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 -
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  25  * files - if they are missing, get the official version at www.gromacs.org.
  26  *
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  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 #ifdef HAVE__FINITE
  44 #include "float.h"
  45 #endif
  46
  47 int gmx_nint(real a)
  48 {
  49   const real half = .5;
  50   int   result;
  51
  52   result = (a < 0.) ? ((int)(a - half)) : ((int)(a + half));
  53   return result;
  54 }
  55
  56 real cuberoot (real x)
  57 {
  58   if (x < 0)
  59     {
  60       return (-pow(-x,1.0/DIM));
  61     }
  62   else
  63     {
  64       return (pow(x,1.0/DIM));
  65     }
  66 }
  67
  68 real sign(real x,real y)
  69 {
  70   if (y < 0)
  71     return -fabs(x);
  72   else
  73     return +fabs(x);
  74 }
  75
  76 /* Double and single precision erf() and erfc() from
  77  * the Sun Freely Distributable Math Library FDLIBM.
  78  * See http://www.netlib.org/fdlibm
  79  * Specific file used: s_erf.c, version 1.3 95/01/18
  80  */
  81 /*
  82  * ====================================================
  83  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  84  *
  85  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  86  * Permission to use, copy, modify, and distribute this
  87  * software is freely granted, provided that this notice
  88  * is preserved.
  89  * ====================================================
  90  */
  91
  92 #if ( (defined SIZEOF_INT && SIZEOF_INT==4) || (SIZEOF_INT_MAX == 2147483647) )
  93    typedef int erf_int32_t;
  94    typedef unsigned int erf_u_int32_t;
  95 #elif (LONG_MAX == 2147483647L)
  96    typedef long erf_int32_t;
  97    typedef unsigned long erf_u_int32_t;
  98 #elif (SHRT_MAX == 2147483647)
  99    typedef short erf_int32_t;
 100    typedef unsigned short erf_u_int32_t;
 101 #else
 102 #  error ERROR: No 32 bit wide integer type found!
 103 #endif
 104
 105
 106 static const double
 107 tiny        = 1e-300,
 108 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 109 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 110 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 111 /* c = (float)0.84506291151 */
 112 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 113 /*
 114  * Coefficients for approximation to  erf on [0,0.84375]
 115  */
 116 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 117 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 118 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 119 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 120 pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
 121 pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
 122 pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
 123 qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
 124 qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
 125 qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
 126 qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
 127 qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
 128 /*
 129  * Coefficients for approximation to  erf  in [0.84375,1.25]
 130  */
 131 pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
 132 pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
 133 pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
 134 pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
 135 pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
 136 pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
 137 pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
 138 qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
 139 qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
 140 qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
