx_math_x86_sse4_1_double.h

   1 /*
   2  * This file is part of the GROMACS molecular simulation package.
   3  *
   4  * Copyright (c) 2012, by the GROMACS development team, led by
   5  * David van der Spoel, Berk Hess, Erik Lindahl, and including many
   6  * others, as listed in the AUTHORS file in the top-level source
   7  * directory and at http://www.gromacs.org.
   8  *
   9  * GROMACS is free software; you can redistribute it and/or
  10  * modify it under the terms of the GNU Lesser General Public License
  11  * as published by the Free Software Foundation; either version 2.1
  12  * of the License, or (at your option) any later version.
  13  *
  14  * GROMACS is distributed in the hope that it will be useful,
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  35 #ifndef _gmx_math_x86_sse4_1_double_h_
  36 #define _gmx_math_x86_sse4_1_double_h_
  37
  38 #include <stdio.h>
  39 #include <math.h>
  40
  41 #include "gmx_x86_sse4_1.h"
  42
  43
  44
  45 #ifndef M_PI
  46 #  define M_PI 3.14159265358979323846264338327950288
  47 #endif
  48
  49 /************************
  50  *                      *
  51  * Simple math routines *
  52  *                      *
  53  ************************/
  54
  55 /* 1.0/sqrt(x) */
  56 static gmx_inline __m128d
  57 gmx_mm_invsqrt_pd(__m128d x)
  58 {
  59     const __m128d half  = _mm_set1_pd(0.5);
  60     const __m128d three = _mm_set1_pd(3.0);
  61
  62     /* Lookup instruction only exists in single precision, convert back and forth... */
  63     __m128d lu = _mm_cvtps_pd(_mm_rsqrt_ps( _mm_cvtpd_ps(x)));
  64
  65     lu = _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu,lu),x)),lu));
  66     return _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu,lu),x)),lu));
  67 }
  68
  69 /* 1.0/sqrt(x), done for a pair of arguments to improve throughput */
  70 static void
  71 gmx_mm_invsqrt_pair_pd(__m128d x1, __m128d x2, __m128d *invsqrt1, __m128d *invsqrt2)
  72 {
  73     const __m128d half   = _mm_set1_pd(0.5);
  74     const __m128d three  = _mm_set1_pd(3.0);
  75     const __m128  halff  = _mm_set1_ps(0.5f);
  76     const __m128  threef = _mm_set1_ps(3.0f);
  77
  78     __m128 xf,luf;
  79     __m128d lu1,lu2;
  80
  81     /* Do first N-R step in float for 2x throughput */
  82     xf  = _mm_shuffle_ps(_mm_cvtpd_ps(x1),_mm_cvtpd_ps(x2),_MM_SHUFFLE(1,0,1,0));
  83     luf = _mm_rsqrt_ps(xf);
  84     luf = _mm_mul_ps(halff,_mm_mul_ps(_mm_sub_ps(threef,_mm_mul_ps(_mm_mul_ps(luf,luf),xf)),luf));
  85
  86     lu2 = _mm_cvtps_pd(_mm_shuffle_ps(luf,luf,_MM_SHUFFLE(3,2,3,2)));
  87     lu1 = _mm_cvtps_pd(luf);
  88
  89     *invsqrt1 = _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu1,lu1),x1)),lu1));
  90     *invsqrt2 = _mm_mul_pd(half,_mm_mul_pd(_mm_sub_pd(three,_mm_mul_pd(_mm_mul_pd(lu2,lu2),x2)),lu2));
  91 }
  92
  93 /* sqrt(x) - Do NOT use this (but rather invsqrt) if you actually need 1.0/sqrt(x) */
  94 static gmx_inline __m128d
  95 gmx_mm_sqrt_pd(__m128d x)
  96 {
  97     __m128d mask;
  98     __m128d res;
  99
 100     mask = _mm_cmpeq_pd(x,_mm_setzero_pd());
 101     res  = _mm_andnot_pd(mask,gmx_mm_invsqrt_pd(x));
 102
 103     res  = _mm_mul_pd(x,res);
 104
 105     return res;
 106 }
 107
 108 /* 1.0/x */
 109 static gmx_inline __m128d
 110 gmx_mm_inv_pd(__m128d x)
 111 {
 112     const __m128d two  = _mm_set1_pd(2.0);
 113
 114     /* Lookup instruction only exists in single precision, convert back and forth... */
 115     __m128d lu = _mm_cvtps_pd(_mm_rcp_ps( _mm_cvtpd_ps(x)));
 116
 117     /* Perform two N-R steps for double precision */
 118     lu         = _mm_mul_pd(lu,_mm_sub_pd(two,_mm_mul_pd(x,lu)));
 119     return _mm_mul_pd(lu,_mm_sub_pd(two,_mm_mul_pd(x,lu)));
 120 }
 121
 122 static gmx_inline __m128d
 123 gmx_mm_abs_pd(__m128d x)
 124 {
 125     const __m128d signmask  = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF, 0xFFFFFFFF, 0x7FFFFFFF, 0xFFFFFFFF) );
 126
 127     return _mm_and_pd(x,signmask);
 128 }
 129
 130
 131 /*
 132  * 2^x function.
 133  *
 134  * The 2^w term is calculated from a (6,0)-th order (no denominator) Minimax polynomia on the interval
 135  * [-0.5,0.5].
 136  *
 137  * The approximation on [-0.5,0.5] is a rational Padé approximation, 1+2*P(x^2)/(Q(x^2)-P(x^2)),
 138  * according to the same algorithm as used in the Cephes/netlib math routines.
 139  */
 140 static __m128d
 141 gmx_mm_exp2_pd(__m128d x)
 142 {
 143     /* Lower bound: We do not allow numbers that would lead to an IEEE fp representation exponent smaller than -126. */
 144     const __m128d arglimit = _mm_set1_pd(1022.0);
 145     const __m128i expbase  = _mm_set1_epi32(1023);
 146
 147     const __m128d P2       = _mm_set1_pd(2.30933477057345225087e-2);
 148     const __m128d P1       = _mm_set1_pd(2.02020656693165307700e1);
 149     const __m128d P0       = _mm_set1_pd(1.51390680115615096133e3);
 150     /* Q2 == 1.0 */
 151     const __m128d Q1       = _mm_set1_pd(2.33184211722314911771e2);
 152     const __m128d Q0       = _mm_set1_pd(4.36821166879210612817e3);
 153     const __m128d one      = _mm_set1_pd(1.0);
 154     const __m128d two      = _mm_set1_pd(2.0);
 155
 156     __m128d valuemask;
 157     __m128i iexppart;
 158     __m128d fexppart;
 159     __m128d intpart;
 160     __m128d z,z2;
 161     __m128d PolyP,PolyQ;
 162
 163     iexppart  = _mm_cvtpd_epi32(x);
 164     intpart   = _mm_round_pd(x,_MM_FROUND_TO_NEAREST_INT);
 165
 166     /* The two lowest elements of iexppart now contains 32-bit numbers with a correctly biased exponent.
 167      * To be able to shift it into the exponent for a double precision number we first need to
 168      * shuffle so that the lower half contains the first element, and the upper half the second.
 169      * This should really be done as a zero-extension, but since the next instructions will shift
 170      * the registers left by 52 bits it doesn't matter what we put there - it will be shifted out.
 171      * (thus we just use element 2 from iexppart).
 172      */
 173     iexppart  = _mm_shuffle_epi32(iexppart,_MM_SHUFFLE(2,1,2,0));
 174
 175     /* Do the shift operation on the 64-bit registers */
 176     iexppart  = _mm_add_epi32(iexppart,expbase);
 177     iexppart  = _mm_slli_epi64(iexppart,52);
 178
 179     valuemask = _mm_cmpge_pd(arglimit,gmx_mm_abs_pd(x));
 180     fexppart  = _mm_and_pd(valuemask,gmx_mm_castsi128_pd(iexppart));
 181
 182     z         = _mm_sub_pd(x,intpart);
 183     z2        = _mm_mul_pd(z,z);
 184
 185     PolyP     = _mm_mul_pd(P2,z2);
 186     PolyP     = _mm_add_pd(PolyP,P1);
 187     PolyQ     = _mm_add_pd(z2,Q1);
 188     PolyP     = _mm_mul_pd(PolyP,z2);
 189     PolyQ     = _mm_mul_pd(PolyQ,z2);
 190     PolyP     = _mm_add_pd(PolyP,P0);
 191     PolyQ     = _mm_add_pd(PolyQ,Q0);
 192     PolyP     = _mm_mul_pd(PolyP,z);
 193
 194     z         = _mm_mul_pd(PolyP,gmx_mm_inv_pd(_mm_sub_pd(PolyQ,PolyP)));
 195     z         = _mm_add_pd(one,_mm_mul_pd(two,z));
 196
 197     z         = _mm_mul_pd(z,fexppart);
 198
 199     return  z;
 200 }
 201
 202 /* Exponential function. This could be calculated from 2^x as Exp(x)=2^(y), where y=log2(e)*x,
 203  * but there will then be a small rounding error since we lose some precision due to the
 204  * multiplication. This will then be magnified a lot by the exponential.
 205  *
 206  * Instead, we calculate the fractional part directly as a Padé approximation of
 207  * Exp(z) on [-0.5,0.5]. We use extended precision arithmetics to calculate the fraction
 208  * remaining after 2^y, which avoids the precision-loss.
 209  */
 210 static __m128d
 211 gmx_mm_exp_pd(__m128d exparg)
 212 {
 213     const __m128d argscale = _mm_set1_pd(1.4426950408889634073599);
 214     /* Lower bound: We do not allow numbers that would lead to an IEEE fp representation exponent smaller than -126. */
 215     const __m128d arglimit = _mm_set1_pd(1022.0);
 216     const __m128i expbase  = _mm_set1_epi32(1023);
 217
 218     const __m128d invargscale0  = _mm_set1_pd(6.93145751953125e-1);
 219     const __m128d invargscale1  = _mm_set1_pd(1.42860682030941723212e-6);
 220
 221     const __m128d P2       = _mm_set1_pd(1.26177193074810590878e-4);
 222     const __m128d P1       = _mm_set1_pd(3.02994407707441961300e-2);
 223     /* P0 == 1.0 */
 224     const __m128d Q3       = _mm_set1_pd(3.00198505138664455042E-6);
 225     const __m128d Q2       = _mm_set1_pd(2.52448340349684104192E-3);
 226     const __m128d Q1       = _mm_set1_pd(2.27265548208155028766E-1);
 227     /* Q0 == 2.0 */
 228     const __m128d one      = _mm_set1_pd(1.0);
 229     const __m128d two      = _mm_set1_pd(2.0);
 230
 231     __m128d valuemask;
 232     __m128i iexppart;
 233     __m128d fexppart;
 234     __m128d intpart;
 235     __m128d x,z,z2;
 236     __m128d PolyP,PolyQ;
 237
 238     x             = _mm_mul_pd(exparg,argscale);
 239
 240     iexppart  = _mm_cvtpd_epi32(x);
 241     intpart   = _mm_round_pd(x,_MM_FROUND_TO_NEAREST_INT);
 242
 243     /* The two lowest elements of iexppart now contains 32-bit numbers with a correctly biased exponent.
