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Class ErrorFunction

Error function

formula here ... Used to calculate the cumulative normal distribution function

Hierarchy

  • ErrorFunction

Implements

Index

Methods

Methods

f

  •             x
         2      |
erf(x)  =  ---------  | exp(-t*t)dt
      sqrt(pi) \|
            0
    erfc(x) =  1-erf(x)

    Note that erf(-x) = -erf(x) erfc(-x) = 2 - erfc(x)

    Method:

    1. For |x| in [0, 0.84375] erf(x) = x + x*R(x^2)
       erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
         = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
      where R = P/Q where P is an odd poly of degree 8 and Q is an odd poly of degree 10.
                    -57.90
 | R - (erf(x)-x)/x | <= 2
    Remark. The formula is derived by noting
     erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
and that
     2/sqrt(pi) = 1.128379167095512573896158903121545171688
is close to one. The interval is chosen because the fix
point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
near 0.6174), and by some experiment, 0.84375 is chosen to
guarantee the error is less than one ulp for erf.

 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
    c = 0.84506291151 rounded to single (24 bits)

    erf(x) = sign(x) * (c + P1(s)/Q1(s)) erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 1+(c+P1(s)/Q1(s)) if x < 0 |P1/Q1 - (erf(|x|)-c)| <= 2*-59.06 Remark: here we use the taylor series expansion at x=1. erf(1+s) = erf(1) + sPoly(s) = 0.845.. + P1(s)/Q1(s) That is, we use rational approximation to approximate erf(1+s) - (c = (single)0.84506291151) Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] where P1(s) = degree 6 poly in s Q1(s) = degree 6 poly in s

     3. For x in [1.25,1/0.35(~2.857143)],

    erfc(x) = (1/x)exp(-xx-0.5625+R1/S1) erf(x) = 1 - erfc(x) where R1(z) = degree 7 poly in z, (z=1/x^2) S1(z) = degree 8 poly in z

     4. For x in [1/0.35,28]

    erfc(x) = (1/x)exp(-xx-0.5625+R2/S2) if x > 0 = 2.0 - (1/x)exp(-x*x-0.5625+R2/S2) if -6<x<0 = 2.0 - tiny (if x <= -6) erf(x) = sign(x)(1.0 - erfc(x)) if x < 6, else erf(x) = sign(x)*(1.0 - tiny) where R2(z) = degree 6 poly in z, (z=1/x^2) S2(z) = degree 7 poly in z

     Note1:
To compute exp(-x*x-0.5625+R/S), let s be a single
precision number and s := x; then
 -x*x = -s*s + (s-x)*(s+x)
     exp(-x*x-0.5626+R/S) =
     exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
 Note2:
Here 4 and 5 make use of the asymptotic series
       exp(-x*x)
 erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
       x*sqrt(pi)
We use rational approximation to approximate

    g(s)=f(1/x^2) = log(erfc(x)x) - xx + 0.5625 Here is the error bound for R1/S1 and R2/S2 |R1/S1 - f(x)| < 2(-62.57) |R2/S2 - f(x)| < 2(-61.52)

     5. For inf > x >= 28

    erf(x) = sign(x) (1 - tiny) (raise inexact) erfc(x) = tinytiny (raise underflow) if x > 0 = 2 - tiny if x<0

     7. Special case:

    erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, erfc/erf(NaN) is NaN

    Parameters

    Returns Real