src/gmxlib/maths.c

   1 /*
   2  *
   3  *                This source code is part of
   4  *
   5  *                 G   R   O   M   A   C   S
   6  *
   7  *          GROningen MAchine for Chemical Simulations
   8  *
   9  *                        VERSION 3.2.0
  10  * Written by David van der Spoel, Erik Lindahl, Berk Hess, and others.
  11  * Copyright (c) 1991-2000, University of Groningen, The Netherlands.
  12  * Copyright (c) 2001-2004, The GROMACS development team,
  13  * check out http://www.gromacs.org for more information.
  14
  15  * This program is free software; you can redistribute it and/or
  16  * modify it under the terms of the GNU General Public License
  17  * as published by the Free Software Foundation; either version 2
  18  * of the License, or (at your option) any later version.
  19  *
  20  * If you want to redistribute modifications, please consider that
  21  * scientific software is very special. Version control is crucial -
  22  * bugs must be traceable. We will be happy to consider code for
  23  * inclusion in the official distribution, but derived work must not
  24  * be called official GROMACS. Details are found in the README & COPYING
  25  * files - if they are missing, get the official version at www.gromacs.org.
  26  *
  27  * To help us fund GROMACS development, we humbly ask that you cite
  28  * the papers on the package - you can find them in the top README file.
  29  *
  30  * For more info, check our website at http://www.gromacs.org
  31  *
  32  * And Hey:
  33  * GROningen Mixture of Alchemy and Childrens' Stories
  34  */
  35 #ifdef HAVE_CONFIG_H
  36 #include <config.h>
  37 #endif
  38
  39
  40 #include <math.h>
  41 #include <limits.h>
  42 #include "maths.h"
  43
  44 int gmx_nint(real a)
  45 {
  46   const real half = .5;
  47   int   result;
  48
  49   result = (a < 0.) ? ((int)(a - half)) : ((int)(a + half));
  50   return result;
  51 }
  52
  53 real cuberoot (real x)
  54 {
  55   if (x < 0)
  56     {
  57       return (-pow(-x,1.0/DIM));
  58     }
  59   else
  60     {
  61       return (pow(x,1.0/DIM));
  62     }
  63 }
  64
  65 real sign(real x,real y)
  66 {
  67   if (y < 0)
  68     return -fabs(x);
  69   else
  70     return +fabs(x);
  71 }
  72
  73 /* Double and single precision erf() and erfc() from
  74  * the Sun Freely Distributable Math Library FDLIBM.
  75  * See http://www.netlib.org/fdlibm
  76  * Specific file used: s_erf.c, version 1.3 95/01/18
  77  */
  78 /*
  79  * ====================================================
  80  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  81  *
  82  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  83  * Permission to use, copy, modify, and distribute this
  84  * software is freely granted, provided that this notice
  85  * is preserved.
  86  * ====================================================
  87  */
  88
  89 #if (INT_MAX == 2147483647)
  90    typedef int erf_int32_t;
  91    typedef unsigned int erf_u_int32_t;
  92 #elif (LONG_MAX == 2147483647L)
  93    typedef long erf_int32_t;
  94    typedef unsigned long erf_u_int32_t;
  95 #elif (SHRT_MAX == 2147483647)
  96    typedef short erf_int32_t;
  97    typedef unsigned short erf_u_int32_t;
  98 #else
  99 #  error ERROR: No 32 bit wide integer type found!