 244      * To be able to shift it into the exponent for a double precision number we first need to
 245      * shuffle so that the lower half contains the first element, and the upper half the second.
 246      * This should really be done as a zero-extension, but since the next instructions will shift
 247      * the registers left by 52 bits it doesn't matter what we put there - it will be shifted out.
 248      * (thus we just use element 2 from iexppart).
 249      */
 250     iexppart  = _mm_shuffle_epi32(iexppart,_MM_SHUFFLE(2,1,2,0));
 251
 252     /* Do the shift operation on the 64-bit registers */
 253     iexppart  = _mm_add_epi32(iexppart,expbase);
 254     iexppart  = _mm_slli_epi64(iexppart,52);
 255
 256     valuemask = _mm_cmpge_pd(arglimit,gmx_mm_abs_pd(x));
 257     fexppart  = _mm_and_pd(valuemask,gmx_mm_castsi128_pd(iexppart));
 258
 259     z         = _mm_sub_pd(exparg,_mm_mul_pd(invargscale0,intpart));
 260     z         = _mm_sub_pd(z,_mm_mul_pd(invargscale1,intpart));
 261
 262     z2        = _mm_mul_pd(z,z);
 263
 264     PolyQ     = _mm_mul_pd(Q3,z2);
 265     PolyQ     = _mm_add_pd(PolyQ,Q2);
 266     PolyP     = _mm_mul_pd(P2,z2);
 267     PolyQ     = _mm_mul_pd(PolyQ,z2);
 268     PolyP     = _mm_add_pd(PolyP,P1);
 269     PolyQ     = _mm_add_pd(PolyQ,Q1);
 270     PolyP     = _mm_mul_pd(PolyP,z2);
 271     PolyQ     = _mm_mul_pd(PolyQ,z2);
 272     PolyP     = _mm_add_pd(PolyP,one);
 273     PolyQ     = _mm_add_pd(PolyQ,two);
 274
 275     PolyP     = _mm_mul_pd(PolyP,z);
 276
 277     z         = _mm_mul_pd(PolyP,gmx_mm_inv_pd(_mm_sub_pd(PolyQ,PolyP)));
 278     z         = _mm_add_pd(one,_mm_mul_pd(two,z));
 279
 280     z         = _mm_mul_pd(z,fexppart);
 281
 282     return  z;
 283 }
 284
 285
 286
 287 static __m128d
 288 gmx_mm_log_pd(__m128d x)
 289 {
 290     /* Same algorithm as cephes library */
 291     const __m128d expmask    = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FF00000, 0x00000000, 0x7FF00000, 0x00000000) );
 292
 293     const __m128i expbase_m1 = _mm_set1_epi32(1023-1); /* We want non-IEEE format */
 294
 295     const __m128d half       = _mm_set1_pd(0.5);
 296     const __m128d one        = _mm_set1_pd(1.0);
 297     const __m128d two        = _mm_set1_pd(2.0);
 298     const __m128d invsq2     = _mm_set1_pd(1.0/sqrt(2.0));
 299
 300     const __m128d corr1      = _mm_set1_pd(-2.121944400546905827679e-4);
 301     const __m128d corr2      = _mm_set1_pd(0.693359375);
 302
 303     const __m128d P5         = _mm_set1_pd(1.01875663804580931796e-4);
 304     const __m128d P4         = _mm_set1_pd(4.97494994976747001425e-1);
 305     const __m128d P3         = _mm_set1_pd(4.70579119878881725854e0);
 306     const __m128d P2         = _mm_set1_pd(1.44989225341610930846e1);
 307     const __m128d P1         = _mm_set1_pd(1.79368678507819816313e1);
 308     const __m128d P0         = _mm_set1_pd(7.70838733755885391666e0);
 309
 310     const __m128d Q4         = _mm_set1_pd(1.12873587189167450590e1);
 311     const __m128d Q3         = _mm_set1_pd(4.52279145837532221105e1);
 312     const __m128d Q2         = _mm_set1_pd(8.29875266912776603211e1);
 313     const __m128d Q1         = _mm_set1_pd(7.11544750618563894466e1);
 314     const __m128d Q0         = _mm_set1_pd(2.31251620126765340583e1);
 315
 316     const __m128d R2         = _mm_set1_pd(-7.89580278884799154124e-1);
 317     const __m128d R1         = _mm_set1_pd(1.63866645699558079767e1);
 318     const __m128d R0         = _mm_set1_pd(-6.41409952958715622951e1);
 319
 320     const __m128d S2         = _mm_set1_pd(-3.56722798256324312549E1);
 321     const __m128d S1         = _mm_set1_pd(3.12093766372244180303E2);
 322     const __m128d S0         = _mm_set1_pd(-7.69691943550460008604E2);
 323
 324     __m128d fexp;
 325     __m128i iexp;
 326
 327     __m128d mask1,mask2;
 328     __m128d corr,t1,t2,q;
 329     __m128d zA,yA,xA,zB,yB,xB,z;
 330     __m128d polyR,polyS;
 331     __m128d polyP1,polyP2,polyQ1,polyQ2;
 332
 333     /* Separate x into exponent and mantissa, with a mantissa in the range [0.5..1[ (not IEEE754 standard!) */
 334     fexp   = _mm_and_pd(x,expmask);
 335     iexp   = gmx_mm_castpd_si128(fexp);
 336     iexp   = _mm_srli_epi64(iexp,52);
 337     iexp   = _mm_sub_epi32(iexp,expbase_m1);
 338     iexp   = _mm_shuffle_epi32(iexp, _MM_SHUFFLE(1,1,2,0) );
 339     fexp   = _mm_cvtepi32_pd(iexp);
 340
 341     x      = _mm_andnot_pd(expmask,x);
 342     x      = _mm_or_pd(x,one);
 343     x      = _mm_mul_pd(x,half);
 344
 345     mask1     = _mm_cmpgt_pd(gmx_mm_abs_pd(fexp),two);
 346     mask2     = _mm_cmplt_pd(x,invsq2);
 347
 348     fexp   = _mm_sub_pd(fexp,_mm_and_pd(mask2,one));
 349
 350     /* If mask1 is set ('A') */
 351     zA     = _mm_sub_pd(x,half);
 352     t1     = _mm_blendv_pd( zA,x,mask2 );
 353     zA     = _mm_sub_pd(t1,half);
 354     t2     = _mm_blendv_pd( x,zA,mask2 );
 355     yA     = _mm_mul_pd(half,_mm_add_pd(t2,one));
 356
 357     xA     = _mm_mul_pd(zA,gmx_mm_inv_pd(yA));
 358     zA     = _mm_mul_pd(xA,xA);
 359
 360     /* EVALUATE POLY */
 361     polyR  = _mm_mul_pd(R2,zA);
 362     polyR  = _mm_add_pd(polyR,R1);
 363     polyR  = _mm_mul_pd(polyR,zA);
 364     polyR  = _mm_add_pd(polyR,R0);
 365
 366     polyS  = _mm_add_pd(zA,S2);
 367     polyS  = _mm_mul_pd(polyS,zA);
 368     polyS  = _mm_add_pd(polyS,S1);
 369     polyS  = _mm_mul_pd(polyS,zA);
 370     polyS  = _mm_add_pd(polyS,S0);
 371
 372     q      = _mm_mul_pd(polyR,gmx_mm_inv_pd(polyS));
 373     zA     = _mm_mul_pd(_mm_mul_pd(xA,zA),q);
 374
 375     zA     = _mm_add_pd(zA,_mm_mul_pd(corr1,fexp));
 376     zA     = _mm_add_pd(zA,xA);
 377     zA     = _mm_add_pd(zA,_mm_mul_pd(corr2,fexp));
 378
 379     /* If mask1 is not set ('B') */
 380     corr   = _mm_and_pd(mask2,x);
 381     xB     = _mm_add_pd(x,corr);
 382     xB     = _mm_sub_pd(xB,one);
 383     zB     = _mm_mul_pd(xB,xB);
 384
 385     polyP1 = _mm_mul_pd(P5,zB);
 386     polyP2 = _mm_mul_pd(P4,zB);
 387     polyP1 = _mm_add_pd(polyP1,P3);
 388     polyP2 = _mm_add_pd(polyP2,P2);
 389     polyP1 = _mm_mul_pd(polyP1,zB);
 390     polyP2 = _mm_mul_pd(polyP2,zB);
 391     polyP1 = _mm_add_pd(polyP1,P1);
 392     polyP2 = _mm_add_pd(polyP2,P0);
 393     polyP1 = _mm_mul_pd(polyP1,xB);
 394     polyP1 = _mm_add_pd(polyP1,polyP2);
 395
 396     polyQ2 = _mm_mul_pd(Q4,zB);
 397     polyQ1 = _mm_add_pd(zB,Q3);
 398     polyQ2 = _mm_add_pd(polyQ2,Q2);
 399     polyQ1 = _mm_mul_pd(polyQ1,zB);
 400     polyQ2 = _mm_mul_pd(polyQ2,zB);
 401     polyQ1 = _mm_add_pd(polyQ1,Q1);
 402     polyQ2 = _mm_add_pd(polyQ2,Q0);
 403     polyQ1 = _mm_mul_pd(polyQ1,xB);
 404     polyQ1 = _mm_add_pd(polyQ1,polyQ2);
 405
 406     fexp   = _mm_and_pd(fexp,_mm_cmpneq_pd(fexp,_mm_setzero_pd()));
 407
 408     q      = _mm_mul_pd(polyP1,gmx_mm_inv_pd(polyQ1));
 409     yB     = _mm_mul_pd(_mm_mul_pd(xB,zB),q);
 410
 411     yB     = _mm_add_pd(yB,_mm_mul_pd(corr1,fexp));
 412     yB     = _mm_sub_pd(yB,_mm_mul_pd(half,zB));
 413     zB     = _mm_add_pd(xB,yB);
 414     zB     = _mm_add_pd(zB,_mm_mul_pd(corr2,fexp));
 415
 416     z      = _mm_blendv_pd( zB,zA,mask1 );
 417
 418     return z;
 419 }
 420
 421
 422 static __m128d
 423 gmx_mm_erf_pd(__m128d x)
 424 {
 425     /* Coefficients for minimax approximation of erf(x)=x*(CAoffset + P(x^2)/Q(x^2)) in range [-0.75,0.75] */
 426     const __m128d CAP4      = _mm_set1_pd(-0.431780540597889301512e-4);
 427     const __m128d CAP3      = _mm_set1_pd(-0.00578562306260059236059);
 428     const __m128d CAP2      = _mm_set1_pd(-0.028593586920219752446);
 429     const __m128d CAP1      = _mm_set1_pd(-0.315924962948621698209);
