Single (float) precision special functions suite

/*							acoshf.c
 *
 *	Inverse hyperbolic cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, acoshf();
 *
 * y = acoshf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns inverse hyperbolic cosine of argument.
 *
 * If 1 <= x < 1.5, a polynomial approximation
 *
 *	sqrt(z) * P(z)
 *
 * where z = x-1, is used. Otherwise,
 *
 * acosh(x) = log( x + sqrt( (x-1)(x+1) ).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 1,3 100000 1.8e-7 3.9e-8
 * IEEE 1,2000 100000 3.0e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * acoshf domain |x| < 1 0.0
 *
 */

/*							airy.c
 *
 *	Airy function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, ai, aip, bi, bip;
 * int airyf();
 *
 * airyf( x, &ai, &aip, &bi, &bip );
 *
 *
 *
 * DESCRIPTION:
 *
 * Solution of the differential equation
 *
 *	y"(x) = xy.
 *
 * The function returns the two independent solutions Ai, Bi
 * and their first derivatives Ai'(x), Bi'(x).
 *
 * Evaluation is by power series summation for small x,
 * by rational minimax approximations for large x.
 *
 *
 *
 * ACCURACY:
 * Error criterion is absolute when function <= 1, relative
 * when function > 1, except * denotes relative error criterion.
 * For large negative x, the absolute error increases as x^1.5.
 * For large positive x, the relative error increases as x^1.5.
 *
 * Arithmetic domain function # trials peak rms
 * IEEE -10, 0 Ai 50000 7.0e-7 1.2e-7
 * IEEE 0, 10 Ai 50000 9.9e-6* 6.8e-7*
 * IEEE -10, 0 Ai' 50000 2.4e-6 3.5e-7
 * IEEE 0, 10 Ai' 50000 8.7e-6* 6.2e-7*
 * IEEE -10, 10 Bi 100000 2.2e-6 2.6e-7
 * IEEE -10, 10 Bi' 50000 2.2e-6 3.5e-7
 *
 */

/*							asinf.c
 *
 *	Inverse circular sine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, asinf();
 *
 * y = asinf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns radian angle between -pi/2 and +pi/2 whose sine is x.
 *
 * A polynomial of the form x + x**3 P(x**2)
 * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is
 * transformed by the identity
 *
 * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -1, 1 100000 2.5e-7 5.0e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * asinf domain |x| > 1 0.0
 *
 */

/*							acosf()
 *
 *	Inverse circular cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, acosf();
 *
 * y = acosf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns radian angle between -pi/2 and +pi/2 whose cosine
 * is x.
 *
 * Analytically, acos(x) = pi/2 - asin(x). However if |x| is
 * near 1, there is cancellation error in subtracting asin(x)
 * from pi/2. Hence if x < -0.5,
 *
 * acos(x) =	 pi - 2.0 * asin( sqrt((1+x)/2) );
 *
 * or if x > +0.5,
 *
 * acos(x) =	 2.0 * asin( sqrt((1-x)/2) ).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -1, 1 100000 1.4e-7 4.2e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * acosf domain |x| > 1 0.0
 */

/*							asinhf.c
 *
 *	Inverse hyperbolic sine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, asinhf();
 *
 * y = asinhf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns inverse hyperbolic sine of argument.
 *
 * If |x| < 0.5, the function is approximated by a rational
 * form x + x**3 P(x)/Q(x). Otherwise,
 *
 * asinh(x) = log( x + sqrt(1 + x*x) ).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -3,3 100000 2.4e-7 4.1e-8
 *
 */

/*							atanf.c
 *
 *	Inverse circular tangent
 * (arctangent)
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, atanf();
 *
 * y = atanf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns radian angle between -pi/2 and +pi/2 whose tangent
 * is x.
 *
 * Range reduction is from four intervals into the interval
 * from zero to tan( pi/8 ). A polynomial approximates
 * the function in this basic interval.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10, 10 100000 1.9e-7 4.1e-8
 *
 */

/*							atan2f()
 *
 *	Quadrant correct inverse circular tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, z, atan2f();
 *
 * z = atan2f( y, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns radian angle whose tangent is y/x.
 * Define compile time symbol ANSIC = 1 for ANSI standard,
 * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range
 * 0 to 2PI, args (x,y).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10, 10 100000 1.9e-7 4.1e-8
 * See atan.c.
 *
 */

/*							atanhf.c
 *
 *	Inverse hyperbolic tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, atanhf();
 *
 * y = atanhf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns inverse hyperbolic tangent of argument in the range
 * MINLOGF to MAXLOGF.
 *
 * If |x| < 0.5, a polynomial approximation is used.
 * Otherwise,
 * atanh(x) = 0.5 * log( (1+x)/(1-x) ).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -1,1 100000 1.4e-7 3.1e-8
 *
 */

/*							bdtrf.c
 *
 *	Binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * float p, y, bdtrf();
 *
 * y = bdtrf( k, n, p );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms 0 through k of the Binomial
 * probability density:
 *
 * k
 * -- ( n ) j n-j
 * > ( ) p (1-p)
 * -- ( j )
 * j=0
 *
 * The terms are not summed directly; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error (p varies from 0 to 1):
 * arithmetic domain # trials peak rms
 * IEEE 0,100 2000 6.9e-5 1.1e-5
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * bdtrf domain k < 0 0.0
 * n < k
 * x < 0, x > 1
 *
 */

/*							bdtrcf()
 *
 *	Complemented binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * float p, y, bdtrcf();
 *
 * y = bdtrcf( k, n, p );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms k+1 through n of the Binomial
 * probability density:
 *
 * n
 * -- ( n ) j n-j
 * > ( ) p (1-p)
 * -- ( j )
 * j=k+1
 *
 * The terms are not summed directly; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error (p varies from 0 to 1):
 * arithmetic domain # trials peak rms
 * IEEE 0,100 2000 6.0e-5 1.2e-5
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * bdtrcf domain x<0, x>1, n<k 0.0
 */

/*							bdtrif()
 *
 *	Inverse binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * float p, y, bdtrif();
 *
 * p = bdtrf( k, n, y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Finds the event probability p such that the sum of the
 * terms 0 through k of the Binomial probability density
 * is equal to the given cumulative probability y.
 *
 * This is accomplished using the inverse beta integral
 * function and the relation
 *
 * 1 - p = incbi( n-k, k+1, y ).
 *
 *
 *
 *
 * ACCURACY:
 *
 * Relative error (p varies from 0 to 1):
 * arithmetic domain # trials peak rms
 * IEEE 0,100 2000 3.5e-5 3.3e-6
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * bdtrif domain k < 0, n <= k 0.0
 * x < 0, x > 1
 *
 */

/*							betaf.c
 *
 *	Beta function
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, b, y, betaf();
 *
 * y = betaf( a, b );
 *
 *
 *
 * DESCRIPTION:
 *
 * - -
 * | (a) | (b)
 * beta( a, b ) = -----------.
 * -
 * | (a+b)
 *
 * For large arguments the logarithm of the function is
 * evaluated using lgam(), then exponentiated.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,30 10000 4.0e-5 6.0e-6
 * IEEE -20,0 10000 4.9e-3 5.4e-5
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * betaf overflow log(beta) > MAXLOG 0.0
 * a or b < 0 integer 0.0
 *
 */

/*							cbrtf.c
 *
 *	Cube root
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, cbrtf();
 *
 * y = cbrtf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the cube root of the argument, which may be negative.
 *
 * Range reduction involves determining the power of 2 of
 * the argument. A polynomial of degree 2 applied to the
 * mantissa, and multiplication by the cube root of 1, 2, or 4
 * approximates the root to within about 0.1%. Then Newton's
 * iteration is used to converge to an accurate result.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,1e38 100000 7.6e-8 2.7e-8
 *
 */

