MSVC_erfc.h

#ifndef OPENMM_MSVC_ERFC_H_
#define OPENMM_MSVC_ERFC_H_

/*
 * At least up to version 8 (VC++ 2005), Microsoft does not support the
 * standard C99 erf() and erfc() functions. For now we're including these
 * definitions for an MSVC compilation; if these are added later then
 * the #ifdef below should change to compare _MSC_VER with a particular
 * version level.
 */

#ifdef _MSC_VER


/***************************
*   erf.cpp
*   author:  Steve Strand
*   written: 29-Jan-04
***************************/

#include <cmath>

static const double rel_error= 1E-12;        //calculate 12 significant figures
//you can adjust rel_error to trade off between accuracy and speed
//but don't ask for > 15 figures (assuming usual 52 bit mantissa in a double)

static double erfc(double x);

static double erf(double x)
//erf(x) = 2/sqrt(pi)*integral(exp(-t^2),t,0,x)
//       = 2/sqrt(pi)*[x - x^3/3 + x^5/5*2! - x^7/7*3! + ...]
//       = 1-erfc(x)
{
    static const double two_sqrtpi=  1.128379167095512574;        // 2/sqrt(pi)
    if (fabs(x) > 2.2) {
        return 1.0 - erfc(x);        //use continued fraction when fabs(x) > 2.2
    }
    double sum= x, term= x, xsqr= x*x;
    int j= 1;
    do {
        term*= xsqr/j;
        sum-= term/(2*j+1);
        ++j;
        term*= xsqr/j;
        sum+= term/(2*j+1);
        ++j;
    } while (fabs(term)/sum > rel_error);
    return two_sqrtpi*sum;
}


static double erfc(double x)
//erfc(x) = 2/sqrt(pi)*integral(exp(-t^2),t,x,inf)
//        = exp(-x^2)/sqrt(pi) * [1/x+ (1/2)/x+ (2/2)/x+ (3/2)/x+ (4/2)/x+ ...]
//        = 1-erf(x)
//expression inside [] is a continued fraction so '+' means add to denominator only
{
    static const double one_sqrtpi=  0.564189583547756287;        // 1/sqrt(pi)
    if (fabs(x) < 2.2) {
        return 1.0 - erf(x);        //use series when fabs(x) < 2.2
    }
    // Don't look for x==0 here!
    if (x < 0) {               //continued fraction only valid for x>0
        return 2.0 - erfc(-x);
    }
    double a=1, b=x;                //last two convergent numerators
    double c=x, d=x*x+0.5;          //last two convergent denominators
    double q1, q2= b/d;             //last two convergents (a/c and b/d)
    double n= 1.0, t;
    do {
        t= a*n+b*x;
        a= b;
        b= t;
        t= c*n+d*x;
        c= d;
        d= t;
        n+= 0.5;
        q1= q2;
        q2= b/d;
      } while (fabs(q1-q2)/q2 > rel_error);
    return one_sqrtpi*exp(-x*x)*q2;
}

#endif // _MSC_VER

#endif // OPENMM_MSVC_ERFC_H_