 430     const __m128d CAP0      = _mm_set1_pd(0.14952975608477029151);
 431
 432     const __m128d CAQ5      = _mm_set1_pd(-0.374089300177174709737e-5);
 433     const __m128d CAQ4      = _mm_set1_pd(0.00015126584532155383535);
 434     const __m128d CAQ3      = _mm_set1_pd(0.00536692680669480725423);
 435     const __m128d CAQ2      = _mm_set1_pd(0.0668686825594046122636);
 436     const __m128d CAQ1      = _mm_set1_pd(0.402604990869284362773);
 437     /* CAQ0 == 1.0 */
 438     const __m128d CAoffset  = _mm_set1_pd(0.9788494110107421875);
 439
 440     /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)*x*(P(x-1)/Q(x-1)) in range [1.0,4.5] */
 441     const __m128d CBP6      = _mm_set1_pd(2.49650423685462752497647637088e-10);
 442     const __m128d CBP5      = _mm_set1_pd(0.00119770193298159629350136085658);
 443     const __m128d CBP4      = _mm_set1_pd(0.0164944422378370965881008942733);
 444     const __m128d CBP3      = _mm_set1_pd(0.0984581468691775932063932439252);
 445     const __m128d CBP2      = _mm_set1_pd(0.317364595806937763843589437418);
 446     const __m128d CBP1      = _mm_set1_pd(0.554167062641455850932670067075);
 447     const __m128d CBP0      = _mm_set1_pd(0.427583576155807163756925301060);
 448     const __m128d CBQ7      = _mm_set1_pd(0.00212288829699830145976198384930);
 449     const __m128d CBQ6      = _mm_set1_pd(0.0334810979522685300554606393425);
 450     const __m128d CBQ5      = _mm_set1_pd(0.2361713785181450957579508850717);
 451     const __m128d CBQ4      = _mm_set1_pd(0.955364736493055670530981883072);
 452     const __m128d CBQ3      = _mm_set1_pd(2.36815675631420037315349279199);
 453     const __m128d CBQ2      = _mm_set1_pd(3.55261649184083035537184223542);
 454     const __m128d CBQ1      = _mm_set1_pd(2.93501136050160872574376997993);
 455     /* CBQ0 == 1.0 */
 456
 457     /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)/x*(P(1/x)/Q(1/x)) in range [4.5,inf] */
 458     const __m128d CCP6      = _mm_set1_pd(-2.8175401114513378771);
 459     const __m128d CCP5      = _mm_set1_pd(-3.22729451764143718517);
 460     const __m128d CCP4      = _mm_set1_pd(-2.5518551727311523996);
 461     const __m128d CCP3      = _mm_set1_pd(-0.687717681153649930619);
 462     const __m128d CCP2      = _mm_set1_pd(-0.212652252872804219852);
 463     const __m128d CCP1      = _mm_set1_pd(0.0175389834052493308818);
 464     const __m128d CCP0      = _mm_set1_pd(0.00628057170626964891937);
 465
 466     const __m128d CCQ6      = _mm_set1_pd(5.48409182238641741584);
 467     const __m128d CCQ5      = _mm_set1_pd(13.5064170191802889145);
 468     const __m128d CCQ4      = _mm_set1_pd(22.9367376522880577224);
 469     const __m128d CCQ3      = _mm_set1_pd(15.930646027911794143);
 470     const __m128d CCQ2      = _mm_set1_pd(11.0567237927800161565);
 471     const __m128d CCQ1      = _mm_set1_pd(2.79257750980575282228);
 472     /* CCQ0 == 1.0 */
 473     const __m128d CCoffset  = _mm_set1_pd(0.5579090118408203125);
 474
 475     const __m128d one       = _mm_set1_pd(1.0);
 476     const __m128d two       = _mm_set1_pd(2.0);
 477
 478     const __m128d signbit   = gmx_mm_castsi128_pd( _mm_set_epi32(0x80000000,0x00000000,0x80000000,0x00000000) );
 479
 480     __m128d xabs,x2,x4,t,t2,w,w2;
 481     __m128d PolyAP0,PolyAP1,PolyAQ0,PolyAQ1;
 482     __m128d PolyBP0,PolyBP1,PolyBQ0,PolyBQ1;
 483     __m128d PolyCP0,PolyCP1,PolyCQ0,PolyCQ1;
 484     __m128d res_erf,res_erfcB,res_erfcC,res_erfc,res;
 485     __m128d mask,expmx2;
 486
 487     /* Calculate erf() */
 488     xabs     = gmx_mm_abs_pd(x);
 489     x2       = _mm_mul_pd(x,x);
 490     x4       = _mm_mul_pd(x2,x2);
 491
 492     PolyAP0  = _mm_mul_pd(CAP4,x4);
 493     PolyAP1  = _mm_mul_pd(CAP3,x4);
 494     PolyAP0  = _mm_add_pd(PolyAP0,CAP2);
 495     PolyAP1  = _mm_add_pd(PolyAP1,CAP1);
 496     PolyAP0  = _mm_mul_pd(PolyAP0,x4);
 497     PolyAP1  = _mm_mul_pd(PolyAP1,x2);
 498     PolyAP0  = _mm_add_pd(PolyAP0,CAP0);
 499     PolyAP0  = _mm_add_pd(PolyAP0,PolyAP1);
 500
 501     PolyAQ1  = _mm_mul_pd(CAQ5,x4);
 502     PolyAQ0  = _mm_mul_pd(CAQ4,x4);
 503     PolyAQ1  = _mm_add_pd(PolyAQ1,CAQ3);
 504     PolyAQ0  = _mm_add_pd(PolyAQ0,CAQ2);
 505     PolyAQ1  = _mm_mul_pd(PolyAQ1,x4);
 506     PolyAQ0  = _mm_mul_pd(PolyAQ0,x4);
 507     PolyAQ1  = _mm_add_pd(PolyAQ1,CAQ1);
 508     PolyAQ0  = _mm_add_pd(PolyAQ0,one);
 509     PolyAQ1  = _mm_mul_pd(PolyAQ1,x2);
 510     PolyAQ0  = _mm_add_pd(PolyAQ0,PolyAQ1);
 511
 512     res_erf  = _mm_mul_pd(PolyAP0,gmx_mm_inv_pd(PolyAQ0));
 513     res_erf  = _mm_add_pd(CAoffset,res_erf);
 514     res_erf  = _mm_mul_pd(x,res_erf);
 515
 516     /* Calculate erfc() in range [1,4.5] */
 517     t       = _mm_sub_pd(xabs,one);
 518     t2      = _mm_mul_pd(t,t);
 519
 520     PolyBP0  = _mm_mul_pd(CBP6,t2);
 521     PolyBP1  = _mm_mul_pd(CBP5,t2);
 522     PolyBP0  = _mm_add_pd(PolyBP0,CBP4);
 523     PolyBP1  = _mm_add_pd(PolyBP1,CBP3);
 524     PolyBP0  = _mm_mul_pd(PolyBP0,t2);
 525     PolyBP1  = _mm_mul_pd(PolyBP1,t2);
 526     PolyBP0  = _mm_add_pd(PolyBP0,CBP2);
 527     PolyBP1  = _mm_add_pd(PolyBP1,CBP1);
 528     PolyBP0  = _mm_mul_pd(PolyBP0,t2);
 529     PolyBP1  = _mm_mul_pd(PolyBP1,t);
 530     PolyBP0  = _mm_add_pd(PolyBP0,CBP0);
 531     PolyBP0  = _mm_add_pd(PolyBP0,PolyBP1);
 532
 533     PolyBQ1 = _mm_mul_pd(CBQ7,t2);
 534     PolyBQ0 = _mm_mul_pd(CBQ6,t2);
 535     PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ5);
 536     PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ4);
 537     PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
 538     PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
 539     PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ3);
 540     PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ2);
 541     PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
 542     PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
 543     PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ1);
 544     PolyBQ0 = _mm_add_pd(PolyBQ0,one);
 545     PolyBQ1 = _mm_mul_pd(PolyBQ1,t);
 546     PolyBQ0 = _mm_add_pd(PolyBQ0,PolyBQ1);
 547
 548     res_erfcB = _mm_mul_pd(PolyBP0,gmx_mm_inv_pd(PolyBQ0));
 549
 550     res_erfcB = _mm_mul_pd(res_erfcB,xabs);
 551
 552     /* Calculate erfc() in range [4.5,inf] */
 553     w       = gmx_mm_inv_pd(xabs);
 554     w2      = _mm_mul_pd(w,w);
 555
 556     PolyCP0  = _mm_mul_pd(CCP6,w2);
 557     PolyCP1  = _mm_mul_pd(CCP5,w2);
 558     PolyCP0  = _mm_add_pd(PolyCP0,CCP4);
 559     PolyCP1  = _mm_add_pd(PolyCP1,CCP3);
 560     PolyCP0  = _mm_mul_pd(PolyCP0,w2);
 561     PolyCP1  = _mm_mul_pd(PolyCP1,w2);
 562     PolyCP0  = _mm_add_pd(PolyCP0,CCP2);
 563     PolyCP1  = _mm_add_pd(PolyCP1,CCP1);
 564     PolyCP0  = _mm_mul_pd(PolyCP0,w2);
 565     PolyCP1  = _mm_mul_pd(PolyCP1,w);
 566     PolyCP0  = _mm_add_pd(PolyCP0,CCP0);
 567     PolyCP0  = _mm_add_pd(PolyCP0,PolyCP1);
 568
 569     PolyCQ0  = _mm_mul_pd(CCQ6,w2);
 570     PolyCQ1  = _mm_mul_pd(CCQ5,w2);
 571     PolyCQ0  = _mm_add_pd(PolyCQ0,CCQ4);
 572     PolyCQ1  = _mm_add_pd(PolyCQ1,CCQ3);
 573     PolyCQ0  = _mm_mul_pd(PolyCQ0,w2);
 574     PolyCQ1  = _mm_mul_pd(PolyCQ1,w2);
 575     PolyCQ0  = _mm_add_pd(PolyCQ0,CCQ2);
 576     PolyCQ1  = _mm_add_pd(PolyCQ1,CCQ1);
 577     PolyCQ0  = _mm_mul_pd(PolyCQ0,w2);
 578     PolyCQ1  = _mm_mul_pd(PolyCQ1,w);
 579     PolyCQ0  = _mm_add_pd(PolyCQ0,one);
 580     PolyCQ0  = _mm_add_pd(PolyCQ0,PolyCQ1);
 581
 582     expmx2   = gmx_mm_exp_pd( _mm_or_pd(signbit, x2) );
 583
 584     res_erfcC = _mm_mul_pd(PolyCP0,gmx_mm_inv_pd(PolyCQ0));
 585     res_erfcC = _mm_add_pd(res_erfcC,CCoffset);