/*							chbevlf.c
 *
 *	Evaluate Chebyshev series
 *
 *
 *
 * SYNOPSIS:
 *
 * int N;
 * float x, y, coef[N], chebevlf();
 *
 * y = chbevlf( x, coef, N );
 *
 *
 *
 * DESCRIPTION:
 *
 * Evaluates the series
 *
 * N-1
 * - '
 * y = > coef[i] T (x/2)
 * - i
 * i=0
 *
 * of Chebyshev polynomials Ti at argument x/2.
 *
 * Coefficients are stored in reverse order, i.e. the zero
 * order term is last in the array. Note N is the number of
 * coefficients, not the order.
 *
 * If coefficients are for the interval a to b, x must
 * have been transformed to x -> 2(2x - b - a)/(b-a) before
 * entering the routine. This maps x from (a, b) to (-1, 1),
 * over which the Chebyshev polynomials are defined.
 *
 * If the coefficients are for the inverted interval, in
 * which (a, b) is mapped to (1/b, 1/a), the transformation
 * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity,
 * this becomes x -> 4a/x - 1.
 *
 *
 *
 * SPEED:
 *
 * Taking advantage of the recurrence properties of the
 * Chebyshev polynomials, the routine requires one more
 * addition per loop than evaluating a nested polynomial of
 * the same degree.
 *
 */

/*							chdtrf.c
 *
 *	Chi-square distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * float df, x, y, chdtrf();
 *
 * y = chdtrf( df, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the area under the left hand tail (from 0 to x)
 * of the Chi square probability density function with
 * v degrees of freedom.
 *
 *
 * inf.
 * -
 * 1 | | v/2-1 -t/2
 * P( x | v ) = ----------- | t e dt
 * v/2 - | |
 * 2 | (v/2) -
 * x
 *
 * where x is the Chi-square variable.
 *
 * The incomplete gamma integral is used, according to the
 * formula
 *
 *	y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).
 *
 *
 * The arguments must both be positive.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 3.2e-5 5.0e-6
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * chdtrf domain x < 0 or v < 1 0.0
 */

/*							chdtrcf()
 *
 *	Complemented Chi-square distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * float v, x, y, chdtrcf();
 *
 * y = chdtrcf( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the area under the right hand tail (from x to
 * infinity) of the Chi square probability density function
 * with v degrees of freedom:
 *
 *
 * inf.
 * -
 * 1 | | v/2-1 -t/2
 * P( x | v ) = ----------- | t e dt
 * v/2 - | |
 * 2 | (v/2) -
 * x
 *
 * where x is the Chi-square variable.
 *
 * The incomplete gamma integral is used, according to the
 * formula
 *
 *	y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).
 *
 *
 * The arguments must both be positive.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 2.7e-5 3.2e-6
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * chdtrc domain x < 0 or v < 1 0.0
 */

/*							chdtrif()
 *
 *	Inverse of complemented Chi-square distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * float df, x, y, chdtrif();
 *
 * x = chdtrif( df, y );
 *
 *
 *
 *
 * DESCRIPTION:
 *
 * Finds the Chi-square argument x such that the integral
 * from x to infinity of the Chi-square density is equal
 * to the given cumulative probability y.
 *
 * This is accomplished using the inverse gamma integral
 * function and the relation
 *
 * x/2 = igami( df/2, y );
 *
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 10000 2.2e-5 8.5e-7
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * chdtri domain y < 0 or y > 1 0.0
 * v < 1
 *
 */

/*							clogf.c
 *
 *	Complex natural logarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * void clogf();
 * cmplxf z, w;
 *
 * clogf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns complex logarithm to the base e (2.718...) of
 * the complex argument x.
 *
 * If z = x + iy, r = sqrt( x**2 + y**2 ),
 * then
 * w = log(r) + i arctan(y/x).
 * 
 * The arctangent ranges from -PI to +PI.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 1.9e-6 6.2e-8
 *
 * Larger relative error can be observed for z near 1 +i0.
 * In IEEE arithmetic the peak absolute error is 3.1e-7.
 *
 */

/*							cexpf()
 *
 *	Complex exponential function
 *
 *
 *
 * SYNOPSIS:
 *
 * void cexpf();
 * cmplxf z, w;
 *
 * cexpf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the exponential of the complex argument z
 * into the complex result w.
 *
 * If
 * z = x + iy,
 * r = exp(x),
 *
 * then
 *
 * w = r cos y + i r sin y.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 1.4e-7 4.5e-8
 *
 */

/*							csinf()
 *
 *	Complex circular sine
 *
 *
 *
 * SYNOPSIS:
 *
 * void csinf();
 * cmplxf z, w;
 *
 * csinf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 * z = x + iy,
 *
 * then
 *
 * w = sin x cosh y + i cos x sinh y.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 1.9e-7 5.5e-8
 *
 */

/*							ccosf()
 *
 *	Complex circular cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * void ccosf();
 * cmplxf z, w;
 *
 * ccosf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 * z = x + iy,
 *
 * then
 *
 * w = cos x cosh y - i sin x sinh y.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 1.8e-7 5.5e-8
 */

/*							ctanf()
 *
 *	Complex circular tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void ctanf();
 * cmplxf z, w;
 *
 * ctanf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 * z = x + iy,
 *
 * then
 *
 * sin 2x + i sinh 2y
 * w = --------------------.
 * cos 2x + cosh 2y
 *
 * On the real axis the denominator is zero at odd multiples
 * of PI/2. The denominator is evaluated by its Taylor
 * series near these points.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 3.3e-7 5.1e-8
 */

/*							ccotf()
 *
 *	Complex circular cotangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void ccotf();
 * cmplxf z, w;
 *
 * ccotf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 * z = x + iy,
 *
 * then
 *
 * sin 2x - i sinh 2y
 * w = --------------------.
 * cosh 2y - cos 2x
 *
 * On the real axis, the denominator has zeros at even
 * multiples of PI/2. Near these points it is evaluated
 * by a Taylor series.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 3.6e-7 5.7e-8
 * Also tested by ctan * ccot = 1 + i0.
 */

/*							casinf()
 *
 *	Complex circular arc sine
 *
 *
 *
 * SYNOPSIS:
 *
 * void casinf();
 * cmplxf z, w;
 *
 * casinf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * Inverse complex sine:
 *
 * 2
 * w = -i clog( iz + csqrt( 1 - z ) ).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 1.1e-5 1.5e-6
 * Larger relative error can be observed for z near zero.
 *
 */

/*							cacosf()
 *
 *	Complex circular arc cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * void cacosf();
 * cmplxf z, w;
 *
 * cacosf( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 * w = arccos z = PI/2 - arcsin z.
 *
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 9.2e-6 1.2e-6
 *
 */

/*							catan()
 *
 *	Complex circular arc tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * void catan();
 * cmplxf z, w;
 *
 * catan( &z, &w );
 *
 *
 *
 * DESCRIPTION:
 *
 * If
 * z = x + iy,
 *
 * then
 * 1 ( 2x )
 * Re w = - arctan(-----------) + k PI
 * 2 ( 2 2)
 * (1 - x - y )
 *
 * ( 2 2)
 * 1 (x + (y+1) )
 * Im w = - log(------------)
 * 4 ( 2 2)
 * (x + (y-1) )
 *
 * Where k is an arbitrary integer.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,+10 30000 2.3e-6 5.2e-8
 *
 */

/*							cmplxf.c
 *
 *	Complex number arithmetic
 *
 *
 *
 * SYNOPSIS:
 *
 * typedef struct {
 * float r; real part
 * float i; imaginary part
 * }cmplxf;
 *
 * cmplxf *a, *b, *c;
 *
 * caddf( a, b, c ); c = b + a
 * csubf( a, b, c ); c = b - a
 * cmulf( a, b, c ); c = b * a
 * cdivf( a, b, c ); c = b / a
 * cnegf( c ); c = -c
 * cmovf( b, c ); c = b
 *
 *
 *
 * DESCRIPTION:
 *
 * Addition:
 * c.r = b.r + a.r
 * c.i = b.i + a.i
 *
 * Subtraction:
 * c.r = b.r - a.r
 * c.i = b.i - a.i
 *
 * Multiplication:
 * c.r = b.r * a.r - b.i * a.i
 * c.i = b.r * a.i + b.i * a.r
 *
 * Division:
 * d = a.r * a.r + a.i * a.i
 * c.r = (b.r * a.r + b.i * a.i)/d
 * c.i = (b.i * a.r - b.r * a.i)/d
 * ACCURACY:
 *
 * In DEC arithmetic, the test (1/z) * z = 1 had peak relative
 * error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had
 * peak relative error 8.3e-17, rms 2.1e-17.
 *
 * Tests in the rectangle {-10,+10}:
 * Relative error:
 * arithmetic function # trials peak rms
 * IEEE cadd 30000 5.9e-8 2.6e-8
 * IEEE csub 30000 6.0e-8 2.6e-8
 * IEEE cmul 30000 1.1e-7 3.7e-8
 * IEEE cdiv 30000 2.1e-7 5.7e-8
 */