 100 #endif
 101
 102
 103 #ifdef GMX_DOUBLE
 104
 105 static const double
 106 tiny        = 1e-300,
 107 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 108 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 109 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 110 /* c = (float)0.84506291151 */
 111 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 112 /*
 113  * Coefficients for approximation to  erf on [0,0.84375]
 114  */
 115 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 116 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 117 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 118 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 119 pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
 120 pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
 121 pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
 122 qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
 123 qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
 124 qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
 125 qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
 126 qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
 127 /*
 128  * Coefficients for approximation to  erf  in [0.84375,1.25]
 129  */
 130 pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
 131 pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
 132 pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
 133 pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
 134 pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
 135 pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
 136 pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
 137 qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
 138 qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
 139 qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
 140 qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
 141 qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
 142 qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
 143 /*
 144  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 145  */
 146 ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
 147 ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
 148 ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
 149 ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
 150 ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
 151 ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
 152 ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
 153 ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
 154 sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
 155 sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
 156 sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
 157 sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
 158 sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
 159 sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
 160 sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
 161 sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
 162 /*
 163  * Coefficients for approximation to  erfc in [1/.35,28]
 164  */
 165 rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
 166 rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
 167 rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
 168 rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
 169 rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
 170 rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
 171 rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
 172 sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
 173 sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
 174 sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
 175 sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
 176 sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 177 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 178 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
 179
 180 double gmx_erf(double x)
 181 {
 182
 183         erf_int32_t hx,ix,i;
 184     double R,S,P,Q,s,y,z,r;
 185
 186     union
 187     {
 188         double d;
 189         int    i[2];
 190     }
 191     conv;
 192
 193     conv.d=x;
 194
 195 #ifdef IEEE754_BIG_ENDIAN_WORD_ORDER
 196     hx=conv.i[0];
 197 #else
 198     hx=conv.i[1];
 199 #endif
 200
 201         ix = hx&0x7fffffff;
 202         if(ix>=0x7ff00000)
 203     {
 204                 /* erf(nan)=nan */
 205             i = ((erf_u_int32_t)hx>>31)<<1;
 206             return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
 207         }
 208
 209         if(ix < 0x3feb0000)
 210     {
 211                 /* |x|<0.84375 */
 212             if(ix < 0x3e300000)
 213         {
 214             /* |x|<2**-28 */
 215                 if (ix < 0x00800000)
 216                 return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