 586     res_erfcC = _mm_mul_pd(res_erfcC,w);
 587
 588     mask = _mm_cmpgt_pd(xabs,_mm_set1_pd(4.5));
 589     res_erfc = _mm_blendv_pd(res_erfcB,res_erfcC,mask);
 590
 591     res_erfc = _mm_mul_pd(res_erfc,expmx2);
 592
 593     /* erfc(x<0) = 2-erfc(|x|) */
 594     mask = _mm_cmplt_pd(x,_mm_setzero_pd());
 595     res_erfc = _mm_blendv_pd(res_erfc,_mm_sub_pd(two,res_erfc),mask);
 596
 597     /* Select erf() or erfc() */
 598     mask = _mm_cmplt_pd(xabs,one);
 599     res  = _mm_blendv_pd(_mm_sub_pd(one,res_erfc),res_erf,mask);
 600
 601     return res;
 602 }
 603
 604
 605 static __m128d
 606 gmx_mm_erfc_pd(__m128d x)
 607 {
 608     /* Coefficients for minimax approximation of erf(x)=x*(CAoffset + P(x^2)/Q(x^2)) in range [-0.75,0.75] */
 609     const __m128d CAP4      = _mm_set1_pd(-0.431780540597889301512e-4);
 610     const __m128d CAP3      = _mm_set1_pd(-0.00578562306260059236059);
 611     const __m128d CAP2      = _mm_set1_pd(-0.028593586920219752446);
 612     const __m128d CAP1      = _mm_set1_pd(-0.315924962948621698209);
 613     const __m128d CAP0      = _mm_set1_pd(0.14952975608477029151);
 614
 615     const __m128d CAQ5      = _mm_set1_pd(-0.374089300177174709737e-5);
 616     const __m128d CAQ4      = _mm_set1_pd(0.00015126584532155383535);
 617     const __m128d CAQ3      = _mm_set1_pd(0.00536692680669480725423);
 618     const __m128d CAQ2      = _mm_set1_pd(0.0668686825594046122636);
 619     const __m128d CAQ1      = _mm_set1_pd(0.402604990869284362773);
 620     /* CAQ0 == 1.0 */
 621     const __m128d CAoffset  = _mm_set1_pd(0.9788494110107421875);
 622
 623     /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)*x*(P(x-1)/Q(x-1)) in range [1.0,4.5] */
 624     const __m128d CBP6      = _mm_set1_pd(2.49650423685462752497647637088e-10);
 625     const __m128d CBP5      = _mm_set1_pd(0.00119770193298159629350136085658);
 626     const __m128d CBP4      = _mm_set1_pd(0.0164944422378370965881008942733);
 627     const __m128d CBP3      = _mm_set1_pd(0.0984581468691775932063932439252);
 628     const __m128d CBP2      = _mm_set1_pd(0.317364595806937763843589437418);
 629     const __m128d CBP1      = _mm_set1_pd(0.554167062641455850932670067075);
 630     const __m128d CBP0      = _mm_set1_pd(0.427583576155807163756925301060);
 631     const __m128d CBQ7      = _mm_set1_pd(0.00212288829699830145976198384930);
 632     const __m128d CBQ6      = _mm_set1_pd(0.0334810979522685300554606393425);
 633     const __m128d CBQ5      = _mm_set1_pd(0.2361713785181450957579508850717);
 634     const __m128d CBQ4      = _mm_set1_pd(0.955364736493055670530981883072);
 635     const __m128d CBQ3      = _mm_set1_pd(2.36815675631420037315349279199);
 636     const __m128d CBQ2      = _mm_set1_pd(3.55261649184083035537184223542);
 637     const __m128d CBQ1      = _mm_set1_pd(2.93501136050160872574376997993);
 638     /* CBQ0 == 1.0 */
 639
 640     /* Coefficients for minimax approximation of erfc(x)=exp(-x^2)/x*(P(1/x)/Q(1/x)) in range [4.5,inf] */
 641     const __m128d CCP6      = _mm_set1_pd(-2.8175401114513378771);
 642     const __m128d CCP5      = _mm_set1_pd(-3.22729451764143718517);
 643     const __m128d CCP4      = _mm_set1_pd(-2.5518551727311523996);
 644     const __m128d CCP3      = _mm_set1_pd(-0.687717681153649930619);
 645     const __m128d CCP2      = _mm_set1_pd(-0.212652252872804219852);
 646     const __m128d CCP1      = _mm_set1_pd(0.0175389834052493308818);
 647     const __m128d CCP0      = _mm_set1_pd(0.00628057170626964891937);
 648
 649     const __m128d CCQ6      = _mm_set1_pd(5.48409182238641741584);
 650     const __m128d CCQ5      = _mm_set1_pd(13.5064170191802889145);
 651     const __m128d CCQ4      = _mm_set1_pd(22.9367376522880577224);
 652     const __m128d CCQ3      = _mm_set1_pd(15.930646027911794143);
 653     const __m128d CCQ2      = _mm_set1_pd(11.0567237927800161565);
 654     const __m128d CCQ1      = _mm_set1_pd(2.79257750980575282228);
 655     /* CCQ0 == 1.0 */
 656     const __m128d CCoffset  = _mm_set1_pd(0.5579090118408203125);
 657
 658     const __m128d one       = _mm_set1_pd(1.0);
 659     const __m128d two       = _mm_set1_pd(2.0);
 660
 661     const __m128d signbit   = gmx_mm_castsi128_pd( _mm_set_epi32(0x80000000,0x00000000,0x80000000,0x00000000) );
 662
 663     __m128d xabs,x2,x4,t,t2,w,w2;
 664     __m128d PolyAP0,PolyAP1,PolyAQ0,PolyAQ1;
 665     __m128d PolyBP0,PolyBP1,PolyBQ0,PolyBQ1;
 666     __m128d PolyCP0,PolyCP1,PolyCQ0,PolyCQ1;
 667     __m128d res_erf,res_erfcB,res_erfcC,res_erfc,res;
 668     __m128d mask,expmx2;
 669
 670     /* Calculate erf() */
 671     xabs     = gmx_mm_abs_pd(x);
 672     x2       = _mm_mul_pd(x,x);
 673     x4       = _mm_mul_pd(x2,x2);
 674
 675     PolyAP0  = _mm_mul_pd(CAP4,x4);
 676     PolyAP1  = _mm_mul_pd(CAP3,x4);
 677     PolyAP0  = _mm_add_pd(PolyAP0,CAP2);
 678     PolyAP1  = _mm_add_pd(PolyAP1,CAP1);
 679     PolyAP0  = _mm_mul_pd(PolyAP0,x4);
 680     PolyAP1  = _mm_mul_pd(PolyAP1,x2);
 681     PolyAP0  = _mm_add_pd(PolyAP0,CAP0);
 682     PolyAP0  = _mm_add_pd(PolyAP0,PolyAP1);
 683
 684     PolyAQ1  = _mm_mul_pd(CAQ5,x4);
 685     PolyAQ0  = _mm_mul_pd(CAQ4,x4);
 686     PolyAQ1  = _mm_add_pd(PolyAQ1,CAQ3);
 687     PolyAQ0  = _mm_add_pd(PolyAQ0,CAQ2);
 688     PolyAQ1  = _mm_mul_pd(PolyAQ1,x4);
 689     PolyAQ0  = _mm_mul_pd(PolyAQ0,x4);
 690     PolyAQ1  = _mm_add_pd(PolyAQ1,CAQ1);
 691     PolyAQ0  = _mm_add_pd(PolyAQ0,one);
 692     PolyAQ1  = _mm_mul_pd(PolyAQ1,x2);
 693     PolyAQ0  = _mm_add_pd(PolyAQ0,PolyAQ1);
 694
 695     res_erf  = _mm_mul_pd(PolyAP0,gmx_mm_inv_pd(PolyAQ0));
 696     res_erf  = _mm_add_pd(CAoffset,res_erf);
 697     res_erf  = _mm_mul_pd(x,res_erf);
 698
 699     /* Calculate erfc() in range [1,4.5] */
 700     t       = _mm_sub_pd(xabs,one);
 701     t2      = _mm_mul_pd(t,t);
 702
 703     PolyBP0  = _mm_mul_pd(CBP6,t2);
 704     PolyBP1  = _mm_mul_pd(CBP5,t2);
 705     PolyBP0  = _mm_add_pd(PolyBP0,CBP4);
 706     PolyBP1  = _mm_add_pd(PolyBP1,CBP3);
 707     PolyBP0  = _mm_mul_pd(PolyBP0,t2);
 708     PolyBP1  = _mm_mul_pd(PolyBP1,t2);
 709     PolyBP0  = _mm_add_pd(PolyBP0,CBP2);
 710     PolyBP1  = _mm_add_pd(PolyBP1,CBP1);
 711     PolyBP0  = _mm_mul_pd(PolyBP0,t2);
 712     PolyBP1  = _mm_mul_pd(PolyBP1,t);
 713     PolyBP0  = _mm_add_pd(PolyBP0,CBP0);
 714     PolyBP0  = _mm_add_pd(PolyBP0,PolyBP1);
 715
 716     PolyBQ1 = _mm_mul_pd(CBQ7,t2);
 717     PolyBQ0 = _mm_mul_pd(CBQ6,t2);
 718     PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ5);
 719     PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ4);
 720     PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
 721     PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
 722     PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ3);
 723     PolyBQ0 = _mm_add_pd(PolyBQ0,CBQ2);
 724     PolyBQ1 = _mm_mul_pd(PolyBQ1,t2);
 725     PolyBQ0 = _mm_mul_pd(PolyBQ0,t2);
 726     PolyBQ1 = _mm_add_pd(PolyBQ1,CBQ1);
 727     PolyBQ0 = _mm_add_pd(PolyBQ0,one);
 728     PolyBQ1 = _mm_mul_pd(PolyBQ1,t);
 729     PolyBQ0 = _mm_add_pd(PolyBQ0,PolyBQ1);
 730
 731     res_erfcB = _mm_mul_pd(PolyBP0,gmx_mm_inv_pd(PolyBQ0));
 732
 733     res_erfcB = _mm_mul_pd(res_erfcB,xabs);
 734
 735     /* Calculate erfc() in range [4.5,inf] */
 736     w       = gmx_mm_inv_pd(xabs);
 737     w2      = _mm_mul_pd(w,w);
 738
 739     PolyCP0  = _mm_mul_pd(CCP6,w2);
 740     PolyCP1  = _mm_mul_pd(CCP5,w2);
 741     PolyCP0  = _mm_add_pd(PolyCP0,CCP4);
 742     PolyCP1  = _mm_add_pd(PolyCP1,CCP3);
 743     PolyCP0  = _mm_mul_pd(PolyCP0,w2);
 744     PolyCP1  = _mm_mul_pd(PolyCP1,w2);
 745     PolyCP0  = _mm_add_pd(PolyCP0,CCP2);