/*							coshf.c
 *
 *	Hyperbolic cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, coshf();
 *
 * y = coshf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns hyperbolic cosine of argument in the range MINLOGF to
 * MAXLOGF.
 *
 * cosh(x) = ( exp(x) + exp(-x) )/2.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-MAXLOGF 100000 1.2e-7 2.8e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * coshf overflow |x| > MAXLOGF MAXNUMF
 *
 *
 */

/*							dawsnf.c
 *
 *	Dawson's Integral
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, dawsnf();
 *
 * y = dawsnf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 * x
 * -
 * 2 | | 2
 * dawsn(x) = exp( -x ) | exp( t ) dt
 * | |
 * -
 * 0
 *
 * Three different rational approximations are employed, for
 * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,10 50000 4.4e-7 6.3e-8
 *
 *
 */

/*							ellief.c
 *
 *	Incomplete elliptic integral of the second kind
 *
 *
 *
 * SYNOPSIS:
 *
 * float phi, m, y, ellief();
 *
 * y = ellief( phi, m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 * phi
 * -
 * | |
 * | 2
 * E(phi\m) = | sqrt( 1 - m sin t ) dt
 * |
 * | | 
 * -
 * 0
 *
 * of amplitude phi and modulus m, using the arithmetic -
 * geometric mean algorithm.
 *
 *
 *
 * ACCURACY:
 *
 * Tested at random arguments with phi in [0, 2] and m in
 * [0, 1].
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,2 10000 4.5e-7 7.4e-8
 *
 *
 */

/*							ellikf.c
 *
 *	Incomplete elliptic integral of the first kind
 *
 *
 *
 * SYNOPSIS:
 *
 * float phi, m, y, ellikf();
 *
 * y = ellikf( phi, m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 *
 * phi
 * -
 * | |
 * | dt
 * F(phi\m) = | ------------------
 * | 2
 * | | sqrt( 1 - m sin t )
 * -
 * 0
 *
 * of amplitude phi and modulus m, using the arithmetic -
 * geometric mean algorithm.
 *
 *
 *
 *
 * ACCURACY:
 *
 * Tested at random points with phi in [0, 2] and m in
 * [0, 1].
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,2 10000 2.9e-7 5.8e-8
 *
 *
 */

/*							ellpef.c
 *
 *	Complete elliptic integral of the second kind
 *
 *
 *
 * SYNOPSIS:
 *
 * float m1, y, ellpef();
 *
 * y = ellpef( m1 );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 * pi/2
 * -
 * | | 2
 * E(m) = | sqrt( 1 - m sin t ) dt
 * | | 
 * -
 * 0
 *
 * Where m = 1 - m1, using the approximation
 *
 * P(x) - x log x Q(x).
 *
 * Though there are no singularities, the argument m1 is used
 * rather than m for compatibility with ellpk().
 *
 * E(1) = 1; E(0) = pi/2.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 1 30000 1.1e-7 3.9e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * ellpef domain x<0, x>1 0.0
 *
 */

/*							ellpjf.c
 *
 *	Jacobian Elliptic Functions
 *
 *
 *
 * SYNOPSIS:
 *
 * float u, m, sn, cn, dn, phi;
 * int ellpj();
 *
 * ellpj( u, m, &sn, &cn, &dn, &phi );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
 * and dn(u|m) of parameter m between 0 and 1, and real
 * argument u.
 *
 * These functions are periodic, with quarter-period on the
 * real axis equal to the complete elliptic integral
 * ellpk(1.0-m).
 *
 * Relation to incomplete elliptic integral:
 * If u = ellik(phi,m), then sn(u|m) = sin(phi),
 * and cn(u|m) = cos(phi). Phi is called the amplitude of u.
 *
 * Computation is by means of the arithmetic-geometric mean
 * algorithm, except when m is within 1e-9 of 0 or 1. In the
 * latter case with m close to 1, the approximation applies
 * only for phi < pi/2.
 *
 * ACCURACY:
 *
 * Tested at random points with u between 0 and 10, m between
 * 0 and 1.
 *
 * Absolute error (* = relative error):
 * arithmetic function # trials peak rms
 * IEEE sn 10000 1.7e-6 2.2e-7
 * IEEE cn 10000 1.6e-6 2.2e-7
 * IEEE dn 100000 3.2e-6 2.6e-7
 * IEEE phi 10000 3.9e-7* 6.7e-8*
 *
 * Larger errors occur for m near 1.
 * Peak error observed in consistency check using addition
 * theorem for sn(u+v) was 4e-16 (absolute). Also tested by
 * the above relation to the incomplete elliptic integral.
 * Accuracy deteriorates when u is large.
 *
 */

/*							ellpkf.c
 *
 *	Complete elliptic integral of the first kind
 *
 *
 *
 * SYNOPSIS:
 *
 * float m1, y, ellpkf();
 *
 * y = ellpkf( m1 );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 *
 * pi/2
 * -
 * | |
 * | dt
 * K(m) = | ------------------
 * | 2
 * | | sqrt( 1 - m sin t )
 * -
 * 0
 *
 * where m = 1 - m1, using the approximation
 *
 * P(x) - log x Q(x).
 *
 * The argument m1 is used rather than m so that the logarithmic
 * singularity at m = 1 will be shifted to the origin; this
 * preserves maximum accuracy.
 *
 * K(0) = pi/2.
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,1 30000 1.3e-7 3.4e-8
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * ellpkf domain x<0, x>1 0.0
 *
 */

/*							exp10f.c
 *
 *	Base 10 exponential function
 * (Common antilogarithm)
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, exp10f();
 *
 * y = exp10f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns 10 raised to the x power.
 *
 * Range reduction is accomplished by expressing the argument
 * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2).
 * A polynomial approximates 10**f.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -38,+38 100000 9.8e-8 2.8e-8
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * exp10 underflow x < -MAXL10 0.0
 * exp10 overflow x > MAXL10 MAXNUM
 *
 * IEEE single arithmetic: MAXL10 = 38.230809449325611792.
 *
 */

/*							exp2f.c
 *
 *	Base 2 exponential function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, exp2f();
 *
 * y = exp2f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns 2 raised to the x power.
 *
 * Range reduction is accomplished by separating the argument
 * into an integer k and fraction f such that
 * x k f
 * 2 = 2 2.
 *
 * A polynomial approximates 2**x in the basic range [-0.5, 0.5].
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -127,+127 100000 1.7e-7 2.8e-8
 *
 *
 * See exp.c for comments on error amplification.
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * exp underflow x < -MAXL2 0.0
 * exp overflow x > MAXL2 MAXNUMF
 *
 * For IEEE arithmetic, MAXL2 = 127.
 */

/*							expf.c
 *
 *	Exponential function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, expf();
 *
 * y = expf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns e (2.71828...) raised to the x power.
 *
 * Range reduction is accomplished by separating the argument
 * into an integer k and fraction f such that
 *
 * x k f
 * e = 2 e.
 *
 * A polynomial is used to approximate exp(f)
 * in the basic range [-0.5, 0.5].
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +- MAXLOG 100000 1.7e-7 2.8e-8
 *
 *
 * Error amplification in the exponential function can be
 * a serious matter. The error propagation involves
 * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
 * which shows that a 1 lsb error in representing X produces
 * a relative error of X times 1 lsb in the function.
 * While the routine gives an accurate result for arguments
 * that are exactly represented by a double precision
 * computer number, the result contains amplified roundoff
 * error for large arguments not exactly represented.
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * expf underflow x < MINLOGF 0.0
 * expf overflow x > MAXLOGF MAXNUMF
 *
 */

/*							expnf.c
 *
 *		Exponential integral En
 *
 *
 *
 * SYNOPSIS:
 *
 * int n;
 * float x, y, expnf();
 *
 * y = expnf( n, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Evaluates the exponential integral
 *
 * inf.
 * -
 * | | -xt
 * | e
 * E (x) = | ---- dt.
 * n | n
 * | | t
 * -
 * 1
 *
 *
 * Both n and x must be nonnegative.
 *
 * The routine employs either a power series, a continued
 * fraction, or an asymptotic formula depending on the
 * relative values of n and x.
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 10000 5.6e-7 1.2e-7
 *
 */