 141 qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
 142 qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
 143 qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
 144 /*
 145  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 146  */
 147 ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
 148 ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
 149 ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
 150 ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
 151 ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
 152 ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
 153 ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
 154 ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
 155 sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
 156 sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
 157 sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
 158 sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
 159 sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
 160 sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
 161 sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
 162 sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
 163 /*
 164  * Coefficients for approximation to  erfc in [1/.35,28]
 165  */
 166 rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
 167 rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
 168 rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
 169 rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
 170 rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
 171 rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
 172 rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
 173 sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
 174 sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
 175 sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
 176 sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
 177 sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 178 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 179 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
 180
 181 double gmx_erfd(double x)
 182 {
 183
 184         erf_int32_t hx,ix,i;
 185     double R,S,P,Q,s,y,z,r;
 186
 187     union
 188     {
 189         double d;
 190         int    i[2];
 191     }
 192     conv;
 193
 194     conv.d=x;
 195
 196         /* In release-4-6 and later branches, only the test for
 197          * GMX_IEEE754_BIG_ENDIAN_WORD_ORDER will be required. */
 198 #if defined(IEEE754_BIG_ENDIAN_WORD_ORDER) || defined(GMX_IEEE754_BIG_ENDIAN_WORD_ORDER)
 199     hx=conv.i[0];
 200 #else
 201     hx=conv.i[1];
 202 #endif
 203
 204         ix = hx&0x7fffffff;
 205         if(ix>=0x7ff00000)
 206     {
 207                 /* erf(nan)=nan */
 208             i = ((erf_u_int32_t)hx>>31)<<1;
 209             return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
 210         }
 211
 212         if(ix < 0x3feb0000)
 213     {
 214                 /* |x|<0.84375 */
 215             if(ix < 0x3e300000)
 216         {
 217             /* |x|<2**-28 */
 218                 if (ix < 0x00800000)
 219                 return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
 220             return x + efx*x;
 221             }
 222             z = x*x;
 223             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 224             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 225             y = r/s;
 226             return x + x*y;
 227         }
 228         if(ix < 0x3ff40000)
 229     {
 230                 /* 0.84375 <= |x| < 1.25 */
 231             s = fabs(x)-one;