 746     PolyCP1  = _mm_add_pd(PolyCP1,CCP1);
 747     PolyCP0  = _mm_mul_pd(PolyCP0,w2);
 748     PolyCP1  = _mm_mul_pd(PolyCP1,w);
 749     PolyCP0  = _mm_add_pd(PolyCP0,CCP0);
 750     PolyCP0  = _mm_add_pd(PolyCP0,PolyCP1);
 751
 752     PolyCQ0  = _mm_mul_pd(CCQ6,w2);
 753     PolyCQ1  = _mm_mul_pd(CCQ5,w2);
 754     PolyCQ0  = _mm_add_pd(PolyCQ0,CCQ4);
 755     PolyCQ1  = _mm_add_pd(PolyCQ1,CCQ3);
 756     PolyCQ0  = _mm_mul_pd(PolyCQ0,w2);
 757     PolyCQ1  = _mm_mul_pd(PolyCQ1,w2);
 758     PolyCQ0  = _mm_add_pd(PolyCQ0,CCQ2);
 759     PolyCQ1  = _mm_add_pd(PolyCQ1,CCQ1);
 760     PolyCQ0  = _mm_mul_pd(PolyCQ0,w2);
 761     PolyCQ1  = _mm_mul_pd(PolyCQ1,w);
 762     PolyCQ0  = _mm_add_pd(PolyCQ0,one);
 763     PolyCQ0  = _mm_add_pd(PolyCQ0,PolyCQ1);
 764
 765     expmx2   = gmx_mm_exp_pd( _mm_or_pd(signbit, x2) );
 766
 767     res_erfcC = _mm_mul_pd(PolyCP0,gmx_mm_inv_pd(PolyCQ0));
 768     res_erfcC = _mm_add_pd(res_erfcC,CCoffset);
 769     res_erfcC = _mm_mul_pd(res_erfcC,w);
 770
 771     mask = _mm_cmpgt_pd(xabs,_mm_set1_pd(4.5));
 772     res_erfc = _mm_blendv_pd(res_erfcB,res_erfcC,mask);
 773
 774     res_erfc = _mm_mul_pd(res_erfc,expmx2);
 775
 776     /* erfc(x<0) = 2-erfc(|x|) */
 777     mask = _mm_cmplt_pd(x,_mm_setzero_pd());
 778     res_erfc = _mm_blendv_pd(res_erfc,_mm_sub_pd(two,res_erfc),mask);
 779
 780     /* Select erf() or erfc() */
 781     mask = _mm_cmplt_pd(xabs,one);
 782     res  = _mm_blendv_pd(res_erfc,_mm_sub_pd(one,res_erf),mask);
 783
 784     return res;
 785 }
 786
 787
 788 /* Calculate the force correction due to PME analytically.
 789  *
 790  * This routine is meant to enable analytical evaluation of the
 791  * direct-space PME electrostatic force to avoid tables.
 792  *
 793  * The direct-space potential should be Erfc(beta*r)/r, but there
 794  * are some problems evaluating that:
 795  *
 796  * First, the error function is difficult (read: expensive) to
 797  * approxmiate accurately for intermediate to large arguments, and
 798  * this happens already in ranges of beta*r that occur in simulations.
 799  * Second, we now try to avoid calculating potentials in Gromacs but
 800  * use forces directly.
 801  *
 802  * We can simply things slight by noting that the PME part is really
 803  * a correction to the normal Coulomb force since Erfc(z)=1-Erf(z), i.e.
 804  *
 805  * V= 1/r - Erf(beta*r)/r
 806  *
 807  * The first term we already have from the inverse square root, so
 808  * that we can leave out of this routine.
 809  *
 810  * For pme tolerances of 1e-3 to 1e-8 and cutoffs of 0.5nm to 1.8nm,
 811  * the argument beta*r will be in the range 0.15 to ~4. Use your
 812  * favorite plotting program to realize how well-behaved Erf(z)/z is
 813  * in this range!
 814  *
 815  * We approximate f(z)=erf(z)/z with a rational minimax polynomial.
 816  * However, it turns out it is more efficient to approximate f(z)/z and
 817  * then only use even powers. This is another minor optimization, since
 818  * we actually WANT f(z)/z, because it is going to be multiplied by
 819  * the vector between the two atoms to get the vectorial force. The
 820  * fastest flops are the ones we can avoid calculating!
 821  *
 822  * So, here's how it should be used:
 823  *
 824  * 1. Calculate r^2.
 825  * 2. Multiply by beta^2, so you get z^2=beta^2*r^2.
 826  * 3. Evaluate this routine with z^2 as the argument.
 827  * 4. The return value is the expression:
 828  *
 829  *
 830  *       2*exp(-z^2)     erf(z)
 831  *       ------------ - --------
 832  *       sqrt(Pi)*z^2      z^3
 833  *
 834  * 5. Multiply the entire expression by beta^3. This will get you
 835  *
 836  *       beta^3*2*exp(-z^2)     beta^3*erf(z)
 837  *       ------------------  - ---------------
 838  *          sqrt(Pi)*z^2            z^3
 839  *
 840  *    or, switching back to r (z=r*beta):
 841  *
 842  *       2*beta*exp(-r^2*beta^2)   erf(r*beta)
 843  *       ----------------------- - -----------
 844  *            sqrt(Pi)*r^2            r^3
 845  *
 846  *
 847  *    With a bit of math exercise you should be able to confirm that
 848  *    this is exactly D[Erf[beta*r]/r,r] divided by r another time.
 849  *
 850  * 6. Add the result to 1/r^3, multiply by the product of the charges,
 851  *    and you have your force (divided by r). A final multiplication
 852  *    with the vector connecting the two particles and you have your
 853  *    vectorial force to add to the particles.
 854  *
 855  */
 856 static __m128d
 857 gmx_mm_pmecorrF_pd(__m128d z2)
 858 {
 859     const __m128d  FN10     = _mm_set1_pd(-8.0072854618360083154e-14);
 860     const __m128d  FN9      = _mm_set1_pd(1.1859116242260148027e-11);
 861     const __m128d  FN8      = _mm_set1_pd(-8.1490406329798423616e-10);
 862     const __m128d  FN7      = _mm_set1_pd(3.4404793543907847655e-8);
 863     const __m128d  FN6      = _mm_set1_pd(-9.9471420832602741006e-7);
 864     const __m128d  FN5      = _mm_set1_pd(0.000020740315999115847456);
 865     const __m128d  FN4      = _mm_set1_pd(-0.00031991745139313364005);
 866     const __m128d  FN3      = _mm_set1_pd(0.0035074449373659008203);
 867     const __m128d  FN2      = _mm_set1_pd(-0.031750380176100813405);
 868     const __m128d  FN1      = _mm_set1_pd(0.13884101728898463426);
 869     const __m128d  FN0      = _mm_set1_pd(-0.75225277815249618847);
 870
 871     const __m128d  FD5      = _mm_set1_pd(0.000016009278224355026701);
 872     const __m128d  FD4      = _mm_set1_pd(0.00051055686934806966046);
 873     const __m128d  FD3      = _mm_set1_pd(0.0081803507497974289008);
 874     const __m128d  FD2      = _mm_set1_pd(0.077181146026670287235);
 875     const __m128d  FD1      = _mm_set1_pd(0.41543303143712535988);
 876     const __m128d  FD0      = _mm_set1_pd(1.0);
 877
 878     __m128d z4;
 879     __m128d polyFN0,polyFN1,polyFD0,polyFD1;
 880
 881     z4             = _mm_mul_pd(z2,z2);
 882
 883     polyFD1        = _mm_mul_pd(FD5,z4);
 884     polyFD0        = _mm_mul_pd(FD4,z4);
 885     polyFD1        = _mm_add_pd(polyFD1,FD3);
 886     polyFD0        = _mm_add_pd(polyFD0,FD2);
 887     polyFD1        = _mm_mul_pd(polyFD1,z4);
 888     polyFD0        = _mm_mul_pd(polyFD0,z4);
 889     polyFD1        = _mm_add_pd(polyFD1,FD1);
 890     polyFD0        = _mm_add_pd(polyFD0,FD0);
 891     polyFD1        = _mm_mul_pd(polyFD1,z2);
 892     polyFD0        = _mm_add_pd(polyFD0,polyFD1);
 893
 894     polyFD0        = gmx_mm_inv_pd(polyFD0);
 895
 896     polyFN0        = _mm_mul_pd(FN10,z4);
 897     polyFN1        = _mm_mul_pd(FN9,z4);
 898     polyFN0        = _mm_add_pd(polyFN0,FN8);
 899     polyFN1        = _mm_add_pd(polyFN1,FN7);
 900     polyFN0        = _mm_mul_pd(polyFN0,z4);
 901     polyFN1        = _mm_mul_pd(polyFN1,z4);
 902     polyFN0        = _mm_add_pd(polyFN0,FN6);
 903     polyFN1        = _mm_add_pd(polyFN1,FN5);
 904     polyFN0        = _mm_mul_pd(polyFN0,z4);
 905     polyFN1        = _mm_mul_pd(polyFN1,z4);
 906     polyFN0        = _mm_add_pd(polyFN0,FN4);
 907     polyFN1        = _mm_add_pd(polyFN1,FN3);
 908     polyFN0        = _mm_mul_pd(polyFN0,z4);
 909     polyFN1        = _mm_mul_pd(polyFN1,z4);
 910     polyFN0        = _mm_add_pd(polyFN0,FN2);
 911     polyFN1        = _mm_add_pd(polyFN1,FN1);
 912     polyFN0        = _mm_mul_pd(polyFN0,z4);
 913     polyFN1        = _mm_mul_pd(polyFN1,z2);
 914     polyFN0        = _mm_add_pd(polyFN0,FN0);
 915     polyFN0        = _mm_add_pd(polyFN0,polyFN1);
 916
 917     return   _mm_mul_pd(polyFN0,polyFD0);
 918 }
 919
 920
 921
 922
 923 /* Calculate the potential correction due to PME analytically.