/*							expx2f.c
 *
 *	Exponential of squared argument
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, expx2f();
 *
 * y = expx2f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes y = exp(x*x) while suppressing error amplification
 * that would ordinarily arise from the inexactness of the argument x*x.
 * 
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -9.4, 9.4 10^7 1.7e-7 4.7e-8
 *
 */

/*							facf.c
 *
 *	Factorial function
 *
 *
 *
 * SYNOPSIS:
 *
 * float y, facf();
 * int i;
 *
 * y = facf( i );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns factorial of i = 1 * 2 * 3 * ... * i.
 * fac(0) = 1.0.
 *
 * Due to machine arithmetic bounds the largest value of
 * i accepted is 33 in single precision arithmetic.
 * Greater values, or negative ones,
 * produce an error message and return MAXNUM.
 *
 *
 *
 * ACCURACY:
 *
 * For i < 34 the values are simply tabulated, and have
 * full machine accuracy.
 *
 */

/*							fdtrf.c
 *
 *	F distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int df1, df2;
 * float x, y, fdtrf();
 *
 * y = fdtrf( df1, df2, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the area from zero to x under the F density
 * function (also known as Snedcor's density or the
 * variance ratio density). This is the density
 * of x = (u1/df1)/(u2/df2), where u1 and u2 are random
 * variables having Chi square distributions with df1
 * and df2 degrees of freedom, respectively.
 *
 * The incomplete beta integral is used, according to the
 * formula
 *
 *	P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
 *
 *
 * The arguments a and b are greater than zero, and x
 * x is nonnegative.
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 2.2e-5 1.1e-6
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * fdtrf domain a<0, b<0, x<0 0.0
 *
 */

/*							fdtrcf()
 *
 *	Complemented F distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int df1, df2;
 * float x, y, fdtrcf();
 *
 * y = fdtrcf( df1, df2, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the area from x to infinity under the F density
 * function (also known as Snedcor's density or the
 * variance ratio density).
 *
 *
 * inf.
 * -
 * 1 | | a-1 b-1
 * 1-P(x) = ------ | t (1-t) dt
 * B(a,b) | |
 * -
 * x
 *
 * (See fdtr.c.)
 *
 * The incomplete beta integral is used, according to the
 * formula
 *
 *	P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 7.3e-5 1.2e-5
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * fdtrcf domain a<0, b<0, x<0 0.0
 *
 */

/*							fdtrif()
 *
 *	Inverse of complemented F distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * float df1, df2, x, y, fdtrif();
 *
 * x = fdtrif( df1, df2, y );
 *
 *
 *
 *
 * DESCRIPTION:
 *
 * Finds the F density argument x such that the integral
 * from x to infinity of the F density is equal to the
 * given probability y.
 *
 * This is accomplished using the inverse beta integral
 * function and the relations
 *
 * z = incbi( df2/2, df1/2, y )
 * x = df2 (1-z) / (df1 z).
 *
 * Note: the following relations hold for the inverse of
 * the uncomplemented F distribution:
 *
 * z = incbi( df1/2, df2/2, y )
 * x = df2 z / (df1 (1-z)).
 *
 *
 *
 * ACCURACY:
 *
 * arithmetic domain # trials peak rms
 * Absolute error:
 * IEEE 0,100 5000 4.0e-5 3.2e-6
 * Relative error:
 * IEEE 0,100 5000 1.2e-3 1.8e-5
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * fdtrif domain y <= 0 or y > 1 0.0
 * v < 1
 *
 */

/*							ceilf()
 *							floorf()
 *							frexpf()
 *							ldexpf()
 *							signbitf()
 *							isnanf()
 *							isfinitef()
 *
 *	Single precision floating point numeric utilities
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y;
 * float ceilf(), floorf(), frexpf(), ldexpf();
 * int signbit(), isnan(), isfinite();
 * int expnt, n;
 *
 * y = floorf(x);
 * y = ceilf(x);
 * y = frexpf( x, &expnt );
 * y = ldexpf( x, n );
 * n = signbit(x);
 * n = isnan(x);
 * n = isfinite(x);
 *
 *
 *
 * DESCRIPTION:
 *
 * All four routines return a single precision floating point
 * result.
 *
 * sfloor() returns the largest integer less than or equal to x.
 * It truncates toward minus infinity.
 *
 * sceil() returns the smallest integer greater than or equal
 * to x. It truncates toward plus infinity.
 *
 * sfrexp() extracts the exponent from x. It returns an integer
 * power of two to expnt and the significand between 0.5 and 1
 * to y. Thus x = y * 2**expn.
 *
 * ldexpf() multiplies x by 2**n.
 *
 * signbit(x) returns 1 if the sign bit of x is 1, else 0.
 *
 * These functions are part of the standard C run time library
 * for many but not all C compilers. The ones supplied are
 * written in C for either DEC or IEEE arithmetic. They should
 * be used only if your compiler library does not already have
 * them.
 *
 * The IEEE versions assume that denormal numbers are implemented
 * in the arithmetic. Some modifications will be required if
 * the arithmetic has abrupt rather than gradual underflow.
 */

/*							fresnlf.c
 *
 *	Fresnel integral
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, S, C;
 * void fresnlf();
 *
 * fresnlf( x, &S, &C );
 *
 *
 * DESCRIPTION:
 *
 * Evaluates the Fresnel integrals
 *
 * x
 * -
 * | |
 * C(x) = | cos(pi/2 t**2) dt,
 * | |
 * -
 * 0
 *
 * x
 * -
 * | |
 * S(x) = | sin(pi/2 t**2) dt.
 * | |
 * -
 * 0
 *
 *
 * The integrals are evaluated by power series for small x.
 * For x >= 1 auxiliary functions f(x) and g(x) are employed
 * such that
 *
 * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
 * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
 *
 *
 *
 * ACCURACY:
 *
 * Relative error.
 *
 * Arithmetic function domain # trials peak rms
 * IEEE S(x) 0, 10 30000 1.1e-6 1.9e-7
 * IEEE C(x) 0, 10 30000 1.1e-6 2.0e-7
 */

/*							gammaf.c
 *
 *	Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, gammaf();
 * extern int sgngamf;
 *
 * y = gammaf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns gamma function of the argument. The result is
 * correctly signed, and the sign (+1 or -1) is also
 * returned in a global (extern) variable named sgngamf.
 * This same variable is also filled in by the logarithmic
 * gamma function lgam().
 *
 * Arguments between 0 and 10 are reduced by recurrence and the
 * function is approximated by a polynomial function covering
 * the interval (2,3). Large arguments are handled by Stirling's
 * formula. Negative arguments are made positive using
 * a reflection formula. 
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,-33 100,000 5.7e-7 1.0e-7
 * IEEE -33,0 100,000 6.1e-7 1.2e-7
 *
 *
 */

/*							lgamf()
 *
 *	Natural logarithm of gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, lgamf();
 * extern int sgngamf;
 *
 * y = lgamf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the base e (2.718...) logarithm of the absolute
 * value of the gamma function of the argument.
 * The sign (+1 or -1) of the gamma function is returned in a
 * global (extern) variable named sgngamf.
 *
 * For arguments greater than 6.5, the logarithm of the gamma
 * function is approximated by the logarithmic version of
 * Stirling's formula. Arguments between 0 and +6.5 are reduced by
 * by recurrence to the interval [.75,1.25] or [1.5,2.5] of a rational
 * approximation. The cosecant reflection formula is employed for
 * arguments less than zero.
 *
 * Arguments greater than MAXLGM = 2.035093e36 return MAXNUM and an
 * error message.
 *
 *
 *
 * ACCURACY:
 *
 *
 *
 * arithmetic domain # trials peak rms
 * IEEE -100,+100 500,000 7.4e-7 6.8e-8
 * The error criterion was relative when the function magnitude
 * was greater than one but absolute when it was less than one.
 * The routine has low relative error for positive arguments.
 *
 * The following test used the relative error criterion.
 * IEEE -2, +3 100000 4.0e-7 5.6e-8
 *
 */