 232             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 233             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 234             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 235         }
 236         if (ix >= 0x40180000)
 237     {
 238         /* inf>|x|>=6 */
 239             if(hx>=0) return one-tiny; else return tiny-one;
 240         }
 241         x = fabs(x);
 242         s = one/(x*x);
 243         if(ix< 0x4006DB6E)
 244     {
 245         /* |x| < 1/0.35 */
 246             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
 247             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
 248         }
 249     else
 250     {
 251         /* |x| >= 1/0.35 */
 252             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
 253             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
 254         }
 255
 256     conv.d = x;
 257
 258         /* In release-4-6 and later branches, only the test for
 259          * GMX_IEEE754_BIG_ENDIAN_WORD_ORDER will be required. */
 260 #if defined(IEEE754_BIG_ENDIAN_WORD_ORDER) || defined(GMX_IEEE754_BIG_ENDIAN_WORD_ORDER)
 261     conv.i[1] = 0;
 262 #else
 263     conv.i[0] = 0;
 264 #endif
 265
 266     z = conv.d;
 267
 268         r  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
 269         if(hx>=0)
 270         return one-r/x;
 271     else
 272         return  r/x-one;
 273 }
 274
 275
 276 double gmx_erfcd(double x)
 277 {
 278         erf_int32_t hx,ix;
 279         double R,S,P,Q,s,y,z,r;
 280
 281     union
 282     {
 283         double d;
 284         int    i[2];
 285     }
 286     conv;
 287
 288     conv.d = x;
 289
 290         /* In release-4-6 and later branches, only the test for
 291          * GMX_IEEE754_BIG_ENDIAN_WORD_ORDER will be required. */
 292 #if defined(IEEE754_BIG_ENDIAN_WORD_ORDER) || defined(GMX_IEEE754_BIG_ENDIAN_WORD_ORDER)
 293     hx=conv.i[0];
 294 #else
 295     hx=conv.i[1];
 296 #endif
 297
 298         ix = hx&0x7fffffff;
 299         if(ix>=0x7ff00000)
 300     {
 301         /* erfc(nan)=nan */
 302         /* erfc(+-inf)=0,2 */
 303             return (double)(((erf_u_int32_t)hx>>31)<<1)+one/x;
 304         }
 305
 306         if(ix < 0x3feb0000)
 307     {
 308                 /* |x|<0.84375 */
 309             double r1,r2,s1,s2,s3,z2,z4;
 310             if(ix < 0x3c700000)         /* |x|<2**-56 */
 311             return one-x;
 312             z = x*x;
 313             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 314             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 315             y = r/s;
 316             if(hx < 0x3fd00000)
 317         {
 318             /* x<1/4 */
 319             return one-(x+x*y);
 320             }
 321         else
 322         {
 323             r = x*y;
 324             r += (x-half);
 325                 return half - r ;
 326             }
 327         }
 328
 329     if(ix < 0x3ff40000)
 330     {
 331                 /* 0.84375 <= |x| < 1.25 */
 332         s = fabs(x)-one;
 333         P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 334         Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 335         if(hx>=0) {
 336             z  = one-erx; return z - P/Q;
 337         } else {
 338             z = erx+P/Q; return one+z;
 339         }
 340         }
 341         if (ix < 0x403c0000)
 342     {
 343         /* |x|<28 */
 344             x = fabs(x);
 345             s = one/(x*x);
 346             if(ix< 0x4006DB6D)
 347         {
 348             /* |x| < 1/.35 ~ 2.857143*/
 349                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
 350                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
 351             }
 352         else
 353         {
 354             /* |x| >= 1/.35 ~ 2.857143 */
 355             if(hx<0&&ix>=0x40180000)