 924  *
 925  * See gmx_mm256_pmecorrF_ps() for details about the approximation.
 926  *
 927  * This routine calculates Erf(z)/z, although you should provide z^2
 928  * as the input argument.
 929  *
 930  * Here's how it should be used:
 931  *
 932  * 1. Calculate r^2.
 933  * 2. Multiply by beta^2, so you get z^2=beta^2*r^2.
 934  * 3. Evaluate this routine with z^2 as the argument.
 935  * 4. The return value is the expression:
 936  *
 937  *
 938  *        erf(z)
 939  *       --------
 940  *          z
 941  *
 942  * 5. Multiply the entire expression by beta and switching back to r (z=r*beta):
 943  *
 944  *       erf(r*beta)
 945  *       -----------
 946  *           r
 947  *
 948  * 6. Subtract the result from 1/r, multiply by the product of the charges,
 949  *    and you have your potential.
 950  *
 951  */
 952 static __m128d
 953 gmx_mm_pmecorrV_pd(__m128d z2)
 954 {
 955     const __m128d  VN9      = _mm_set1_pd(-9.3723776169321855475e-13);
 956     const __m128d  VN8      = _mm_set1_pd(1.2280156762674215741e-10);
 957     const __m128d  VN7      = _mm_set1_pd(-7.3562157912251309487e-9);
 958     const __m128d  VN6      = _mm_set1_pd(2.6215886208032517509e-7);
 959     const __m128d  VN5      = _mm_set1_pd(-4.9532491651265819499e-6);
 960     const __m128d  VN4      = _mm_set1_pd(0.00025907400778966060389);
 961     const __m128d  VN3      = _mm_set1_pd(0.0010585044856156469792);
 962     const __m128d  VN2      = _mm_set1_pd(0.045247661136833092885);
 963     const __m128d  VN1      = _mm_set1_pd(0.11643931522926034421);
 964     const __m128d  VN0      = _mm_set1_pd(1.1283791671726767970);
 965
 966     const __m128d  VD5      = _mm_set1_pd(0.000021784709867336150342);
 967     const __m128d  VD4      = _mm_set1_pd(0.00064293662010911388448);
 968     const __m128d  VD3      = _mm_set1_pd(0.0096311444822588683504);
 969     const __m128d  VD2      = _mm_set1_pd(0.085608012351550627051);
 970     const __m128d  VD1      = _mm_set1_pd(0.43652499166614811084);
 971     const __m128d  VD0      = _mm_set1_pd(1.0);
 972
 973     __m128d z4;
 974     __m128d polyVN0,polyVN1,polyVD0,polyVD1;
 975
 976     z4             = _mm_mul_pd(z2,z2);
 977
 978     polyVD1        = _mm_mul_pd(VD5,z4);
 979     polyVD0        = _mm_mul_pd(VD4,z4);
 980     polyVD1        = _mm_add_pd(polyVD1,VD3);
 981     polyVD0        = _mm_add_pd(polyVD0,VD2);
 982     polyVD1        = _mm_mul_pd(polyVD1,z4);
 983     polyVD0        = _mm_mul_pd(polyVD0,z4);
 984     polyVD1        = _mm_add_pd(polyVD1,VD1);
 985     polyVD0        = _mm_add_pd(polyVD0,VD0);
 986     polyVD1        = _mm_mul_pd(polyVD1,z2);
 987     polyVD0        = _mm_add_pd(polyVD0,polyVD1);
 988
 989     polyVD0        = gmx_mm_inv_pd(polyVD0);
 990
 991     polyVN1        = _mm_mul_pd(VN9,z4);
 992     polyVN0        = _mm_mul_pd(VN8,z4);
 993     polyVN1        = _mm_add_pd(polyVN1,VN7);
 994     polyVN0        = _mm_add_pd(polyVN0,VN6);
 995     polyVN1        = _mm_mul_pd(polyVN1,z4);
 996     polyVN0        = _mm_mul_pd(polyVN0,z4);
 997     polyVN1        = _mm_add_pd(polyVN1,VN5);
 998     polyVN0        = _mm_add_pd(polyVN0,VN4);
 999     polyVN1        = _mm_mul_pd(polyVN1,z4);
1000     polyVN0        = _mm_mul_pd(polyVN0,z4);
1001     polyVN1        = _mm_add_pd(polyVN1,VN3);
1002     polyVN0        = _mm_add_pd(polyVN0,VN2);
1003     polyVN1        = _mm_mul_pd(polyVN1,z4);
1004     polyVN0        = _mm_mul_pd(polyVN0,z4);
1005     polyVN1        = _mm_add_pd(polyVN1,VN1);
1006     polyVN0        = _mm_add_pd(polyVN0,VN0);
1007     polyVN1        = _mm_mul_pd(polyVN1,z2);
1008     polyVN0        = _mm_add_pd(polyVN0,polyVN1);
1009
1010     return   _mm_mul_pd(polyVN0,polyVD0);
1011 }
1012
1013
1014 static int
1015 gmx_mm_sincos_pd(__m128d x,
1016                  __m128d *sinval,
1017                  __m128d *cosval)
1018 {
1019 #ifdef _MSC_VER
1020     __declspec(align(16))
1021     const double sintable[34] =
1022     {
1023         1.00000000000000000e+00 , 0.00000000000000000e+00 ,
1024         9.95184726672196929e-01 , 9.80171403295606036e-02 ,
1025         9.80785280403230431e-01 , 1.95090322016128248e-01 ,
1026         9.56940335732208824e-01 , 2.90284677254462331e-01 ,
1027         9.23879532511286738e-01 , 3.82683432365089782e-01 ,
1028         8.81921264348355050e-01 , 4.71396736825997642e-01 ,
1029         8.31469612302545236e-01 , 5.55570233019602178e-01 ,
1030         7.73010453362736993e-01 , 6.34393284163645488e-01 ,
1031         7.07106781186547573e-01 , 7.07106781186547462e-01 ,
1032         6.34393284163645599e-01 , 7.73010453362736882e-01 ,
1033         5.55570233019602289e-01 , 8.31469612302545125e-01 ,
1034         4.71396736825997809e-01 , 8.81921264348354939e-01 ,
1035         3.82683432365089837e-01 , 9.23879532511286738e-01 ,
1036         2.90284677254462276e-01 , 9.56940335732208935e-01 ,
1037         1.95090322016128304e-01 , 9.80785280403230431e-01 ,
1038         9.80171403295607702e-02 , 9.95184726672196818e-01 ,
1039         0.0 , 1.00000000000000000e+00
1040     };
1041 #else
1042     const __m128d sintable[17] =
1043     {
1044         _mm_set_pd( 0.0 , 1.0 ),
1045         _mm_set_pd( sin(  1.0 * (M_PI/2.0) / 16.0) , cos(  1.0 * (M_PI/2.0) / 16.0) ),
1046         _mm_set_pd( sin(  2.0 * (M_PI/2.0) / 16.0) , cos(  2.0 * (M_PI/2.0) / 16.0) ),
1047         _mm_set_pd( sin(  3.0 * (M_PI/2.0) / 16.0) , cos(  3.0 * (M_PI/2.0) / 16.0) ),
1048         _mm_set_pd( sin(  4.0 * (M_PI/2.0) / 16.0) , cos(  4.0 * (M_PI/2.0) / 16.0) ),
1049         _mm_set_pd( sin(  5.0 * (M_PI/2.0) / 16.0) , cos(  5.0 * (M_PI/2.0) / 16.0) ),
1050         _mm_set_pd( sin(  6.0 * (M_PI/2.0) / 16.0) , cos(  6.0 * (M_PI/2.0) / 16.0) ),
1051         _mm_set_pd( sin(  7.0 * (M_PI/2.0) / 16.0) , cos(  7.0 * (M_PI/2.0) / 16.0) ),
1052         _mm_set_pd( sin(  8.0 * (M_PI/2.0) / 16.0) , cos(  8.0 * (M_PI/2.0) / 16.0) ),
1053         _mm_set_pd( sin(  9.0 * (M_PI/2.0) / 16.0) , cos(  9.0 * (M_PI/2.0) / 16.0) ),
1054         _mm_set_pd( sin( 10.0 * (M_PI/2.0) / 16.0) , cos( 10.0 * (M_PI/2.0) / 16.0) ),
1055         _mm_set_pd( sin( 11.0 * (M_PI/2.0) / 16.0) , cos( 11.0 * (M_PI/2.0) / 16.0) ),
1056         _mm_set_pd( sin( 12.0 * (M_PI/2.0) / 16.0) , cos( 12.0 * (M_PI/2.0) / 16.0) ),
1057         _mm_set_pd( sin( 13.0 * (M_PI/2.0) / 16.0) , cos( 13.0 * (M_PI/2.0) / 16.0) ),
1058         _mm_set_pd( sin( 14.0 * (M_PI/2.0) / 16.0) , cos( 14.0 * (M_PI/2.0) / 16.0) ),
1059         _mm_set_pd( sin( 15.0 * (M_PI/2.0) / 16.0) , cos( 15.0 * (M_PI/2.0) / 16.0) ),
1060         _mm_set_pd(  1.0 , 0.0 )
1061     };
1062 #endif
1063
1064     const __m128d signmask    = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1065     const __m128i signbit_epi32  = _mm_set1_epi32(0x80000000);
1066
1067     const __m128d tabscale      = _mm_set1_pd(32.0/M_PI);
1068     const __m128d invtabscale0  = _mm_set1_pd(9.81747508049011230469e-02);
1069     const __m128d invtabscale1  = _mm_set1_pd(1.96197799156550576057e-08);
1070     const __m128i ione          = _mm_set1_epi32(1);
1071     const __m128i i32           = _mm_set1_epi32(32);
1072     const __m128i i16           = _mm_set1_epi32(16);
1073     const __m128i tabmask       = _mm_set1_epi32(0x3F);
1074     const __m128d sinP7         = _mm_set1_pd(-1.0/5040.0);
1075     const __m128d sinP5         = _mm_set1_pd(1.0/120.0);
1076     const __m128d sinP3         = _mm_set1_pd(-1.0/6.0);
1077     const __m128d sinP1         = _mm_set1_pd(1.0);
1078
1079     const __m128d cosP6         = _mm_set1_pd(-1.0/720.0);