 217             return x + efx*x;
 218             }
 219             z = x*x;
 220             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 221             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 222             y = r/s;
 223             return x + x*y;
 224         }
 225         if(ix < 0x3ff40000)
 226     {
 227                 /* 0.84375 <= |x| < 1.25 */
 228             s = fabs(x)-one;
 229             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 230             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 231             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 232         }
 233         if (ix >= 0x40180000)
 234     {
 235         /* inf>|x|>=6 */
 236             if(hx>=0) return one-tiny; else return tiny-one;
 237         }
 238         x = fabs(x);
 239         s = one/(x*x);
 240         if(ix< 0x4006DB6E)
 241     {
 242         /* |x| < 1/0.35 */
 243             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
 244             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
 245         }
 246     else
 247     {
 248         /* |x| >= 1/0.35 */
 249             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
 250             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
 251         }
 252
 253     conv.d = x;
 254
 255 #ifdef IEEE754_BIG_ENDIAN_WORD_ORDER
 256     conv.i[1] = 0;
 257 #else
 258     conv.i[0] = 0;
 259 #endif
 260
 261     z = conv.d;
 262
 263         r  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
 264         if(hx>=0)
 265         return one-r/x;
 266     else
 267         return  r/x-one;
 268 }
 269
 270
 271 double gmx_erfc(double x)
 272 {
 273         erf_int32_t hx,ix;
 274         double R,S,P,Q,s,y,z,r;
 275
 276     union
 277     {
 278         double d;
 279         int    i[2];
 280     }
 281     conv;
 282
 283     conv.d = x;
 284
 285 #ifdef IEEE754_BIG_ENDIAN_WORD_ORDER
 286     hx=conv.i[0];
 287 #else
 288     hx=conv.i[1];
 289 #endif
 290
 291         ix = hx&0x7fffffff;
 292         if(ix>=0x7ff00000)
 293     {
 294         /* erfc(nan)=nan */
 295         /* erfc(+-inf)=0,2 */
 296             return (double)(((erf_u_int32_t)hx>>31)<<1)+one/x;
 297         }
 298
 299         if(ix < 0x3feb0000)
 300     {
 301                 /* |x|<0.84375 */
 302             double r1,r2,s1,s2,s3,z2,z4;
 303             if(ix < 0x3c700000)         /* |x|<2**-56 */
 304             return one-x;
 305             z = x*x;
 306             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 307             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 308             y = r/s;
 309             if(hx < 0x3fd00000)
 310         {
 311             /* x<1/4 */
 312             return one-(x+x*y);
 313             }
 314         else
 315         {
 316             r = x*y;
 317             r += (x-half);
 318                 return half - r ;
 319             }
 320         }
 321
 322     if(ix < 0x3ff40000)
 323     {
 324                 /* 0.84375 <= |x| < 1.25 */
 325         s = fabs(x)-one;
 326         P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 327         Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 328         if(hx>=0) {
 329             z  = one-erx; return z - P/Q;
 330         } else {
 331             z = erx+P/Q; return one+z;
 332         }
 333         }
 334         if (ix < 0x403c0000)
 335     {
 336         /* |x|<28 */
 337             x = fabs(x);
 338             s = one/(x*x);
 339             if(ix< 0x4006DB6D)
 340         {
 341             /* |x| < 1/.35 ~ 2.857143*/
 342                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
 343                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
 344             }
 345         else
 346         {
 347             /* |x| >= 1/.35 ~ 2.857143 */
 348             if(hx<0&&ix>=0x40180000)
 349                 return two-tiny; /* x < -6 */
 350                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
 351                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
 352             }
 353
 354         conv.d = x;
 355
 356 #ifdef IEEE754_BIG_ENDIAN_WORD_ORDER
 357         conv.i[1] = 0;
 358 #else
 359         conv.i[0] = 0;
 360 #endif
 361
 362         z = conv.d;
 363
 364         r  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
 365
 366         if(hx>0)
 367             return r/x;
 368         else
 369             return two-r/x;
 370     }
 371     else
 372     {
 373             if(hx>0)
 374             return tiny*tiny;
 375         else
 376             return two-tiny;
 377         }
 378 }
 379
 380 #else /* single precision */
 381
 382
 383
 384 static const float
 385 tiny        = 1e-30,
 386 half=  5.0000000000e-01, /* 0x3F000000 */
