ProbFuncs

(private, static) _underflow(a) → {Number}

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 322

(static) erf(x) → {Number}

Error function

SYNOPSIS:

double x, y, erf();

y = erf( x );



DESCRIPTION:

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
   DEC       0,1         14000       4.7e-17     1.5e-17
   IEEE      0,1         30000       3.7e-16     1.0e-16

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 487

(static) erfc(a) → {Number}

Complementary error function

SYNOPSIS:

double x, y, erfc();

y = erfc( x );



DESCRIPTION:


 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.

A special function expx2.c is used to suppress error amplification
in computing exp(-x^2).


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,26.6417   30000       1.3e-15     2.2e-16


ERROR MESSAGES:

  message         condition              value returned
erfc underflow    x > 9.231948545 (DEC)       0.0

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 380

(private, static) erfce(x) → {Number}

Exponentially scaled erfc function

  exp(x^2) erfc(x)
  valid for x > 1.
  Use with ndtr and expx2.

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 434

(static) erfi(x) → {Number}

Inverse of error function erf.

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 639

(private, static) expx2(x, sign) → {Number}

Exponential of squared argument

SYNOPSIS:

double x, y, expx2();
int sign;

y = expx2( x, sign );

DESCRIPTION:

Computes y = exp(x*x) while suppressing error amplification
that would ordinarily arise from the inexactness of the
exponential argument x*x.

If sign < 0, the result is inverted; i.e., y = exp(-x*x) .


ACCURACY:

                     Relative error:
arithmetic    domain     # trials      peak         rms
  IEEE      -26.6, 26.6    10^7       3.9e-16     8.9e-17

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 128

(static) ndtr(a) → {Number}

Normal distribution function

SYNOPSIS:

y = ndtr( x );

DESCRIPTION:

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 with care to avoid error amplification in computing exp(-x^2).


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -13,0        30000       1.3e-15     2.2e-16


ERROR MESSAGES:

  message         condition         value returned
erfc underflow    x > 37.519379347       0.0

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 295

(static) ndtri(y0) → {Number}

Inverse of Normal distribution function

SYNOPSIS:

double x, y, ndtri();

x = ndtri( y );

DESCRIPTION:

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
   DEC      0.125, 1         5500       9.5e-17     2.1e-17
   DEC      6e-39, 0.135     3500       5.7e-17     1.3e-17
   IEEE     0.125, 1        20000       7.2e-16     1.3e-16
   IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17


ERROR MESSAGES:

  message         condition    value returned
ndtri domain       x <= 0        -MAXNUM
ndtri domain       x >= 1         MAXNUM

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 587

(private, static) p1evl(x, coef, N) → {Number}

Evaluate polynomial when coefficient of x is 1.0. Otherwise same as polevl.

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 234

(private, static) polevl(x, coef, N) → {Number}

Evaluate polynomial

SYNOPSIS:

int N;
double x, y, coef[N+1], polevl[];

y = polevl( x, coef, N );

DESCRIPTION:

Evaluates polynomial of degree N:

                    2          N
y  =  C  + C x + C x  +...+ C x
       0    1     2          N

Coefficients are stored in reverse order:

coef[0] = C  , ..., coef[N] = C  .
           N                   0

 The function p1evl() assumes that coef[N] = 1.0 and is
omitted from the array.  Its calling arguments are
otherwise the same as polevl().


SPEED:

In the interest of speed, there are no checks for out
of bounds arithmetic.  This routine is used by most of
the functions in the library.  Depending on available
equipment features, the user may wish to rewrite the
program in microcode or assembly language.

Parameters

Returns

Type

Number

Details

Source

math/probfuncs.js, line 208