1080     const __m128d cosP4         = _mm_set1_pd(1.0/24.0);
1081     const __m128d cosP2         = _mm_set1_pd(-1.0/2.0);
1082     const __m128d cosP0         = _mm_set1_pd(1.0);
1083
1084     __m128d scalex;
1085     __m128i tabidx,corridx;
1086     __m128d xabs,z,z2,polySin,polyCos;
1087     __m128d xpoint;
1088     __m128d ypoint0,ypoint1;
1089
1090     __m128d sinpoint,cospoint;
1091     __m128d xsign,ssign,csign;
1092     __m128i imask,sswapsign,cswapsign;
1093     __m128d minusone;
1094
1095     xsign    = _mm_andnot_pd(signmask,x);
1096     xabs     = _mm_and_pd(x,signmask);
1097
1098     scalex   = _mm_mul_pd(tabscale,xabs);
1099     tabidx   = _mm_cvtpd_epi32(scalex);
1100
1101     xpoint   = _mm_round_pd(scalex,_MM_FROUND_TO_NEAREST_INT);
1102
1103     /* Extended precision arithmetics */
1104     z        = _mm_sub_pd(xabs,_mm_mul_pd(invtabscale0,xpoint));
1105     z        = _mm_sub_pd(z,_mm_mul_pd(invtabscale1,xpoint));
1106
1107     /* Range reduction to 0..2*Pi */
1108     tabidx   = _mm_and_si128(tabidx,tabmask);
1109
1110     /* tabidx is now in range [0,..,64] */
1111     imask     = _mm_cmpgt_epi32(tabidx,i32);
1112     sswapsign = imask;
1113     cswapsign = imask;
1114     corridx   = _mm_and_si128(imask,i32);
1115     tabidx    = _mm_sub_epi32(tabidx,corridx);
1116
1117     /* tabidx is now in range [0..32] */
1118     imask     = _mm_cmpgt_epi32(tabidx,i16);
1119     cswapsign = _mm_xor_si128(cswapsign,imask);
1120     corridx   = _mm_sub_epi32(i32,tabidx);
1121     tabidx    = _mm_blendv_epi8(tabidx,corridx,imask);
1122     /* tabidx is now in range [0..16] */
1123     ssign     = _mm_cvtepi32_pd( _mm_or_si128( sswapsign , ione ) );
1124     csign     = _mm_cvtepi32_pd( _mm_or_si128( cswapsign , ione ) );
1125
1126 #ifdef _MSC_VER
1127     ypoint0  = _mm_load_pd(sintable + 2*_mm_extract_epi32(tabidx,0));
1128     ypoint1  = _mm_load_pd(sintable + 2*_mm_extract_epi32(tabidx,1));
1129 #else
1130     ypoint0  = sintable[_mm_extract_epi32(tabidx,0)];
1131     ypoint1  = sintable[_mm_extract_epi32(tabidx,1)];
1132 #endif
1133     sinpoint = _mm_unpackhi_pd(ypoint0,ypoint1);
1134     cospoint = _mm_unpacklo_pd(ypoint0,ypoint1);
1135
1136     sinpoint = _mm_mul_pd(sinpoint,ssign);
1137     cospoint = _mm_mul_pd(cospoint,csign);
1138
1139     z2       = _mm_mul_pd(z,z);
1140
1141     polySin  = _mm_mul_pd(sinP7,z2);
1142     polySin  = _mm_add_pd(polySin,sinP5);
1143     polySin  = _mm_mul_pd(polySin,z2);
1144     polySin  = _mm_add_pd(polySin,sinP3);
1145     polySin  = _mm_mul_pd(polySin,z2);
1146     polySin  = _mm_add_pd(polySin,sinP1);
1147     polySin  = _mm_mul_pd(polySin,z);
1148
1149     polyCos  = _mm_mul_pd(cosP6,z2);
1150     polyCos  = _mm_add_pd(polyCos,cosP4);
1151     polyCos  = _mm_mul_pd(polyCos,z2);
1152     polyCos  = _mm_add_pd(polyCos,cosP2);
1153     polyCos  = _mm_mul_pd(polyCos,z2);
1154     polyCos  = _mm_add_pd(polyCos,cosP0);
1155
1156     *sinval  = _mm_xor_pd(_mm_add_pd( _mm_mul_pd(sinpoint,polyCos) , _mm_mul_pd(cospoint,polySin) ),xsign);
1157     *cosval  = _mm_sub_pd( _mm_mul_pd(cospoint,polyCos) , _mm_mul_pd(sinpoint,polySin) );
1158
1159     return 0;
1160 }
1161
1162 /*
1163  * IMPORTANT: Do NOT call both sin & cos if you need both results, since each of them
1164  * will then call the sincos() routine and waste a factor 2 in performance!
1165  */
1166 static __m128d
1167 gmx_mm_sin_pd(__m128d x)
1168 {
1169     __m128d s,c;
1170     gmx_mm_sincos_pd(x,&s,&c);
1171     return s;
1172 }
1173
1174 /*
1175  * IMPORTANT: Do NOT call both sin & cos if you need both results, since each of them
1176  * will then call the sincos() routine and waste a factor 2 in performance!
1177  */
1178 static __m128d
1179 gmx_mm_cos_pd(__m128d x)
1180 {
1181     __m128d s,c;
1182     gmx_mm_sincos_pd(x,&s,&c);
1183     return c;
1184 }
1185
1186
1187
1188 static __m128d
1189 gmx_mm_tan_pd(__m128d x)
1190 {
1191     __m128d sinval,cosval;
1192     __m128d tanval;
1193
1194     gmx_mm_sincos_pd(x,&sinval,&cosval);
1195
1196     tanval = _mm_mul_pd(sinval,gmx_mm_inv_pd(cosval));
1197
1198     return tanval;
1199 }
1200
1201
1202
1203 static __m128d
1204 gmx_mm_asin_pd(__m128d x)
1205 {
1206     /* Same algorithm as cephes library */
1207     const __m128d signmask  = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1208     const __m128d limit1    = _mm_set1_pd(0.625);
1209     const __m128d limit2    = _mm_set1_pd(1e-8);
1210     const __m128d one       = _mm_set1_pd(1.0);
1211     const __m128d halfpi    = _mm_set1_pd(M_PI/2.0);
1212     const __m128d quarterpi = _mm_set1_pd(M_PI/4.0);
1213     const __m128d morebits  = _mm_set1_pd(6.123233995736765886130e-17);
1214
1215     const __m128d P5        = _mm_set1_pd(4.253011369004428248960e-3);
1216     const __m128d P4        = _mm_set1_pd(-6.019598008014123785661e-1);
1217     const __m128d P3        = _mm_set1_pd(5.444622390564711410273e0);
1218     const __m128d P2        = _mm_set1_pd(-1.626247967210700244449e1);
1219     const __m128d P1        = _mm_set1_pd(1.956261983317594739197e1);
1220     const __m128d P0        = _mm_set1_pd(-8.198089802484824371615e0);
1221
1222     const __m128d Q4        = _mm_set1_pd(-1.474091372988853791896e1);
1223     const __m128d Q3        = _mm_set1_pd(7.049610280856842141659e1);
1224     const __m128d Q2        = _mm_set1_pd(-1.471791292232726029859e2);
1225     const __m128d Q1        = _mm_set1_pd(1.395105614657485689735e2);
1226     const __m128d Q0        = _mm_set1_pd(-4.918853881490881290097e1);
1227
1228     const __m128d R4        = _mm_set1_pd(2.967721961301243206100e-3);
1229     const __m128d R3        = _mm_set1_pd(-5.634242780008963776856e-1);
1230     const __m128d R2        = _mm_set1_pd(6.968710824104713396794e0);
1231     const __m128d R1        = _mm_set1_pd(-2.556901049652824852289e1);
1232     const __m128d R0        = _mm_set1_pd(2.853665548261061424989e1);
1233
1234     const __m128d S3        = _mm_set1_pd(-2.194779531642920639778e1);
1235     const __m128d S2        = _mm_set1_pd(1.470656354026814941758e2);
1236     const __m128d S1        = _mm_set1_pd(-3.838770957603691357202e2);
1237     const __m128d S0        = _mm_set1_pd(3.424398657913078477438e2);
1238
1239     __m128d sign;
1240     __m128d mask;
1241     __m128d xabs;
1242     __m128d zz,ww,z,q,w,y,zz2,ww2;
1243     __m128d PA,PB;
1244     __m128d QA,QB;
1245     __m128d RA,RB;
1246     __m128d SA,SB;
1247     __m128d nom,denom;
1248
1249     sign  = _mm_andnot_pd(signmask,x);
1250     xabs  = _mm_and_pd(x,signmask);
1251
1252     mask  = _mm_cmpgt_pd(xabs,limit1);
1253
1254     zz    = _mm_sub_pd(one,xabs);
1255     ww    = _mm_mul_pd(xabs,xabs);
1256     zz2   = _mm_mul_pd(zz,zz);
1257     ww2   = _mm_mul_pd(ww,ww);
1258
1259     /* R */
1260     RA    = _mm_mul_pd(R4,zz2);
1261     RB    = _mm_mul_pd(R3,zz2);
1262     RA    = _mm_add_pd(RA,R2);
1263     RB    = _mm_add_pd(RB,R1);
1264     RA    = _mm_mul_pd(RA,zz2);
1265     RB    = _mm_mul_pd(RB,zz);
1266     RA    = _mm_add_pd(RA,R0);
1267     RA    = _mm_add_pd(RA,RB);
1268
1269     /* S, SA = zz2 */
1270     SB    = _mm_mul_pd(S3,zz2);
1271     SA    = _mm_add_pd(zz2,S2);
1272     SB    = _mm_add_pd(SB,S1);
1273     SA    = _mm_mul_pd(SA,zz2);
1274     SB    = _mm_mul_pd(SB,zz);
1275     SA    = _mm_add_pd(SA,S0);
1276     SA    = _mm_add_pd(SA,SB);
1277
1278     /* P */
1279     PA    = _mm_mul_pd(P5,ww2);
1280     PB    = _mm_mul_pd(P4,ww2);
1281     PA    = _mm_add_pd(PA,P3);