/*							gdtrf.c
 *
 *	Gamma distribution function
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, b, x, y, gdtrf();
 *
 * y = gdtrf( a, b, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the integral from zero to x of the gamma probability
 * density function:
 *
 *
 * x
 * b -
 * a | | b-1 -at
 * y = ----- | t e dt
 * - | |
 * | (b) -
 * 0
 *
 * The incomplete gamma integral is used, according to the
 * relation
 *
 * y = igam( b, ax ).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 5.8e-5 3.0e-6
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * gdtrf domain x < 0 0.0
 *
 */

/*							gdtrcf.c
 *
 *	Complemented gamma distribution function
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, b, x, y, gdtrcf();
 *
 * y = gdtrcf( a, b, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the integral from x to infinity of the gamma
 * probability density function:
 *
 *
 * inf.
 * b -
 * a | | b-1 -at
 * y = ----- | t e dt
 * - | |
 * | (b) -
 * x
 *
 * The incomplete gamma integral is used, according to the
 * relation
 *
 * y = igamc( b, ax ).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 9.1e-5 1.5e-5
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * gdtrcf domain x < 0 0.0
 *
 */

/*							hyp2f1f.c
 *
 *	Gauss hypergeometric function F
 *	 2 1
 *
 *
 * SYNOPSIS:
 *
 * float a, b, c, x, y, hyp2f1f();
 *
 * y = hyp2f1f( a, b, c, x );
 *
 *
 * DESCRIPTION:
 *
 *
 * hyp2f1( a, b, c, x ) = F ( a, b; c; x )
 * 2 1
 *
 * inf.
 * - a(a+1)...(a+k) b(b+1)...(b+k) k+1
 * = 1 + > ----------------------------- x .
 * - c(c+1)...(c+k) (k+1)!
 * k = 0
 *
 * Cases addressed are
 *	Tests and escapes for negative integer a, b, or c
 *	Linear transformation if c - a or c - b negative integer
 *	Special case c = a or c = b
 *	Linear transformation for x near +1
 *	Transformation for x < -0.5
 *	Psi function expansion if x > 0.5 and c - a - b integer
 * Conditionally, a recurrence on c to make c-a-b > 0
 *
 * |x| > 1 is rejected.
 *
 * The parameters a, b, c are considered to be integer
 * valued if they are within 1.0e-6 of the nearest integer.
 *
 * ACCURACY:
 *
 * Relative error (-1 < x < 1):
 * arithmetic domain # trials peak rms
 * IEEE 0,3 30000 5.8e-4 4.3e-6
 */

/*							hypergf.c
 *
 *	Confluent hypergeometric function
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, b, x, y, hypergf();
 *
 * y = hypergf( a, b, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes the confluent hypergeometric function
 *
 * 1 2
 * a x a(a+1) x
 * F ( a,b;x ) = 1 + ---- + --------- + ...
 * 1 1 b 1! b(b+1) 2!
 *
 * Many higher transcendental functions are special cases of
 * this power series.
 *
 * As is evident from the formula, b must not be a negative
 * integer or zero unless a is an integer with 0 >= a > b.
 *
 * The routine attempts both a direct summation of the series
 * and an asymptotic expansion. In each case error due to
 * roundoff, cancellation, and nonconvergence is estimated.
 * The result with smaller estimated error is returned.
 *
 *
 *
 * ACCURACY:
 *
 * Tested at random points (a, b, x), all three variables
 * ranging from 0 to 30.
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,5 10000 6.6e-7 1.3e-7
 * IEEE 0,30 30000 1.1e-5 6.5e-7
 *
 * Larger errors can be observed when b is near a negative
 * integer or zero. Certain combinations of arguments yield
 * serious cancellation error in the power series summation
 * and also are not in the region of near convergence of the
 * asymptotic series. An error message is printed if the
 * self-estimated relative error is greater than 1.0e-3.
 *
 */

/*							i0f.c
 *
 *	Modified Bessel function of order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, i0();
 *
 * y = i0f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order zero of the
 * argument.
 *
 * The function is defined as i0(x) = j0( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity). Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,30 100000 4.0e-7 7.9e-8
 *
 */

/*							i0ef.c
 *
 *	Modified Bessel function of order zero,
 *	exponentially scaled
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, i0ef();
 *
 * y = i0ef( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of order zero of the argument.
 *
 * The function is defined as i0e(x) = exp(-|x|) j0( ix ).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,30 100000 3.7e-7 7.0e-8
 * See i0f().
 *
 */

/*							i1f.c
 *
 *	Modified Bessel function of order one
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, i1f();
 *
 * y = i1f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order one of the
 * argument.
 *
 * The function is defined as i1(x) = -i j1( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity). Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 100000 1.5e-6 1.6e-7
 *
 *
 */

/*							i1ef.c
 *
 *	Modified Bessel function of order one,
 *	exponentially scaled
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, i1ef();
 *
 * y = i1ef( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of order one of the argument.
 *
 * The function is defined as i1(x) = -i exp(-|x|) j1( ix ).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 30000 1.5e-6 1.5e-7
 * See i1().
 *
 */

/*							igamf.c
 *
 *	Incomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, x, y, igamf();
 *
 * y = igamf( a, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 * x
 * -
 * 1 | | -t a-1
 * igam(a,x) = ----- | e t dt.
 * - | |
 * | (a) -
 * 0
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,30 20000 7.8e-6 5.9e-7
 *
 */

/*							igamcf()
 *
 *	Complemented incomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, x, y, igamcf();
 *
 * y = igamcf( a, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *
 * igamc(a,x) = 1 - igam(a,x)
 *
 * inf.
 * -
 * 1 | | -t a-1
 * = ----- | e t dt.
 * - | |
 * | (a) -
 * x
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,30 30000 7.8e-6 5.9e-7
 *
 */

/*							igamif()
 *
 * Inverse of complemented imcomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, x, y, igamif();
 *
 * x = igamif( a, y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Given y, the function finds x such that
 *
 * igamc( a, x ) = y.
 *
 * It is valid in the right-hand tail of the distribution, y < 0.5. * Starting with the approximate value * * 3 * x = a t * * where * * t = 1 - d - ndtri(y) sqrt(d) * * and * * d = 1/9a, * * the routine performs up to 10 Newton iterations to find the * root of igamc(a,x) - y = 0. * * * ACCURACY: * * Tested for a ranging from 0 to 100 and x from 0 to 1. * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,100 5000 1.0e-5 1.5e-6 * */

/*							incbetf.c
 *
 *	Incomplete beta integral
 *
 *
 * SYNOPSIS:
 *
 * float a, b, x, y, incbetf();
 *
 * y = incbetf( a, b, x );
 *
 *
 * DESCRIPTION:
 *
 * Returns incomplete beta integral of the arguments, evaluated
 * from zero to x. The function is defined as
 *
 * x
 * - -
 * | (a+b) | | a-1 b-1
 * ----------- | t (1-t) dt.
 * - - | |
 * | (a) | (b) -
 * 0
 *
 * The domain of definition is 0 <= x <= 1. In this
 * implementation a and b are restricted to positive values.
 * The integral from x to 1 may be obtained by the symmetry
 * relation
 *
 * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ).
 *
 * The integral is evaluated by a continued fraction expansion.
 * If a < 1, the function calls itself recursively after a
 * transformation to increase a to a+1.
 *
 * ACCURACY:
 *
 * Tested at random points (a,b,x) with a and b in the indicated
 * interval and x between 0 and 1.
 *
 * arithmetic domain # trials peak rms
 * Relative error:
 * IEEE 0,30 10000 3.7e-5 5.1e-6
 * IEEE 0,100 10000 1.7e-4 2.5e-5
 * The useful domain for relative error is limited by underflow
 * of the single precision exponential function.
 * Absolute error:
 * IEEE 0,30 100000 2.2e-5 9.6e-7
 * IEEE 0,100 10000 6.5e-5 3.7e-6
 *
 * Larger errors may occur for extreme ratios of a and b.
 *
 * ERROR MESSAGES:
 * message condition value returned
 * incbetf domain x<0, x>1 0.0
 */