 356                 return two-tiny; /* x < -6 */
 357                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
 358                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
 359             }
 360
 361         conv.d = x;
 362
 363         /* In release-4-6 and later branches, only the test for
 364          * GMX_IEEE754_BIG_ENDIAN_WORD_ORDER will be required. */
 365 #if defined(IEEE754_BIG_ENDIAN_WORD_ORDER) || defined(GMX_IEEE754_BIG_ENDIAN_WORD_ORDER)
 366         conv.i[1] = 0;
 367 #else
 368         conv.i[0] = 0;
 369 #endif
 370
 371         z = conv.d;
 372
 373         r  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
 374
 375         if(hx>0)
 376             return r/x;
 377         else
 378             return two-r/x;
 379     }
 380     else
 381     {
 382             if(hx>0)
 383             return tiny*tiny;
 384         else
 385             return two-tiny;
 386         }
 387 }
 388
 389
 390 static const float
 391 tinyf=  1e-30,
 392 halff=  5.0000000000e-01, /* 0x3F000000 */
 393 onef =  1.0000000000e+00, /* 0x3F800000 */
 394 twof =  2.0000000000e+00, /* 0x40000000 */
 395         /* c = (subfloat)0.84506291151 */
 396 erxf =  8.4506291151e-01, /* 0x3f58560b */
 397 /*
 398  * Coefficients for approximation to  erf on [0,0.84375]
 399  */
 400 efxf =  1.2837916613e-01, /* 0x3e0375d4 */
 401 efx8f=  1.0270333290e+00, /* 0x3f8375d4 */
 402 pp0f =  1.2837916613e-01, /* 0x3e0375d4 */
 403 pp1f = -3.2504209876e-01, /* 0xbea66beb */
 404 pp2f = -2.8481749818e-02, /* 0xbce9528f */
 405 pp3f = -5.7702702470e-03, /* 0xbbbd1489 */
 406 pp4f = -2.3763017452e-05, /* 0xb7c756b1 */
 407 qq1f =  3.9791721106e-01, /* 0x3ecbbbce */
 408 qq2f =  6.5022252500e-02, /* 0x3d852a63 */
 409 qq3f =  5.0813062117e-03, /* 0x3ba68116 */
 410 qq4f =  1.3249473704e-04, /* 0x390aee49 */
 411 qq5f = -3.9602282413e-06, /* 0xb684e21a */
 412 /*
 413  * Coefficients for approximation to  erf  in [0.84375,1.25]
 414  */
 415 pa0f = -2.3621185683e-03, /* 0xbb1acdc6 */
 416 pa1f =  4.1485610604e-01, /* 0x3ed46805 */
 417 pa2f = -3.7220788002e-01, /* 0xbebe9208 */
 418 pa3f =  3.1834661961e-01, /* 0x3ea2fe54 */
 419 pa4f = -1.1089469492e-01, /* 0xbde31cc2 */
 420 pa5f =  3.5478305072e-02, /* 0x3d1151b3 */
 421 pa6f = -2.1663755178e-03, /* 0xbb0df9c0 */
 422 qa1f =  1.0642088205e-01, /* 0x3dd9f331 */
 423 qa2f =  5.4039794207e-01, /* 0x3f0a5785 */
 424 qa3f =  7.1828655899e-02, /* 0x3d931ae7 */
 425 qa4f =  1.2617121637e-01, /* 0x3e013307 */
 426 qa5f =  1.3637083583e-02, /* 0x3c5f6e13 */
 427 qa6f =  1.1984500103e-02, /* 0x3c445aa3 */
 428 /*
 429  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 430  */
 431 ra0f = -9.8649440333e-03, /* 0xbc21a093 */
 432 ra1f = -6.9385856390e-01, /* 0xbf31a0b7 */
 433 ra2f = -1.0558626175e+01, /* 0xc128f022 */
 434 ra3f = -6.2375331879e+01, /* 0xc2798057 */
 435 ra4f = -1.6239666748e+02, /* 0xc322658c */
 436 ra5f = -1.8460508728e+02, /* 0xc3389ae7 */
 437 ra6f = -8.1287437439e+01, /* 0xc2a2932b */
 438 ra7f = -9.8143291473e+00, /* 0xc11d077e */
 439 sa1f =  1.9651271820e+01, /* 0x419d35ce */
 440 sa2f =  1.3765776062e+02, /* 0x4309a863 */
 441 sa3f =  4.3456588745e+02, /* 0x43d9486f */
 442 sa4f =  6.4538726807e+02, /* 0x442158c9 */
 443 sa5f =  4.2900814819e+02, /* 0x43d6810b */
 444 sa6f =  1.0863500214e+02, /* 0x42d9451f */
 445 sa7f =  6.5702495575e+00, /* 0x40d23f7c */
 446 sa8f = -6.0424413532e-02, /* 0xbd777f97 */
 447 /*
 448  * Coefficients for approximation to  erfc in [1/.35,28]
 449  */
 450 rb0f = -9.8649431020e-03, /* 0xbc21a092 */
 451 rb1f = -7.9928326607e-01, /* 0xbf4c9dd4 */
 452 rb2f = -1.7757955551e+01, /* 0xc18e104b */
 453 rb3f = -1.6063638306e+02, /* 0xc320a2ea */
 454 rb4f = -6.3756646729e+02, /* 0xc41f6441 */
 455 rb5f = -1.0250950928e+03, /* 0xc480230b */