 387 one =  1.0000000000e+00, /* 0x3F800000 */
 388 two =  2.0000000000e+00, /* 0x40000000 */
 389         /* c = (subfloat)0.84506291151 */
 390 erx =  8.4506291151e-01, /* 0x3f58560b */
 391 /*
 392  * Coefficients for approximation to  erf on [0,0.84375]
 393  */
 394 efx =  1.2837916613e-01, /* 0x3e0375d4 */
 395 efx8=  1.0270333290e+00, /* 0x3f8375d4 */
 396 pp0  =  1.2837916613e-01, /* 0x3e0375d4 */
 397 pp1  = -3.2504209876e-01, /* 0xbea66beb */
 398 pp2  = -2.8481749818e-02, /* 0xbce9528f */
 399 pp3  = -5.7702702470e-03, /* 0xbbbd1489 */
 400 pp4  = -2.3763017452e-05, /* 0xb7c756b1 */
 401 qq1  =  3.9791721106e-01, /* 0x3ecbbbce */
 402 qq2  =  6.5022252500e-02, /* 0x3d852a63 */
 403 qq3  =  5.0813062117e-03, /* 0x3ba68116 */
 404 qq4  =  1.3249473704e-04, /* 0x390aee49 */
 405 qq5  = -3.9602282413e-06, /* 0xb684e21a */
 406 /*
 407  * Coefficients for approximation to  erf  in [0.84375,1.25]
 408  */
 409 pa0  = -2.3621185683e-03, /* 0xbb1acdc6 */
 410 pa1  =  4.1485610604e-01, /* 0x3ed46805 */
 411 pa2  = -3.7220788002e-01, /* 0xbebe9208 */
 412 pa3  =  3.1834661961e-01, /* 0x3ea2fe54 */
 413 pa4  = -1.1089469492e-01, /* 0xbde31cc2 */
 414 pa5  =  3.5478305072e-02, /* 0x3d1151b3 */
 415 pa6  = -2.1663755178e-03, /* 0xbb0df9c0 */
 416 qa1  =  1.0642088205e-01, /* 0x3dd9f331 */
 417 qa2  =  5.4039794207e-01, /* 0x3f0a5785 */
 418 qa3  =  7.1828655899e-02, /* 0x3d931ae7 */
 419 qa4  =  1.2617121637e-01, /* 0x3e013307 */
 420 qa5  =  1.3637083583e-02, /* 0x3c5f6e13 */
 421 qa6  =  1.1984500103e-02, /* 0x3c445aa3 */
 422 /*
 423  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 424  */
 425 ra0  = -9.8649440333e-03, /* 0xbc21a093 */
 426 ra1  = -6.9385856390e-01, /* 0xbf31a0b7 */
 427 ra2  = -1.0558626175e+01, /* 0xc128f022 */
 428 ra3  = -6.2375331879e+01, /* 0xc2798057 */
 429 ra4  = -1.6239666748e+02, /* 0xc322658c */
 430 ra5  = -1.8460508728e+02, /* 0xc3389ae7 */
 431 ra6  = -8.1287437439e+01, /* 0xc2a2932b */
 432 ra7  = -9.8143291473e+00, /* 0xc11d077e */
 433 sa1  =  1.9651271820e+01, /* 0x419d35ce */
 434 sa2  =  1.3765776062e+02, /* 0x4309a863 */
 435 sa3  =  4.3456588745e+02, /* 0x43d9486f */
 436 sa4  =  6.4538726807e+02, /* 0x442158c9 */
 437 sa5  =  4.2900814819e+02, /* 0x43d6810b */
 438 sa6  =  1.0863500214e+02, /* 0x42d9451f */
 439 sa7  =  6.5702495575e+00, /* 0x40d23f7c */
 440 sa8  = -6.0424413532e-02, /* 0xbd777f97 */
 441 /*
 442  * Coefficients for approximation to  erfc in [1/.35,28]
 443  */
 444 rb0  = -9.8649431020e-03, /* 0xbc21a092 */
 445 rb1  = -7.9928326607e-01, /* 0xbf4c9dd4 */
 446 rb2  = -1.7757955551e+01, /* 0xc18e104b */
 447 rb3  = -1.6063638306e+02, /* 0xc320a2ea */
 448 rb4  = -6.3756646729e+02, /* 0xc41f6441 */
 449 rb5  = -1.0250950928e+03, /* 0xc480230b */
 450 rb6  = -4.8351919556e+02, /* 0xc3f1c275 */
 451 sb1  =  3.0338060379e+01, /* 0x41f2b459 */
 452 sb2  =  3.2579251099e+02, /* 0x43a2e571 */
 453 sb3  =  1.5367296143e+03, /* 0x44c01759 */
 454 sb4  =  3.1998581543e+03, /* 0x4547fdbb */
 455 sb5  =  2.5530502930e+03, /* 0x451f90ce */
 456 sb6  =  4.7452853394e+02, /* 0x43ed43a7 */
 457 sb7  = -2.2440952301e+01; /* 0xc1b38712 */
 458
 459
 460 typedef union
 461 {
 462   float value;
 463   erf_u_int32_t word;
 464 } ieee_float_shape_type;
 465
 466 #define GET_FLOAT_WORD(i,d)                                     \
 467 do {                                                            \
 468   ieee_float_shape_type gf_u;                                   \
 469   gf_u.value = (d);                                             \
 470   (i) = gf_u.word;                                              \
 471 } while (0)
 472
 473
 474 #define SET_FLOAT_WORD(d,i)                                     \
 475 do {                                                            \
 476   ieee_float_shape_type sf_u;                                   \
 477   sf_u.word = (i);                                              \
 478   (d) = sf_u.value;                                             \
 479 } while (0)
 480
 481
 482 float gmx_erf(float x)
 483 {
 484         erf_int32_t hx,ix,i;
 485         float R,S,P,Q,s,y,z,r;
 486
 487     union
 488     {
 489         float  f;
 490         int    i;
 491     }
 492     conv;
 493
 494     conv.f=x;
 495     hx=conv.i;
 496
 497         ix = hx&0x7fffffff;
 498         if(ix>=0x7f800000)
 499     {
 500                 /* erf(nan)=nan */
 501             i = ((erf_u_int32_t)hx>>31)<<1;
 502             return (float)(1-i)+one/x;  /* erf(+-inf)=+-1 */
 503         }
 504
 505         if(ix < 0x3f580000)
 506     {
 507                 /* |x|<0.84375 */
 508             if(ix < 0x31800000)
 509         {
 510             /* |x|<2**-28 */
 511                 if (ix < 0x04000000)
 512                 return (float)0.125*((float)8.0*x+efx8*x);                  /*avoid underflow */