/*							incbif()
 *
 * Inverse of imcomplete beta integral
 *
 *
 *
 * SYNOPSIS:
 *
 * float a, b, x, y, incbif();
 *
 * x = incbif( a, b, y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Given y, the function finds x such that
 *
 * incbet( a, b, x ) = y.
 *
 * the routine performs up to 10 Newton iterations to find the
 * root of incbet(a,b,x) - y = 0.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * x a,b
 * arithmetic domain domain # trials peak rms
 * IEEE 0,1 0,100 5000 2.8e-4 8.3e-6
 *
 * Overflow and larger errors may occur for one of a or b near zero
 * and the other large.
 */

/*							ivf.c
 *
 *	Modified Bessel function of noninteger order
 *
 *
 *
 * SYNOPSIS:
 *
 * float v, x, y, ivf();
 *
 * y = ivf( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order v of the
 * argument. If x is negative, v must be integer valued.
 *
 * The function is defined as Iv(x) = Jv( ix ). It is
 * here computed in terms of the confluent hypergeometric
 * function, according to the formula
 *
 * v -x
 * Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
 *
 * If v is a negative integer, then v is replaced by -v.
 *
 *
 * ACCURACY:
 *
 * Tested at random points (v, x), with v between 0 and
 * 30, x between 0 and 28.
 * arithmetic domain # trials peak rms
 * Relative error:
 * IEEE 0,15 3000 4.7e-6 5.4e-7
 * Absolute error (relative when function > 1)
 * IEEE 0,30 5000 8.5e-6 1.3e-6
 *
 * Accuracy is diminished if v is near a negative integer.
 * The useful domain for relative error is limited by overflow
 * of the single precision exponential function.
 *
 * See also hyperg.c.
 *
 */

/*							j0f.c
 *
 *	Bessel function of order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, j0f();
 *
 * y = j0f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order zero of the argument.
 *
 * The domain is divided into the intervals [0, 2] and
 * (2, infinity). In the first interval the following polynomial
 * approximation is used:
 *
 *
 * 2 2 2
 * (w - r ) (w - r ) (w - r ) P(w)
 * 1 2 3 
 *
 * 2
 * where w = x and the three r's are zeros of the function.
 *
 * In the second interval, the modulus and phase are approximated
 * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
 * and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is
 *
 * j0(x) = Modulus(x) cos( Phase(x) ).
 *
 *
 *
 * ACCURACY:
 *
 * Absolute error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 2 100000 1.3e-7 3.6e-8
 * IEEE 2, 32 100000 1.9e-7 5.4e-8
 *
 */

/*							y0f.c
 *
 *	Bessel function of the second kind, order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, y0f();
 *
 * y = y0f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of the second kind, of order
 * zero, of the argument.
 *
 * The domain is divided into the intervals [0, 2] and
 * (2, infinity). In the first interval a rational approximation
 * R(x) is employed to compute
 *
 * 2 2 2
 * y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x).
 * 1 2 3 
 *
 * Thus a call to j0() is required. The three zeros are removed
 * from R(x) to improve its numerical stability.
 *
 * In the second interval, the modulus and phase are approximated
 * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
 * and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is
 *
 * y0(x) = Modulus(x) sin( Phase(x) ).
 *
 *
 *
 *
 * ACCURACY:
 *
 * Absolute error, when y0(x) < 1; else relative error:
 *
 * arithmetic domain # trials peak rms
 * IEEE 0, 2 100000 2.4e-7 3.4e-8
 * IEEE 2, 32 100000 1.8e-7 5.3e-8
 *
 */

/*							j1f.c
 *
 *	Bessel function of order one
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, j1f();
 *
 * y = j1f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order one of the argument.
 *
 * The domain is divided into the intervals [0, 2] and
 * (2, infinity). In the first interval a polynomial approximation
 * 2 
 * (w - r ) x P(w)
 * 1 
 * 2 
 * is used, where w = x and r is the first zero of the function.
 *
 * In the second interval, the modulus and phase are approximated
 * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
 * and Phase(x) = x + 1/x R(1/x^2) - 3pi/4. The function is
 *
 * j0(x) = Modulus(x) cos( Phase(x) ).
 *
 *
 *
 * ACCURACY:
 *
 * Absolute error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 2 100000 1.2e-7 2.5e-8
 * IEEE 2, 32 100000 2.0e-7 5.3e-8
 *
 *
 */

/*							y1
 *
 *	Bessel function of second kind of order one
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, y1();
 *
 * y = y1( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of the second kind of order one
 * of the argument.
 *
 * The domain is divided into the intervals [0, 2] and
 * (2, infinity). In the first interval a rational approximation
 * R(x) is employed to compute
 *
 * 2
 * y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) .
 * 1
 *
 * Thus a call to j1() is required.
 *
 * In the second interval, the modulus and phase are approximated
 * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
 * and Phase(x) = x + 1/x S(1/x^2) - 3pi/4. Then the function is
 *
 * y0(x) = Modulus(x) sin( Phase(x) ).
 *
 *
 *
 *
 * ACCURACY:
 *
 * Absolute error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 2 100000 2.2e-7 4.6e-8
 * IEEE 2, 32 100000 1.9e-7 5.3e-8
 *
 * (error criterion relative when |y1| > 1).
 *
 */

/*							jnf.c
 *
 *	Bessel function of integer order
 *
 *
 *
 * SYNOPSIS:
 *
 * int n;
 * float x, y, jnf();
 *
 * y = jnf( n, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order n, where n is a
 * (possibly negative) integer.
 *
 * The ratio of jn(x) to j0(x) is computed by backward
 * recurrence. First the ratio jn/jn-1 is found by a
 * continued fraction expansion. Then the recurrence
 * relating successive orders is applied until j0 or j1 is
 * reached.
 *
 * If n = 0 or 1 the routine for j0 or j1 is called
 * directly.
 *
 *
 *
 * ACCURACY:
 *
 * Absolute error:
 * arithmetic range # trials peak rms
 * IEEE 0, 15 30000 3.6e-7 3.6e-8
 *
 *
 * Not suitable for large n or x. Use jvf() instead.
 *
 */

/*							jvf.c
 *
 *	Bessel function of noninteger order
 *
 *
 *
 * SYNOPSIS:
 *
 * float v, x, y, jvf();
 *
 * y = jvf( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order v of the argument,
 * where v is real. Negative x is allowed if v is an integer.
 *
 * Several expansions are included: the ascending power
 * series, the Hankel expansion, and two transitional
 * expansions for large v. If v is not too large, it
 * is reduced by recurrence to a region of best accuracy.
 *
 * The single precision routine accepts negative v, but with
 * reduced accuracy.
 *
 *
 *
 * ACCURACY:
 * Results for integer v are indicated by *.
 * Error criterion is absolute, except relative when |jv()| > 1.
 *
 * arithmetic domain # trials peak rms
 * v x
 * IEEE 0,125 0,125 30000 2.0e-6 2.0e-7
 * IEEE -17,0 0,125 30000 1.1e-5 4.0e-7
 * IEEE -100,0 0,125 3000 1.5e-4 7.8e-6
 */

/*							k0f.c
 *
 *	Modified Bessel function, third kind, order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, k0f();
 *
 * y = k0f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of the third kind
 * of order zero of the argument.
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity). Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 *
 * ACCURACY:
 *
 * Tested at 2000 random points between 0 and 8. Peak absolute
 * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 30000 7.8e-7 8.5e-8
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * K0 domain x <= 0 MAXNUM
 *
 */

/*							k0ef()
 *
 *	Modified Bessel function, third kind, order zero,
 *	exponentially scaled
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, k0ef();
 *
 * y = k0ef( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of the third kind of order zero of the argument.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 30000 8.1e-7 7.8e-8
 * See k0().
 *
 */

/*							k1f.c
 *
 *	Modified Bessel function, third kind, order one
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, k1f();
 *
 * y = k1f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes the modified Bessel function of the third kind
 * of order one of the argument.
 *
 * The range is partitioned into the two intervals [0,2] and
 * (2, infinity). Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 30000 4.6e-7 7.6e-8
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * k1 domain x <= 0 MAXNUM
 *
 */

/*							k1ef.c
 *
 *	Modified Bessel function, third kind, order one,
 *	exponentially scaled
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, k1ef();
 *
 * y = k1ef( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of the third kind of order one of the argument:
 *
 * k1e(x) = exp(x) * k1(x).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 30000 4.9e-7 6.7e-8
 * See k1().
 *
 */