 456 rb6f = -4.8351919556e+02, /* 0xc3f1c275 */
 457 sb1f =  3.0338060379e+01, /* 0x41f2b459 */
 458 sb2f =  3.2579251099e+02, /* 0x43a2e571 */
 459 sb3f =  1.5367296143e+03, /* 0x44c01759 */
 460 sb4f =  3.1998581543e+03, /* 0x4547fdbb */
 461 sb5f =  2.5530502930e+03, /* 0x451f90ce */
 462 sb6f =  4.7452853394e+02, /* 0x43ed43a7 */
 463 sb7f = -2.2440952301e+01; /* 0xc1b38712 */
 464
 465
 466 typedef union
 467 {
 468   float value;
 469   erf_u_int32_t word;
 470 } ieee_float_shape_type;
 471
 472 #define GET_FLOAT_WORD(i,d)                                     \
 473 do {                                                            \
 474   ieee_float_shape_type gf_u;                                   \
 475   gf_u.value = (d);                                             \
 476   (i) = gf_u.word;                                              \
 477 } while (0)
 478
 479
 480 #define SET_FLOAT_WORD(d,i)                                     \
 481 do {                                                            \
 482   ieee_float_shape_type sf_u;                                   \
 483   sf_u.word = (i);                                              \
 484   (d) = sf_u.value;                                             \
 485 } while (0)
 486
 487
 488 float gmx_erff(float x)
 489 {
 490         erf_int32_t hx,ix,i;
 491         float R,S,P,Q,s,y,z,r;
 492
 493     union
 494     {
 495         float  f;
 496         int    i;
 497     }
 498     conv;
 499
 500     conv.f=x;
 501     hx=conv.i;
 502
 503         ix = hx&0x7fffffff;
 504         if(ix>=0x7f800000)
 505     {
 506                 /* erf(nan)=nan */
 507             i = ((erf_u_int32_t)hx>>31)<<1;
 508             return (float)(1-i)+onef/x; /* erf(+-inf)=+-1 */
 509         }
 510
 511         if(ix < 0x3f580000)
 512     {
 513                 /* |x|<0.84375 */
 514             if(ix < 0x31800000)
 515         {
 516             /* |x|<2**-28 */
 517                 if (ix < 0x04000000)
 518                 return (float)0.125*((float)8.0*x+efx8f*x);                 /*avoid underflow */
 519             return x + efxf*x;
 520             }
 521         z = x*x;
 522         r = pp0f+z*(pp1f+z*(pp2f+z*(pp3f+z*pp4f)));
 523             s = onef+z*(qq1f+z*(qq2f+z*(qq3f+z*(qq4f+z*qq5f))));
 524             y = r/s;
 525             return x + x*y;
 526         }
 527         if(ix < 0x3fa00000)
 528     {
 529         /* 0.84375 <= |x| < 1.25 */
 530             s = fabs(x)-onef;
 531             P = pa0f+s*(pa1f+s*(pa2f+s*(pa3f+s*(pa4f+s*(pa5f+s*pa6f)))));
 532             Q = onef+s*(qa1f+s*(qa2f+s*(qa3f+s*(qa4f+s*(qa5f+s*qa6f)))));
 533             if(hx>=0) return erxf + P/Q; else return -erxf - P/Q;
 534         }
 535     if (ix >= 0x40c00000)
 536     {
 537                 /* inf>|x|>=6 */
 538         if(hx>=0) return onef-tinyf; else return tinyf-onef;
 539         }
 540         x = fabs(x);
 541         s = onef/(x*x);
 542     if(ix< 0x4036DB6E)
 543     {
 544         /* |x| < 1/0.35 */
 545             R=ra0f+s*(ra1f+s*(ra2f+s*(ra3f+s*(ra4f+s*(ra5f+s*(ra6f+s*ra7f))))));
 546             S=onef+s*(sa1f+s*(sa2f+s*(sa3f+s*(sa4f+s*(sa5f+s*(sa6f+s*(sa7f+s*sa8f)))))));
 547         }
 548     else
 549     {
 550         /* |x| >= 1/0.35 */
 551             R=rb0f+s*(rb1f+s*(rb2f+s*(rb3f+s*(rb4f+s*(rb5f+s*rb6f)))));
 552             S=onef+s*(sb1f+s*(sb2f+s*(sb3f+s*(sb4f+s*(sb5f+s*(sb6f+s*sb7f))))));
 553         }
 554
 555     conv.f = x;
 556     conv.i = conv.i & 0xfffff000;
 557     z = conv.f;
 558
 559         r  =  exp(-z*z-(float)0.5625)*exp((z-x)*(z+x)+R/S);
 560         if(hx>=0) return onef-r/x; else return  r/x-onef;
 561 }
 562
 563 float gmx_erfcf(float x)
 564 {
 565         erf_int32_t hx,ix;
