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37 #include "gromacs/math/utilities.h"
53 result = (a < 0.) ? ((int)(a - half)) : ((int)(a + half));
61 return (-pow(-x, 1.0/3.0));
65 return (pow(x, 1.0/3.0));
69 real sign(real x, real y)
81 /* Double and single precision erf() and erfc() from
82 * the Sun Freely Distributable Math Library FDLIBM.
83 * See http://www.netlib.org/fdlibm
84 * Specific file used: s_erf.c, version 1.3 95/01/18
87 * ====================================================
88 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
90 * Developed at SunSoft, a Sun Microsystems, Inc. business.
91 * Permission to use, copy, modify, and distribute this
92 * software is freely granted, provided that this notice
94 * ====================================================
99 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
100 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
101 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
102 /* c = (float)0.84506291151 */
103 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
105 * Coefficients for approximation to erf on [0,0.84375]
107 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
108 efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
109 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
110 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
111 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
112 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
113 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
114 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
115 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
116 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
117 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
118 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
120 * Coefficients for approximation to erf in [0.84375,1.25]
122 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
123 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
124 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
125 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
126 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
127 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
128 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
129 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
130 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
131 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
132 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
133 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
134 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
136 * Coefficients for approximation to erfc in [1.25,1/0.35]
138 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
139 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
140 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
141 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
142 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
143 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
144 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
145 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
146 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
147 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
148 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
149 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
150 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
151 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
152 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
153 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
155 * Coefficients for approximation to erfc in [1/.35,28]
157 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
158 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
159 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
160 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
161 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
162 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
163 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
164 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
165 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
166 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
167 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
168 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
169 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
170 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
172 double gmx_erfd(double x)