 513             return x + efx*x;
 514             }
 515         z = x*x;
 516         r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 517             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 518             y = r/s;
 519             return x + x*y;
 520         }
 521         if(ix < 0x3fa00000)
 522     {
 523         /* 0.84375 <= |x| < 1.25 */
 524             s = fabs(x)-one;
 525             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 526             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 527             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 528         }
 529     if (ix >= 0x40c00000)
 530     {
 531                 /* inf>|x|>=6 */
 532         if(hx>=0) return one-tiny; else return tiny-one;
 533         }
 534         x = fabs(x);
 535         s = one/(x*x);
 536     if(ix< 0x4036DB6E)
 537     {
 538         /* |x| < 1/0.35 */
 539             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
 540             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
 541         }
 542     else
 543     {
 544         /* |x| >= 1/0.35 */
 545             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
 546             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
 547         }
 548
 549     conv.f = x;
 550     conv.i = conv.i & 0xfffff000;
 551     z = conv.f;
 552
 553         r  =  exp(-z*z-(float)0.5625)*exp((z-x)*(z+x)+R/S);
 554         if(hx>=0) return one-r/x; else return  r/x-one;
 555 }
 556
 557 float gmx_erfc(float x)
 558 {
 559         erf_int32_t hx,ix;
 560         float R,S,P,Q,s,y,z,r;
 561
 562     union
 563     {
 564         float  f;
 565         int    i;
 566     }
 567     conv;
 568
 569     conv.f=x;
 570     hx=conv.i;
 571
 572         ix = hx&0x7fffffff;
 573         if(ix>=0x7f800000)
 574     {
 575         /* erfc(nan)=nan */
 576         /* erfc(+-inf)=0,2 */
 577             return (float)(((erf_u_int32_t)hx>>31)<<1)+one/x;
 578         }
 579
 580         if(ix < 0x3f580000)
 581     {
 582                 /* |x|<0.84375 */
 583             if(ix < 0x23800000)
 584             return one-x;       /* |x|<2**-56 */
 585             z = x*x;
 586             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 587             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 588             y = r/s;
 589             if(hx < 0x3e800000)
 590         {
 591             /* x<1/4 */
 592             return one-(x+x*y);
 593             } else {
 594             r = x*y;
 595             r += (x-half);
 596                 return half - r ;
 597             }
 598         }
 599         if(ix < 0x3fa00000)
 600     {
 601         /* 0.84375 <= |x| < 1.25 */
 602             s = fabs(x)-one;
 603             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 604             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 605             if(hx>=0) {
 606                 z  = one-erx; return z - P/Q;
 607             } else {
 608             z = erx+P/Q; return one+z;
 609             }
 610         }
 611         if (ix < 0x41e00000)
 612     {
 613                 /* |x|<28 */
 614         x = fabs(x);
 615             s = one/(x*x);
 616             if(ix< 0x4036DB6D)
 617         {
 618             /* |x| < 1/.35 ~ 2.857143*/
 619             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
 620                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
 621             } else {
 622             /* |x| >= 1/.35 ~ 2.857143 */
 623             if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */
 624                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
 625                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
 626             }
 627
 628         conv.f = x;
 629         conv.i = conv.i & 0xfffff000;
 630         z = conv.f;
 631
 632             r  =  exp(-z*z-(float)0.5625)*exp((z-x)*(z+x)+R/S);
 633             if(hx>0) return r/x; else return two-r/x;
 634         } else {
 635             if(hx>0) return tiny*tiny; else return two-tiny;
 636         }
 637 }
 638
 639 #endif
 640
 641 float fast_float_erf(float x)
 642 {
 643         float t,ans;
 644
 645         t=1.0/(1.0+0.5*x);
 646         ans=t*exp(-x*x-1.26551223+t*(1.00002368+t*(0.37409196+t*(0.09678418+
 647                 t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+
 648                 t*(-0.82215223+t*0.17087277)))))))));
 649         return 1.0-ans;
 650 }
 651
 652 float fast_float_erfc(float x)
 653 {
 654         float t,ans;
 655
 656         t=1.0/(1.0+0.5*x);
 657         ans=t*exp(-x*x-1.26551223+t*(1.00002368+t*(0.37409196+t*(0.09678418+
 658                 t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+
 659                 t*(-0.82215223+t*0.17087277)))))))));
 660         return ans;
 661 }