 566         float R,S,P,Q,s,y,z,r;
 567
 568     union
 569     {
 570         float  f;
 571         int    i;
 572     }
 573     conv;
 574
 575     conv.f=x;
 576     hx=conv.i;
 577
 578         ix = hx&0x7fffffff;
 579         if(ix>=0x7f800000)
 580     {
 581         /* erfc(nan)=nan */
 582         /* erfc(+-inf)=0,2 */
 583             return (float)(((erf_u_int32_t)hx>>31)<<1)+onef/x;
 584         }
 585
 586         if(ix < 0x3f580000)
 587     {
 588                 /* |x|<0.84375 */
 589             if(ix < 0x23800000)
 590             return onef-x;      /* |x|<2**-56 */
 591             z = x*x;
 592             r = pp0f+z*(pp1f+z*(pp2f+z*(pp3f+z*pp4f)));
 593             s = onef+z*(qq1f+z*(qq2f+z*(qq3f+z*(qq4f+z*qq5f))));
 594             y = r/s;
 595             if(hx < 0x3e800000)
 596         {
 597             /* x<1/4 */
 598             return onef-(x+x*y);
 599             } else {
 600             r = x*y;
 601             r += (x-halff);
 602                 return halff - r ;
 603             }
 604         }
 605         if(ix < 0x3fa00000)
 606     {
 607         /* 0.84375 <= |x| < 1.25 */
 608             s = fabs(x)-onef;
 609             P = pa0f+s*(pa1f+s*(pa2f+s*(pa3f+s*(pa4f+s*(pa5f+s*pa6f)))));
 610             Q = onef+s*(qa1f+s*(qa2f+s*(qa3f+s*(qa4f+s*(qa5f+s*qa6f)))));
 611             if(hx>=0) {
 612                 z  = onef-erxf; return z - P/Q;
 613             } else {
 614             z = erxf+P/Q; return onef+z;
 615             }
 616         }
 617         if (ix < 0x41e00000)
 618     {
 619                 /* |x|<28 */
 620         x = fabs(x);
 621             s = onef/(x*x);
 622             if(ix< 0x4036DB6D)
 623         {
 624             /* |x| < 1/.35 ~ 2.857143*/
 625                 R=ra0f+s*(ra1f+s*(ra2f+s*(ra3f+s*(ra4f+s*(ra5f+s*(ra6f+s*ra7f))))));
 626                 S=onef+s*(sa1f+s*(sa2f+s*(sa3f+s*(sa4f+s*(sa5f+s*(sa6f+s*(sa7f+s*sa8f)))))));
 627             } else {
 628             /* |x| >= 1/.35 ~ 2.857143 */
 629             if(hx<0&&ix>=0x40c00000) return twof-tinyf;/* x < -6 */
 630                 R=rb0f+s*(rb1f+s*(rb2f+s*(rb3f+s*(rb4f+s*(rb5f+s*rb6f)))));
 631                 S=onef+s*(sb1f+s*(sb2f+s*(sb3f+s*(sb4f+s*(sb5f+s*(sb6f+s*sb7f))))));
 632             }
 633
 634         conv.f = x;
 635         conv.i = conv.i & 0xfffff000;
 636         z = conv.f;
 637
 638             r  =  exp(-z*z-(float)0.5625)*exp((z-x)*(z+x)+R/S);
 639             if(hx>0) return r/x; else return twof-r/x;
 640         } else {
 641             if(hx>0) return tinyf*tinyf; else return twof-tinyf;
 642         }
 643 }
 644
 645
 646 gmx_bool gmx_isfinite(real x)
 647 {
 648     gmx_bool returnval = TRUE;
 649     /* If no suitable function was found, assume the value is
 650      * finite. */
 651
 652 #ifdef HAVE_ISFINITE
 653     returnval = isfinite(x);
 654 #elif defined HAVE__ISFINITE
 655     returnval = _isfinite(x);
 656 #elif defined HAVE__FINITE
 657     returnval = _finite(x);
 658 #endif
 659     return returnval;
 660 }
 661
 662 gmx_bool
 663 check_int_multiply_for_overflow(gmx_large_int_t a,
 664                                 gmx_large_int_t b,
 665                                 gmx_large_int_t *result)
 666 {
 667     gmx_large_int_t sign = 1;
 668     if((0 == a) || (0 == b))
 669     {
 670         *result = 0;
 671         return TRUE;
 672     }
 673     if(a < 0)
 674     {
 675         a = -a;
 676         sign = -sign;
 677     }
 678     if(b < 0)
 679     {
 680         b = -b;
 681         sign = -sign;
 682     }
 683     if(GMX_LARGE_INT_MAX / b < a)
 684     {
 685         *result = (sign > 0) ? GMX_LARGE_INT_MAX : GMX_LARGE_INT_MIN;
 686         return FALSE;
 687     }
 688     *result = sign * a * b;
 689     return TRUE;
 690 }