174 gmx_int32_t hx, ix, i;
175 double R, S, P, Q, s, y, z, r;
186 #ifdef GMX_IEEE754_BIG_ENDIAN_WORD_ORDER
193 if (ix >= 0x7ff00000)
196 i = ((gmx_uint32_t)hx>>31)<<1;
197 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
208 return 0.125*(8.0*x+efx8*x); /*avoid underflow */
213 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
214 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
220 /* 0.84375 <= |x| < 1.25 */
222 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
223 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
233 if (ix >= 0x40180000)
250 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
251 S = one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
256 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
257 S = one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
262 #ifdef GMX_IEEE754_BIG_ENDIAN_WORD_ORDER
270 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
282 double gmx_erfcd(double x)
285 double R, S, P, Q, s, y, z, r;
296 #ifdef GMX_IEEE754_BIG_ENDIAN_WORD_ORDER
303 if (ix >= 0x7ff00000)
306 /* erfc(+-inf)=0,2 */
307 return (double)(((gmx_uint32_t)hx>>31)<<1)+one/x;
313 double r1, r2, s1, s2, s3, z2, z4;
314 if (ix < 0x3c700000) /* |x|<2**-56 */
319 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
320 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
337 /* 0.84375 <= |x| < 1.25 */
339 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
340 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
343 z = one-erx; return z - P/Q;
347 z = erx+P/Q; return one+z;
357 /* |x| < 1/.35 ~ 2.857143*/
358 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
359 S = one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
363 /* |x| >= 1/.35 ~ 2.857143 */
364 if (hx < 0 && ix >= 0x40180000)
366 return two-tiny; /* x < -6 */
368 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
369 S = one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
374 #ifdef GMX_IEEE754_BIG_ENDIAN_WORD_ORDER
382 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
409 halff = 5.0000000000e-01, /* 0x3F000000 */
410 onef = 1.0000000000e+00, /* 0x3F800000 */
411 twof = 2.0000000000e+00, /* 0x40000000 */
412 /* c = (subfloat)0.84506291151 */
413 erxf = 8.4506291151e-01, /* 0x3f58560b */
415 * Coefficients for approximation to erf on [0,0.84375]
417 efxf = 1.2837916613e-01, /* 0x3e0375d4 */
418 efx8f = 1.0270333290e+00, /* 0x3f8375d4 */
419 pp0f = 1.2837916613e-01, /* 0x3e0375d4 */
420 pp1f = -3.2504209876e-01, /* 0xbea66beb */
421 pp2f = -2.8481749818e-02, /* 0xbce9528f */
422 pp3f = -5.7702702470e-03, /* 0xbbbd1489 */
423 pp4f = -2.3763017452e-05, /* 0xb7c756b1 */
424 qq1f = 3.9791721106e-01, /* 0x3ecbbbce */
425 qq2f = 6.5022252500e-02, /* 0x3d852a63 */
426 qq3f = 5.0813062117e-03, /* 0x3ba68116 */
427 qq4f = 1.3249473704e-04, /* 0x390aee49 */
428 qq5f = -3.9602282413e-06, /* 0xb684e21a */
430 * Coefficients for approximation to erf in [0.84375,1.25]
432 pa0f = -2.3621185683e-03, /* 0xbb1acdc6 */
433 pa1f = 4.1485610604e-01, /* 0x3ed46805 */
434 pa2f = -3.7220788002e-01, /* 0xbebe9208 */
435 pa3f = 3.1834661961e-01, /* 0x3ea2fe54 */
436 pa4f = -1.1089469492e-01, /* 0xbde31cc2 */
437 pa5f = 3.5478305072e-02, /* 0x3d1151b3 */
438 pa6f = -2.1663755178e-03, /* 0xbb0df9c0 */
439 qa1f = 1.0642088205e-01, /* 0x3dd9f331 */
440 qa2f = 5.4039794207e-01, /* 0x3f0a5785 */
441 qa3f = 7.1828655899e-02, /* 0x3d931ae7 */
442 qa4f = 1.2617121637e-01, /* 0x3e013307 */
443 qa5f = 1.3637083583e-02, /* 0x3c5f6e13 */
444 qa6f = 1.1984500103e-02, /* 0x3c445aa3 */
446 * Coefficients for approximation to erfc in [1.25,1/0.35]
448 ra0f = -9.8649440333e-03, /* 0xbc21a093 */
449 ra1f = -6.9385856390e-01, /* 0xbf31a0b7 */
450 ra2f = -1.0558626175e+01, /* 0xc128f022 */
451 ra3f = -6.2375331879e+01, /* 0xc2798057 */
452 ra4f = -1.6239666748e+02, /* 0xc322658c */
453 ra5f = -1.8460508728e+02, /* 0xc3389ae7 */
454 ra6f = -8.1287437439e+01, /* 0xc2a2932b */
455 ra7f = -9.8143291473e+00, /* 0xc11d077e */
456 sa1f = 1.9651271820e+01, /* 0x419d35ce */
457 sa2f = 1.3765776062e+02, /* 0x4309a863 */
458 sa3f = 4.3456588745e+02, /* 0x43d9486f */
459 sa4f = 6.4538726807e+02, /* 0x442158c9 */
460 sa5f = 4.2900814819e+02, /* 0x43d6810b */
461 sa6f = 1.0863500214e+02, /* 0x42d9451f */
462 sa7f = 6.5702495575e+00, /* 0x40d23f7c */
463 sa8f = -6.0424413532e-02, /* 0xbd777f97 */
465 * Coefficients for approximation to erfc in [1/.35,28]
467 rb0f = -9.8649431020e-03, /* 0xbc21a092 */
468 rb1f = -7.9928326607e-01, /* 0xbf4c9dd4 */
469 rb2f = -1.7757955551e+01, /* 0xc18e104b */