/*							knf.c
 *
 *	Modified Bessel function, third kind, integer order
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, knf();
 * int n;
 *
 * y = knf( n, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of the third kind
 * of order n of the argument.
 *
 * The range is partitioned into the two intervals [0,9.55] and
 * (9.55, infinity). An ascending power series is used in the
 * low range, and an asymptotic expansion in the high range.
 *
 *
 *
 * ACCURACY:
 *
 * Absolute error, relative when function > 1:
 * arithmetic domain # trials peak rms
 * IEEE 0,30 10000 2.0e-4 3.8e-6
 *
 * Error is high only near the crossover point x = 9.55
 * between the two expansions used.
 */

/*							log10f.c
 *
 *	Common logarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, log10f();
 *
 * y = log10f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns logarithm to the base 10 of x.
 *
 * The argument is separated into its exponent and fractional
 * parts. The logarithm of the fraction is approximated by
 *
 * log(1+x) = x - 0.5 x**2 + x**3 P(x).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0.5, 2.0 100000 1.3e-7 3.4e-8
 * IEEE 0, MAXNUMF 100000 1.3e-7 2.6e-8
 *
 * In the tests over the interval [0, MAXNUM], the logarithms
 * of the random arguments were uniformly distributed over
 * [-MAXL10, MAXL10].
 *
 * ERROR MESSAGES:
 *
 * log10f singularity: x = 0; returns -MAXL10
 * log10f domain: x < 0; returns -MAXL10
 * MAXL10 = 38.230809449325611792
 */

/*							log2f.c
 *
 *	Base 2 logarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, log2f();
 *
 * y = log2f( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the base 2 logarithm of x.
 *
 * The argument is separated into its exponent and fractional
 * parts. If the exponent is between -1 and +1, the base e
 * logarithm of the fraction is approximated by
 *
 * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
 *
 * Otherwise, setting z = 2(x-1)/x+1),
 * 
 * log(x) = z + z**3 P(z)/Q(z).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE exp(+-88) 100000 1.1e-7 2.4e-8
 * IEEE 0.5, 2.0 100000 1.1e-7 3.0e-8
 *
 * In the tests over the interval [exp(+-88)], the logarithms
 * of the random arguments were uniformly distributed.
 *
 * ERROR MESSAGES:
 *
 * log singularity: x = 0; returns MINLOGF/log(2)
 * log domain: x < 0; returns MINLOGF/log(2)
 */

/*							logf.c
 *
 *	Natural logarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, logf();
 *
 * y = logf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the base e (2.718...) logarithm of x.
 *
 * The argument is separated into its exponent and fractional
 * parts. If the exponent is between -1 and +1, the logarithm
 * of the fraction is approximated by
 *
 * log(1+x) = x - 0.5 x**2 + x**3 P(x)
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0.5, 2.0 100000 7.6e-8 2.7e-8
 * IEEE 1, MAXNUMF 100000 2.6e-8
 *
 * In the tests over the interval [1, MAXNUM], the logarithms
 * of the random arguments were uniformly distributed over
 * [0, MAXLOGF].
 *
 * ERROR MESSAGES:
 *
 * logf singularity: x = 0; returns MINLOG
 * logf domain: x < 0; returns MINLOG
 */

/*							mtherr.c
 *
 *	Library common error handling routine
 *
 *
 *
 * SYNOPSIS:
 *
 * char *fctnam;
 * int code;
 * void mtherr();
 *
 * mtherr( fctnam, code );
 *
 *
 *
 * DESCRIPTION:
 *
 * This routine may be called to report one of the following
 * error conditions (in the include file mconf.h).
 * 
 * Mnemonic Value Significance
 *
 * DOMAIN 1 argument domain error
 * SING 2 function singularity
 * OVERFLOW 3 overflow range error
 * UNDERFLOW 4 underflow range error
 * TLOSS 5 total loss of precision
 * PLOSS 6 partial loss of precision
 * EDOM 33 Unix domain error code
 * ERANGE 34 Unix range error code
 *
 * The default version of the file prints the function name,
 * passed to it by the pointer fctnam, followed by the
 * error condition. The display is directed to the standard
 * output device. The routine then returns to the calling
 * program. Users may wish to modify the program to abort by
 * calling exit() under severe error conditions such as domain
 * errors.
 *
 * Since all error conditions pass control to this function,
 * the display may be easily changed, eliminated, or directed
 * to an error logging device.
 *
 * SEE ALSO:
 *
 * mconf.h
 *
 */

/*							nbdtrf.c
 *
 *	Negative binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * float p, y, nbdtrf();
 *
 * y = nbdtrf( k, n, p );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms 0 through k of the negative
 * binomial distribution:
 *
 * k
 * -- ( n+j-1 ) n j
 * > ( ) p (1-p)
 * -- ( j )
 * j=0
 *
 * In a sequence of Bernoulli trials, this is the probability
 * that k or fewer failures precede the nth success.
 *
 * The terms are not computed individually; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = nbdtr( k, n, p ) = incbet( n, k+1, p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 1.5e-4 1.9e-5
 *
 */

/*							nbdtrcf.c
 *
 *	Complemented negative binomial distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k, n;
 * float p, y, nbdtrcf();
 *
 * y = nbdtrcf( k, n, p );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms k+1 to infinity of the negative
 * binomial distribution:
 *
 * inf
 * -- ( n+j-1 ) n j
 * > ( ) p (1-p)
 * -- ( j )
 * j=k+1
 *
 * The terms are not computed individually; instead the incomplete
 * beta integral is employed, according to the formula
 *
 * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
 *
 * The arguments must be positive, with p ranging from 0 to 1.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 1.4e-4 2.0e-5
 *
 */

/*							ndtrf.c
 *
 *	Normal distribution function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, ndtrf();
 *
 * y = ndtrf( 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.
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -13,0 50000 1.5e-5 2.6e-6
 *
 *
 * ERROR MESSAGES:
 *
 * See erfcf().
 *
 */

/*							erff.c
 *
 *	Error function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, erff();
 *
 * y = erff( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * The integral is
 *
 * x 
 * -
 * 2 | | 2
 * erf(x) = -------- | exp( - t ) dt.
 * sqrt(pi) | |
 * -
 * 0
 *
 * The magnitude of x is limited to 9.231948545 for DEC
 * arithmetic; 1 or -1 is returned outside this range.
 *
 * For 0 <= |x| < 1, erf(x) = x * P(x**2); otherwise
 * erf(x) = 1 - erfc(x).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -9.3,9.3 50000 1.7e-7 2.8e-8
 *
 */

/*							erfcf.c
 *
 *	Complementary error function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, erfcf();
 *
 * y = erfcf( 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 polynomial
 * approximations 1/x P(1/x**2) are computed.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -9.3,9.3 50000 3.9e-6 7.2e-7
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * erfcf underflow x**2 > MAXLOGF 0.0
 *
 *
 */

/*							ndtrif.c
 *
 *	Inverse of Normal distribution function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, ndtrif();
 *
 * x = ndtrif( 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
 * IEEE 1e-38, 1 30000 3.6e-7 5.0e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * ndtrif domain x <= 0 -MAXNUM
 * ndtrif domain x >= 1 MAXNUM
 *
 */

/*							pdtrf.c
 *
 *	Poisson distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k;
 * float m, y, pdtrf();
 *
 * y = pdtrf( k, m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the sum of the first k terms of the Poisson
 * distribution:
 *
 * k j
 * -- -m m
 * > e --
 * -- j!
 * j=0
 *
 * The terms are not summed directly; instead the incomplete
 * gamma integral is employed, according to the relation
 *
 * y = pdtr( k, m ) = igamc( k+1, m ).
 *
 * The arguments must both be positive.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 6.9e-5 8.0e-6
 *
 */

/*							pdtrcf()
 *
 *	Complemented poisson distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k;
 * float m, y, pdtrcf();
 *
 * y = pdtrcf( k, m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the sum of the terms k+1 to infinity of the Poisson
 * distribution:
 *
 * inf. j
 * -- -m m
 * > e --
 * -- j!
 * j=k+1
 *
 * The terms are not summed directly; instead the incomplete
 * gamma integral is employed, according to the formula
 *
 * y = pdtrc( k, m ) = igam( k+1, m ).
 *
 * The arguments must both be positive.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 8.4e-5 1.2e-5
 *
 */

