function value = error_f ( x )

%*****************************************************************************80
%
%% error_f() evaluates the error function ERF(X).
%
%  Discussion:
%
%    Since some compilers already supply a routine named ERF which evaluates
%    the error function, this routine has been given a distinct, if
%    somewhat unnatural, name.
%    
%    The function is defined by:
%
%      ERF(X) = ( 2 / sqrt ( PI ) ) * Integral ( 0 <= T <= X ) EXP ( - T^2 ) dT.
%
%    Properties of the function include:
%
%      Limit ( X -> -Infinity ) ERF(X) =          -1.0;
%                               ERF(0) =           0.0;
%                               ERF(0.476936...) = 0.5;
%      Limit ( X -> +Infinity ) ERF(X) =          +1.0.
%
%      0.5 * ( ERF(X/sqrt(2)) + 1 ) = Normal_01_CDF(X)
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    17 November 2006
%
%  Author:
%
%    Original FORTRAN77 version by William Cody.
%    This version by John Burkardt.
%
%  Reference:
%
%    William Cody,
%    Rational Chebyshev Approximations for the Error Function,
%    Mathematics of Computation,
%    1969, pages 631-638.
%
%  Input:
%
%    real X, the argument of the error function.
%
%  Output:
%
%    real VALUE, the value of the error function.
%
  a = [ ...
    3.16112374387056560E+00, ...
    1.13864154151050156E+02, ...
    3.77485237685302021E+02, ...
    3.20937758913846947E+03, ...
    1.85777706184603153E-01 ];
  b = [ ...
    2.36012909523441209E+01, ...
    2.44024637934444173E+02, ...
    1.28261652607737228E+03, ...
    2.84423683343917062E+03 ];
  c = [ ...
    5.64188496988670089E-01, ...
    8.88314979438837594E+00, ...
    6.61191906371416295E+01, ...
    2.98635138197400131E+02, ...
    8.81952221241769090E+02, ...
    1.71204761263407058E+03, ...
    2.05107837782607147E+03, ...
    1.23033935479799725E+03, ...
    2.15311535474403846E-08 ];
  d = [ ...
    1.57449261107098347E+01, ...
    1.17693950891312499E+02, ...
    5.37181101862009858E+02, ...
    1.62138957456669019E+03, ...
    3.29079923573345963E+03, ...
    4.36261909014324716E+03, ...
    3.43936767414372164E+03, ...
    1.23033935480374942E+03 ];
  p = [ ...
    3.05326634961232344E-01, ...
    3.60344899949804439E-01, ...
    1.25781726111229246E-01, ...
    1.60837851487422766E-02, ...
    6.58749161529837803E-04, ...
    1.63153871373020978E-02 ];
  q = [ ...
    2.56852019228982242E+00, ...
    1.87295284992346047E+00, ...
    5.27905102951428412E-01, ...
    6.05183413124413191E-02, ...
    2.33520497626869185E-03 ];
  sqrpi = 0.56418958354775628695;
  thresh = 0.46875;
  xbig = 26.543;
  xsmall = 1.11E-16;

  xabs = abs ( ( x ) );
%
%  Evaluate ERF(X) for |X| <= 0.46875.
%
  if ( xabs <= thresh )

    if ( xsmall < xabs )
      xsq = xabs * xabs;
    else
      xsq = 0.0;
    end

    xnum = a(5) * xsq;
    xden = xsq;
    for i = 1 : 3
      xnum = ( xnum + a(i) ) * xsq;
      xden = ( xden + b(i) ) * xsq;
    end

    erfxd = x * ( xnum + a(4) ) / ( xden + b(4) );
%
%  Evaluate ERFC(X) for 0.46875 <= |X| <= 4.0.
%
  elseif ( xabs <= 4.0 )

    xnum = c(9) * xabs;
    xden = xabs;
    for i = 1 : 7
      xnum = ( xnum + c(i) ) * xabs;
      xden = ( xden + d(i) ) * xabs;
    end

    erfxd = ( xnum + c(8) ) / ( xden + d(8) );
    xsq = floor ( xabs * 16.0 ) / 16.0;
    del = ( xabs - xsq ) * ( xabs + xsq );
    erfxd = exp ( - xsq * xsq ) * exp ( - del ) * erfxd;

    erfxd = ( 0.5 - erfxd ) + 0.5;

    if ( x < 0.0 )
      erfxd = - erfxd;
    end
%
%  Evaluate ERFC(X) for 4.0 < |X|.
%
  else

    if ( xbig <= xabs )

      if ( 0.0 < x )
        erfxd = 1.0;
      else
        erfxd = -1.0;
      end

    else

      xsq = 1.0 / ( xabs * xabs )

      xnum = p(6) * xsq;
      xden = xsq;
      for i = 1 : 4
        xnum = ( xnum + p(i) ) * xsq;
        xden = ( xden + q(i) ) * xsq;
      end

      erfxd = xsq * ( xnum + p(5) ) / ( xden + q(5) );
      erfxd = ( sqrpi - erfxd ) / xabs;
      xsq = floor ( xabs * 16.0 ) / 16.0;
      del = ( xabs - xsq ) * ( xabs + xsq );
      erfxd = exp ( - xsq * xsq ) * exp ( - del ) * erfxd;

      erfxd = ( 0.5 - erfxd ) + 0.5;

      if ( x < 0.0 )
        erfxd = - erfxd;
      end

    end

  end

  value = erfxd;

  return
end