470 rb3f = -1.6063638306e+02, /* 0xc320a2ea */
471 rb4f = -6.3756646729e+02, /* 0xc41f6441 */
472 rb5f = -1.0250950928e+03, /* 0xc480230b */
473 rb6f = -4.8351919556e+02, /* 0xc3f1c275 */
474 sb1f = 3.0338060379e+01, /* 0x41f2b459 */
475 sb2f = 3.2579251099e+02, /* 0x43a2e571 */
476 sb3f = 1.5367296143e+03, /* 0x44c01759 */
477 sb4f = 3.1998581543e+03, /* 0x4547fdbb */
478 sb5f = 2.5530502930e+03, /* 0x451f90ce */
479 sb6f = 4.7452853394e+02, /* 0x43ed43a7 */
480 sb7f = -2.2440952301e+01; /* 0xc1b38712 */
487 } ieee_float_shape_type;
489 #define GET_FLOAT_WORD(i, d) \
491 ieee_float_shape_type gf_u; \
497 #define SET_FLOAT_WORD(d, i) \
499 ieee_float_shape_type sf_u; \
505 float gmx_erff(float x)
507 gmx_int32_t hx, ix, i;
508 float R, S, P, Q, s, y, z, r;
521 if (ix >= 0x7f800000)
524 i = ((gmx_uint32_t)hx>>31)<<1;
525 return (float)(1-i)+onef/x; /* erf(+-inf)=+-1 */
536 return (float)0.125*((float)8.0*x+efx8f*x); /*avoid underflow */
541 r = pp0f+z*(pp1f+z*(pp2f+z*(pp3f+z*pp4f)));
542 s = onef+z*(qq1f+z*(qq2f+z*(qq3f+z*(qq4f+z*qq5f))));
548 /* 0.84375 <= |x| < 1.25 */
550 P = pa0f+s*(pa1f+s*(pa2f+s*(pa3f+s*(pa4f+s*(pa5f+s*pa6f)))));
551 Q = onef+s*(qa1f+s*(qa2f+s*(qa3f+s*(qa4f+s*(qa5f+s*qa6f)))));
561 if (ix >= 0x40c00000)
578 R = ra0f+s*(ra1f+s*(ra2f+s*(ra3f+s*(ra4f+s*(ra5f+s*(ra6f+s*ra7f))))));
579 S = onef+s*(sa1f+s*(sa2f+s*(sa3f+s*(sa4f+s*(sa5f+s*(sa6f+s*(sa7f+s*sa8f)))))));
584 R = rb0f+s*(rb1f+s*(rb2f+s*(rb3f+s*(rb4f+s*(rb5f+s*rb6f)))));
585 S = onef+s*(sb1f+s*(sb2f+s*(sb3f+s*(sb4f+s*(sb5f+s*(sb6f+s*sb7f))))));
589 conv.i = conv.i & 0xfffff000;
592 r = exp(-z*z-(float)0.5625)*exp((z-x)*(z+x)+R/S);
603 float gmx_erfcf(float x)
606 float R, S, P, Q, s, y, z, r;
619 if (ix >= 0x7f800000)
622 /* erfc(+-inf)=0,2 */
623 return (float)(((gmx_uint32_t)hx>>31)<<1)+onef/x;
631 return onef-x; /* |x|<2**-56 */
634 r = pp0f+z*(pp1f+z*(pp2f+z*(pp3f+z*pp4f)));
635 s = onef+z*(qq1f+z*(qq2f+z*(qq3f+z*(qq4f+z*qq5f))));
651 /* 0.84375 <= |x| < 1.25 */
653 P = pa0f+s*(pa1f+s*(pa2f+s*(pa3f+s*(pa4f+s*(pa5f+s*pa6f)))));
654 Q = onef+s*(qa1f+s*(qa2f+s*(qa3f+s*(qa4f+s*(qa5f+s*qa6f)))));
657 z = onef-erxf; return z - P/Q;
661 z = erxf+P/Q; return onef+z;
671 /* |x| < 1/.35 ~ 2.857143*/
672 R = ra0f+s*(ra1f+s*(ra2f+s*(ra3f+s*(ra4f+s*(ra5f+s*(ra6f+s*ra7f))))));
673 S = onef+s*(sa1f+s*(sa2f+s*(sa3f+s*(sa4f+s*(sa5f+s*(sa6f+s*(sa7f+s*sa8f)))))));
677 /* |x| >= 1/.35 ~ 2.857143 */
678 if (hx < 0 && ix >= 0x40c00000)
680 return twof-tinyf; /* x < -6 */
682 R = rb0f+s*(rb1f+s*(rb2f+s*(rb3f+s*(rb4f+s*(rb5f+s*rb6f)))));
683 S = onef+s*(sb1f+s*(sb2f+s*(sb3f+s*(sb4f+s*(sb5f+s*(sb6f+s*sb7f))))));
687 conv.i = conv.i & 0xfffff000;
690 r = exp(-z*z-(float)0.5625)*exp((z-x)*(z+x)+R/S);
714 gmx_bool gmx_isfinite(real gmx_unused x)
719 returnval = _finite(x);
720 #elif defined HAVE_ISFINITE
721 returnval = isfinite(x);
722 #elif defined HAVE__ISFINITE
723 returnval = _isfinite(x);
725 /* If no suitable function was found, assume the value is
732 gmx_bool gmx_isnan(real x)
738 gmx_within_tol(double f1,
742 /* The or-equal is important - otherwise we return false if f1==f2==0 */
743 if (fabs(f1-f2) <= tol*0.5*(fabs(f1)+fabs(f2)) )
754 gmx_numzero(double a)
756 return gmx_within_tol(a, 0.0, GMX_REAL_MIN/GMX_REAL_EPS);
760 gmx_log2i(unsigned int n)
762 assert(n != 0); /* behavior differs for 0 */
763 #if defined(__INTEL_COMPILER)
764 return _bit_scan_reverse(n);
765 #elif defined(__GNUC__) && UINT_MAX == 4294967295U /*also for clang*/
766 return __builtin_clz(n) ^ 31U; /* xor gets optimized out */
767 #elif defined(_MSC_VER) && _MSC_VER >= 1400
770 _BitScanReverse(&i, n);
773 #elif defined(__xlC__)
774 return 31 - __cntlz4(n);
776 /* http://graphics.stanford.edu/~seander/bithacks.html#IntegerLogLookup */
777 #define LT(n) n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n
778 static const char LogTable256[256] = {
779 -1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3,
780 LT(4), LT(5), LT(5), LT(6), LT(6), LT(6), LT(6),
781 LT(7), LT(7), LT(7), LT(7), LT(7), LT(7), LT(7), LT(7)
785 unsigned int r; /* r will be lg(n) */
786 unsigned int t, tt; /* temporaries */
788 if ((tt = n >> 16) != 0)
790 r = ((t = tt >> 8) != 0) ? 24 + LogTable256[t] : 16 + LogTable256[tt];
794 r = ((t = n >> 8) != 0) ? 8 + LogTable256[t] : LogTable256[n];
801 check_int_multiply_for_overflow(gmx_int64_t a,
805 gmx_int64_t sign = 1;
806 if ((0 == a) || (0 == b))
821 if (GMX_INT64_MAX / b < a)
823 *result = (sign > 0) ? GMX_INT64_MAX : GMX_INT64_MIN;
826 *result = sign * a * b;
830 int gmx_greatest_common_divisor(int p, int q)