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erf.h
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//From http://www.digitalmars.com/archives/cplusplus/3634.html (Jan 2004)
//Author Steve Strand. No intellectual propery rights claimed.
#ifndef OB_ERF_H
#define OB_ERF_H
#include <math.h>
// Hide this from doxygen -- internal only code
namespace temperf
{
double erf(double x);
double erfc(double x);
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)
inline 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 (abs(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;
}
inline 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 (abs(x) < 2.2)
return 1.0 - erf(x); //use series when fabs(x) < 2.2
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;
}
}//namespace
#endif // OB_ERF_H