📄 normaldistr.cs
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/*************************************************************************
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
Contributors:
* Sergey Bochkanov (ALGLIB project). Translation from C to
pseudocode.
See subroutines comments for additional copyrights.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
- Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
- Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer listed
in this license in the documentation and/or other materials
provided with the distribution.
- Neither the name of the copyright holders nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*************************************************************************/
using System;
class normaldistr
{
/*************************************************************************
Error function
The integral is
x
-
2 | | 2
erf(x) = -------- | exp( - t ) dt.
sqrt(pi) | |
-
0
For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,1 30000 3.7e-16 1.0e-16
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
public static double erf(double x)
{
double result = 0;
double xsq = 0;
double s = 0;
double p = 0;
double q = 0;
s = Math.Sign(x);
x = Math.Abs(x);
if( x<0.5 )
{
xsq = x*x;
p = 0.007547728033418631287834;
p = 0.288805137207594084924010+xsq*p;
p = 14.3383842191748205576712+xsq*p;
p = 38.0140318123903008244444+xsq*p;
p = 3017.82788536507577809226+xsq*p;
p = 7404.07142710151470082064+xsq*p;
p = 80437.3630960840172832162+xsq*p;
q = 0.0;
q = 1.00000000000000000000000+xsq*q;
q = 38.0190713951939403753468+xsq*q;
q = 658.070155459240506326937+xsq*q;
q = 6379.60017324428279487120+xsq*q;
q = 34216.5257924628539769006+xsq*q;
q = 80437.3630960840172826266+xsq*q;
result = s*1.1283791670955125738961589031*x*p/q;
return result;
}
if( x>=10 )
{
result = s;
return result;
}
result = s*(1-erfc(x));
return result;
}
/*************************************************************************
Complementary error function
1 - erf(x) =
inf.
-
2 | | 2
erfc(x) = -------- | exp( - t ) dt
sqrt(pi) | |
-
x
For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,26.6417 30000 5.7e-14 1.5e-14
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
public static double erfc(double x)
{
double result = 0;
double p = 0;
double q = 0;
if( x<0 )
{
result = 2-erfc(-x);
return result;
}
if( x<0.5 )
{
result = 1.0-erf(x);
return result;
}
if( x>=10 )
{
result = 0;
return result;
}
p = 0.0;
p = 0.5641877825507397413087057563+x*p;
p = 9.675807882987265400604202961+x*p;
p = 77.08161730368428609781633646+x*p;
p = 368.5196154710010637133875746+x*p;
p = 1143.262070703886173606073338+x*p;
p = 2320.439590251635247384768711+x*p;
p = 2898.0293292167655611275846+x*p;
p = 1826.3348842295112592168999+x*p;
q = 1.0;
q = 17.14980943627607849376131193+x*q;
q = 137.1255960500622202878443578+x*q;
q = 661.7361207107653469211984771+x*q;
q = 2094.384367789539593790281779+x*q;
q = 4429.612803883682726711528526+x*q;
q = 6089.5424232724435504633068+x*q;
q = 4958.82756472114071495438422+x*q;
q = 1826.3348842295112595576438+x*q;
result = Math.Exp(-AP.Math.Sqr(x))*p/q;
return result;
}
/*************************************************************************
Normal distribution function
Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:
x
-
1 | | 2
ndtr(x) = --------- | exp( - t /2 ) dt
sqrt(2pi) | |
-
-inf.
= ( 1 + erf(z) ) / 2
= erfc(z) / 2
where z = x/sqrt(2). Computation is via the functions
erf and erfc.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -13,0 30000 3.4e-14 6.7e-15
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
public static double normaldistribution(double x)
{
double result = 0;
result = 0.5*(erf(x/1.41421356237309504880)+1);
return result;
}
/*************************************************************************
Inverse of the error function
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
public static double inverf(double e)
{
double result = 0;
result = invnormaldistribution(0.5*(e+1))/Math.Sqrt(2);
return result;
}
/*************************************************************************
Inverse of Normal distribution function
Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.