1282     PB    = _mm_add_pd(PB,P2);
1283     PA    = _mm_mul_pd(PA,ww2);
1284     PB    = _mm_mul_pd(PB,ww2);
1285     PA    = _mm_add_pd(PA,P1);
1286     PB    = _mm_add_pd(PB,P0);
1287     PA    = _mm_mul_pd(PA,ww);
1288     PA    = _mm_add_pd(PA,PB);
1289
1290     /* Q, QA = ww2 */
1291     QB    = _mm_mul_pd(Q4,ww2);
1292     QA    = _mm_add_pd(ww2,Q3);
1293     QB    = _mm_add_pd(QB,Q2);
1294     QA    = _mm_mul_pd(QA,ww2);
1295     QB    = _mm_mul_pd(QB,ww2);
1296     QA    = _mm_add_pd(QA,Q1);
1297     QB    = _mm_add_pd(QB,Q0);
1298     QA    = _mm_mul_pd(QA,ww);
1299     QA    = _mm_add_pd(QA,QB);
1300
1301     RA    = _mm_mul_pd(RA,zz);
1302     PA    = _mm_mul_pd(PA,ww);
1303
1304     nom   = _mm_blendv_pd( PA,RA,mask );
1305     denom = _mm_blendv_pd( QA,SA,mask );
1306
1307     q     = _mm_mul_pd( nom, gmx_mm_inv_pd(denom) );
1308
1309     zz    = _mm_add_pd(zz,zz);
1310     zz    = gmx_mm_sqrt_pd(zz);
1311     z     = _mm_sub_pd(quarterpi,zz);
1312     zz    = _mm_mul_pd(zz,q);
1313     zz    = _mm_sub_pd(zz,morebits);
1314     z     = _mm_sub_pd(z,zz);
1315     z     = _mm_add_pd(z,quarterpi);
1316
1317     w     = _mm_mul_pd(xabs,q);
1318     w     = _mm_add_pd(w,xabs);
1319
1320     z     = _mm_blendv_pd( w,z,mask );
1321
1322     mask  = _mm_cmpgt_pd(xabs,limit2);
1323     z     = _mm_blendv_pd( xabs,z,mask );
1324
1325     z = _mm_xor_pd(z,sign);
1326
1327     return z;
1328 }
1329
1330
1331 static __m128d
1332 gmx_mm_acos_pd(__m128d x)
1333 {
1334     const __m128d signmask  = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1335     const __m128d one        = _mm_set1_pd(1.0);
1336     const __m128d half       = _mm_set1_pd(0.5);
1337     const __m128d pi         = _mm_set1_pd(M_PI);
1338     const __m128d quarterpi0 = _mm_set1_pd(7.85398163397448309616e-1);
1339     const __m128d quarterpi1 = _mm_set1_pd(6.123233995736765886130e-17);
1340
1341
1342     __m128d mask1;
1343
1344     __m128d z,z1,z2;
1345
1346     mask1 = _mm_cmpgt_pd(x,half);
1347     z1    = _mm_mul_pd(half,_mm_sub_pd(one,x));
1348     z1    = gmx_mm_sqrt_pd(z1);
1349     z     = _mm_blendv_pd( x,z1,mask1 );
1350
1351     z     = gmx_mm_asin_pd(z);
1352
1353     z1    = _mm_add_pd(z,z);
1354
1355     z2    = _mm_sub_pd(quarterpi0,z);
1356     z2    = _mm_add_pd(z2,quarterpi1);
1357     z2    = _mm_add_pd(z2,quarterpi0);
1358
1359     z     = _mm_blendv_pd(z2,z1,mask1);
1360
1361     return z;
1362 }
1363
1364 static __m128d
1365 gmx_mm_atan_pd(__m128d x)
1366 {
1367     /* Same algorithm as cephes library */
1368     const __m128d signmask  = gmx_mm_castsi128_pd( _mm_set_epi32(0x7FFFFFFF,0xFFFFFFFF,0x7FFFFFFF,0xFFFFFFFF) );
1369     const __m128d limit1    = _mm_set1_pd(0.66);
1370     const __m128d limit2    = _mm_set1_pd(2.41421356237309504880);
1371     const __m128d quarterpi = _mm_set1_pd(M_PI/4.0);
1372     const __m128d halfpi    = _mm_set1_pd(M_PI/2.0);
1373     const __m128d mone      = _mm_set1_pd(-1.0);
1374     const __m128d morebits1 = _mm_set1_pd(0.5*6.123233995736765886130E-17);
1375     const __m128d morebits2 = _mm_set1_pd(6.123233995736765886130E-17);
1376
1377     const __m128d P4        = _mm_set1_pd(-8.750608600031904122785E-1);
1378     const __m128d P3        = _mm_set1_pd(-1.615753718733365076637E1);
1379     const __m128d P2        = _mm_set1_pd(-7.500855792314704667340E1);
1380     const __m128d P1        = _mm_set1_pd(-1.228866684490136173410E2);
1381     const __m128d P0        = _mm_set1_pd(-6.485021904942025371773E1);
1382
1383     const __m128d Q4        = _mm_set1_pd(2.485846490142306297962E1);
1384     const __m128d Q3        = _mm_set1_pd(1.650270098316988542046E2);
1385     const __m128d Q2        = _mm_set1_pd(4.328810604912902668951E2);
1386     const __m128d Q1        = _mm_set1_pd(4.853903996359136964868E2);
1387     const __m128d Q0        = _mm_set1_pd(1.945506571482613964425E2);
1388
1389     __m128d sign;
1390     __m128d mask1,mask2;
1391     __m128d y,t1,t2;
1392     __m128d z,z2;
1393     __m128d P_A,P_B,Q_A,Q_B;
1394
1395     sign   = _mm_andnot_pd(signmask,x);
1396     x      = _mm_and_pd(x,signmask);
1397
1398     mask1  = _mm_cmpgt_pd(x,limit1);
1399     mask2  = _mm_cmpgt_pd(x,limit2);
1400
1401     t1     = _mm_mul_pd(_mm_add_pd(x,mone),gmx_mm_inv_pd(_mm_sub_pd(x,mone)));
1402     t2     = _mm_mul_pd(mone,gmx_mm_inv_pd(x));
1403
1404     y      = _mm_and_pd(mask1,quarterpi);
1405     y      = _mm_or_pd( _mm_and_pd(mask2,halfpi) , _mm_andnot_pd(mask2,y) );
1406
1407     x      = _mm_or_pd( _mm_and_pd(mask1,t1) , _mm_andnot_pd(mask1,x) );
1408     x      = _mm_or_pd( _mm_and_pd(mask2,t2) , _mm_andnot_pd(mask2,x) );
1409
1410     z      = _mm_mul_pd(x,x);
1411     z2     = _mm_mul_pd(z,z);
1412
1413     P_A    = _mm_mul_pd(P4,z2);
1414     P_B    = _mm_mul_pd(P3,z2);
1415     P_A    = _mm_add_pd(P_A,P2);
1416     P_B    = _mm_add_pd(P_B,P1);
1417     P_A    = _mm_mul_pd(P_A,z2);
1418     P_B    = _mm_mul_pd(P_B,z);
1419     P_A    = _mm_add_pd(P_A,P0);
1420     P_A    = _mm_add_pd(P_A,P_B);
1421
1422     /* Q_A = z2 */
1423     Q_B    = _mm_mul_pd(Q4,z2);
1424     Q_A    = _mm_add_pd(z2,Q3);
1425     Q_B    = _mm_add_pd(Q_B,Q2);
1426     Q_A    = _mm_mul_pd(Q_A,z2);
1427     Q_B    = _mm_mul_pd(Q_B,z2);
1428     Q_A    = _mm_add_pd(Q_A,Q1);
1429     Q_B    = _mm_add_pd(Q_B,Q0);
1430     Q_A    = _mm_mul_pd(Q_A,z);
1431     Q_A    = _mm_add_pd(Q_A,Q_B);
1432
1433     z      = _mm_mul_pd(z,P_A);
1434     z      = _mm_mul_pd(z,gmx_mm_inv_pd(Q_A));
1435     z      = _mm_mul_pd(z,x);
1436     z      = _mm_add_pd(z,x);
1437
1438     t1     = _mm_and_pd(mask1,morebits1);
1439     t1     = _mm_or_pd( _mm_and_pd(mask2,morebits2) , _mm_andnot_pd(mask2,t1) );
1440
1441     z      = _mm_add_pd(z,t1);
1442     y      = _mm_add_pd(y,z);
1443
1444     y      = _mm_xor_pd(y,sign);
1445
1446     return y;
1447 }
1448
1449
1450 static __m128d
1451 gmx_mm_atan2_pd(__m128d y, __m128d x)
1452 {
1453     const __m128d pi          = _mm_set1_pd(M_PI);
1454     const __m128d minuspi     = _mm_set1_pd(-M_PI);
1455     const __m128d halfpi      = _mm_set1_pd(M_PI/2.0);
1456     const __m128d minushalfpi = _mm_set1_pd(-M_PI/2.0);
1457
1458     __m128d z,z1,z3,z4;
1459     __m128d w;
1460     __m128d maskx_lt,maskx_eq;
1461     __m128d masky_lt,masky_eq;
1462     __m128d mask1,mask2,mask3,mask4,maskall;
1463
1464     maskx_lt  = _mm_cmplt_pd(x,_mm_setzero_pd());
1465     masky_lt  = _mm_cmplt_pd(y,_mm_setzero_pd());
1466     maskx_eq  = _mm_cmpeq_pd(x,_mm_setzero_pd());
1467     masky_eq  = _mm_cmpeq_pd(y,_mm_setzero_pd());
1468
1469     z         = _mm_mul_pd(y,gmx_mm_inv_pd(x));
1470     z         = gmx_mm_atan_pd(z);
1471
1472     mask1     = _mm_and_pd(maskx_eq,masky_lt);
1473     mask2     = _mm_andnot_pd(maskx_lt,masky_eq);
1474     mask3     = _mm_andnot_pd( _mm_or_pd(masky_lt,masky_eq) , maskx_eq);
1475     mask4     = _mm_and_pd(masky_eq,maskx_lt);
1476
1477     maskall   = _mm_or_pd( _mm_or_pd(mask1,mask2), _mm_or_pd(mask3,mask4) );
1478
1479     z         = _mm_andnot_pd(maskall,z);
1480     z1        = _mm_and_pd(mask1,minushalfpi);
1481     z3        = _mm_and_pd(mask3,halfpi);
1482     z4        = _mm_and_pd(mask4,pi);
1483
1484     z         = _mm_or_pd( _mm_or_pd(z,z1), _mm_or_pd(z3,z4) );
1485
1486     w         = _mm_blendv_pd(pi,minuspi,masky_lt);
1487     w         = _mm_and_pd(w,maskx_lt);
1488
1489     w         = _mm_andnot_pd(maskall,w);
1490
1491     z         = _mm_add_pd(z,w);
1492
1493     return z;
1494 }
1495
1496 #endif /*_gmx_math_x86_sse4_1_double_h_ */