For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) ); then the approximation is
x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2). For larger arguments,
w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0.125, 1 20000 7.2e-16 1.3e-16
IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
public static double invnormaldistribution(double y0)
{
double result = 0;
double expm2 = 0;
double s2pi = 0;
double x = 0;
double y = 0;
double z = 0;
double y2 = 0;
double x0 = 0;
double x1 = 0;
int code = 0;
double p0 = 0;
double q0 = 0;
double p1 = 0;
double q1 = 0;
double p2 = 0;
double q2 = 0;
expm2 = 0.13533528323661269189;
s2pi = 2.50662827463100050242;
if( y0<=0 )
{
result = -AP.Math.MaxRealNumber;
return result;
}
if( y0>=1 )
{
result = AP.Math.MaxRealNumber;
return result;
}
code = 1;
y = y0;
if( y>1.0-expm2 )
{
y = 1.0-y;
code = 0;
}
if( y>expm2 )
{
y = y-0.5;
y2 = y*y;
p0 = -59.9633501014107895267;
p0 = 98.0010754185999661536+y2*p0;
p0 = -56.6762857469070293439+y2*p0;
p0 = 13.9312609387279679503+y2*p0;
p0 = -1.23916583867381258016+y2*p0;
q0 = 1;
q0 = 1.95448858338141759834+y2*q0;
q0 = 4.67627912898881538453+y2*q0;
q0 = 86.3602421390890590575+y2*q0;
q0 = -225.462687854119370527+y2*q0;
q0 = 200.260212380060660359+y2*q0;
q0 = -82.0372256168333339912+y2*q0;
q0 = 15.9056225126211695515+y2*q0;
q0 = -1.18331621121330003142+y2*q0;
x = y+y*y2*p0/q0;
x = x*s2pi;
result = x;
return result;
}
x = Math.Sqrt(-(2.0*Math.Log(y)));
x0 = x-Math.Log(x)/x;
z = 1.0/x;
if( x<8.0 )
{
p1 = 4.05544892305962419923;
p1 = 31.5251094599893866154+z*p1;
p1 = 57.1628192246421288162+z*p1;
p1 = 44.0805073893200834700+z*p1;
p1 = 14.6849561928858024014+z*p1;
p1 = 2.18663306850790267539+z*p1;
p1 = -(1.40256079171354495875*0.1)+z*p1;
p1 = -(3.50424626827848203418*0.01)+z*p1;
p1 = -(8.57456785154685413611*0.0001)+z*p1;
q1 = 1;
q1 = 15.7799883256466749731+z*q1;
q1 = 45.3907635128879210584+z*q1;
q1 = 41.3172038254672030440+z*q1;
q1 = 15.0425385692907503408+z*q1;
q1 = 2.50464946208309415979+z*q1;
q1 = -(1.42182922854787788574*0.1)+z*q1;
q1 = -(3.80806407691578277194*0.01)+z*q1;
q1 = -(9.33259480895457427372*0.0001)+z*q1;
x1 = z*p1/q1;
}
else
{
p2 = 3.23774891776946035970;
p2 = 6.91522889068984211695+z*p2;
p2 = 3.93881025292474443415+z*p2;
p2 = 1.33303460815807542389+z*p2;
p2 = 2.01485389549179081538*0.1+z*p2;
p2 = 1.23716634817820021358*0.01+z*p2;
p2 = 3.01581553508235416007*0.0001+z*p2;
p2 = 2.65806974686737550832*0.000001+z*p2;
p2 = 6.23974539184983293730*0.000000001+z*p2;
q2 = 1;
q2 = 6.02427039364742014255+z*q2;
q2 = 3.67983563856160859403+z*q2;
q2 = 1.37702099489081330271+z*q2;
q2 = 2.16236993594496635890*0.1+z*q2;
q2 = 1.34204006088543189037*0.01+z*q2;
q2 = 3.28014464682127739104*0.0001+z*q2;
q2 = 2.89247864745380683936*0.000001+z*q2;
q2 = 6.79019408009981274425*0.000000001+z*q2;
x1 = z*p2/q2;
}
x = x0-x1;
if( code!=0 )
{
x = -x;
}
result = x;
return result;
}
}
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