/*							pdtrif()
 *
 *	Inverse Poisson distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * int k;
 * float m, y, pdtrf();
 *
 * m = pdtrif( k, y );
 *
 *
 *
 *
 * DESCRIPTION:
 *
 * Finds the Poisson variable x such that the integral
 * from 0 to x of the Poisson density is equal to the
 * given probability y.
 *
 * This is accomplished using the inverse gamma integral
 * function and the relation
 *
 * m = igami( k+1, y ).
 *
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,100 5000 8.7e-6 1.4e-6
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * pdtri domain y < 0 or y >= 1 0.0
 * k < 0
 *
 */

/*							polevlf.c
 *							p1evlf.c
 *
 *	Evaluate polynomial
 *
 *
 *
 * SYNOPSIS:
 *
 * int N;
 * float x, y, coef[N+1], polevlf[];
 *
 * y = polevlf( 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.
 *
 */

/*							polynf.c
 *							polyrf.c
 * Arithmetic operations on polynomials
 *
 * In the following descriptions a, b, c are polynomials of degree
 * na, nb, nc respectively. The degree of a polynomial cannot
 * exceed a run-time value MAXPOLF. An operation that attempts
 * to use or generate a polynomial of higher degree may produce a
 * result that suffers truncation at degree MAXPOL. The value of
 * MAXPOL is set by calling the function
 *
 * polinif( maxpol );
 *
 * where maxpol is the desired maximum degree. This must be
 * done prior to calling any of the other functions in this module.
 * Memory for internal temporary polynomial storage is allocated
 * by polinif().
 *
 * Each polynomial is represented by an array containing its
 * coefficients, together with a separately declared integer equal
 * to the degree of the polynomial. The coefficients appear in
 * ascending order; that is,
 *
 * 2 na
 * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x .
 *
 *
 *
 * sum = poleva( a, na, x );	Evaluate polynomial a(t) at t = x.
 * polprtf( a, na, D );		Print the coefficients of a to D digits.
 * polclrf( a, na );		Set a identically equal to zero, up to a[na].
 * polmovf( a, na, b );		Set b = a.
 * poladdf( a, na, b, nb, c );	c = b + a, nc = max(na,nb)
 * polsubf( a, na, b, nb, c );	c = b - a, nc = max(na,nb)
 * polmulf( a, na, b, nb, c );	c = b * a, nc = na+nb
 *
 *
 * Division:
 *
 * i = poldivf( a, na, b, nb, c );	c = b / a, nc = MAXPOL
 *
 * returns i = the degree of the first nonzero coefficient of a.
 * The computed quotient c must be divided by x^i. An error message
 * is printed if a is identically zero.
 *
 *
 * Change of variables:
 * If a and b are polynomials, and t = a(x), then
 * c(t) = b(a(x))
 * is a polynomial found by substituting a(x) for t. The
 * subroutine call for this is
 *
 * polsbtf( a, na, b, nb, c );
 *
 *
 * Notes:
 * poldivf() is an integer routine; polevaf() is float.
 * Any of the arguments a, b, c may refer to the same array.
 *
 */

/*							powf.c
 *
 *	Power function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, z, powf();
 *
 * z = powf( x, y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes x raised to the yth power. Analytically,
 *
 * x**y = exp( y log(x) ).
 *
 * Following Cody and Waite, this program uses a lookup table
 * of 2**-i/16 and pseudo extended precision arithmetic to
 * obtain an extra three bits of accuracy in both the logarithm
 * and the exponential.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -10,10 100,000 1.4e-7 3.6e-8
 * 1/10 < x < 10, x uniformly distributed.
 * -10 < y < 10, y uniformly distributed.
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * powf overflow x**y > MAXNUMF MAXNUMF
 * powf underflow x**y < 1/MAXNUMF 0.0
 * powf domain x<0 and y noninteger 0.0
 *
 */

/*							powif.c
 *
 *	Real raised to integer power
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, powif();
 * int n;
 *
 * y = powif( x, n );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns argument x raised to the nth power.
 * The routine efficiently decomposes n as a sum of powers of
 * two. The desired power is a product of two-to-the-kth
 * powers of x. Thus to compute the 32767 power of x requires
 * 28 multiplications instead of 32767 multiplications.
 *
 *
 *
 * ACCURACY:
 *
 *
 * Relative error:
 * arithmetic x domain n domain # trials peak rms
 * IEEE .04,26 -26,26 100000 1.1e-6 2.0e-7
 * IEEE 1,2 -128,128 100000 1.1e-5 1.0e-6
 *
 * Returns MAXNUMF on overflow, zero on underflow.
 *
 */

/*							psif.c
 *
 *	Psi (digamma) function
 *
 *
 * SYNOPSIS:
 *
 * float x, y, psif();
 *
 * y = psif( x );
 *
 *
 * DESCRIPTION:
 *
 * d -
 * psi(x) = -- ln | (x)
 * dx
 *
 * is the logarithmic derivative of the gamma function.
 * For integer x,
 * n-1
 * -
 * psi(n) = -EUL + > 1/k.
 * -
 * k=1
 *
 * This formula is used for 0 < n <= 10. If x is negative, it
 * is transformed to a positive argument by the reflection
 * formula psi(1-x) = psi(x) + pi cot(pi x).
 * For general positive x, the argument is made greater than 10
 * using the recurrence psi(x+1) = psi(x) + 1/x.
 * Then the following asymptotic expansion is applied:
 *
 * inf. B
 * - 2k
 * psi(x) = log(x) - 1/2x - > -------
 * - 2k
 * k=1 2k x
 *
 * where the B2k are Bernoulli numbers.
 *
 * ACCURACY:
 * Absolute error, relative when |psi| > 1 :
 * arithmetic domain # trials peak rms
 * IEEE -33,0 30000 8.2e-7 1.2e-7
 * IEEE 0,33 100000 7.3e-7 7.7e-8
 *
 * ERROR MESSAGES:
 * message condition value returned
 * psi singularity x integer <=0 MAXNUMF
 */

/*						rgammaf.c
 *
 *	Reciprocal gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, rgammaf();
 *
 * y = rgammaf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns one divided by the gamma function of the argument.
 *
 * The function is approximated by a Chebyshev expansion in
 * the interval [0,1]. Range reduction is by recurrence
 * for arguments between -34.034 and +34.84425627277176174.
 * 1/MAXNUMF is returned for positive arguments outside this
 * range.
 *
 * The reciprocal gamma function has no singularities,
 * but overflow and underflow may occur for large arguments.
 * These conditions return either MAXNUMF or 1/MAXNUMF with
 * appropriate sign.
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -34,+34 100000 8.9e-7 1.1e-7
 */

/*							shichif.c
 *
 *	Hyperbolic sine and cosine integrals
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, Chi, Shi;
 *
 * shichi( x, &Chi, &Shi );
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integrals
 *
 * x
 * -
 * | | cosh t - 1
 * Chi(x) = eul + ln x + | ----------- dt,
 * | | t
 * -
 * 0
 *
 * x
 * -
 * | | sinh t
 * Shi(x) = | ------ dt
 * | | t
 * -
 * 0
 *
 * where eul = 0.57721566490153286061 is Euler's constant.
 * The integrals are evaluated by power series for x < 8
 * and by Chebyshev expansions for x between 8 and 88.
 * For large x, both functions approach exp(x)/2x.
 * Arguments greater than 88 in magnitude return MAXNUM.
 *
 *
 * ACCURACY:
 *
 * Test interval 0 to 88.
 * Relative error:
 * arithmetic function # trials peak rms
 * IEEE Shi 20000 3.5e-7 7.0e-8
 * Absolute error, except relative when |Chi| > 1:
 * IEEE Chi 20000 3.8e-7 7.6e-8
 */

/*							sicif.c
 *
 *	Sine and cosine integrals
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, Ci, Si;
 *
 * sicif( x, &Si, &Ci );
 *
 *
 * DESCRIPTION:
 *
 * Evaluates the integrals
 *
 * x
 * -
 * | cos t - 1
 * Ci(x) = eul + ln x + | --------- dt,
 * | t
 * -
 * 0
 * x
 * -
 * | sin t
 * Si(x) = | ----- dt
 * | t
 * -
 * 0
 *
 * where eul = 0.57721566490153286061 is Euler's constant.
 * The integrals are approximated by rational functions.
 * For x > 8 auxiliary functions f(x) and g(x) are employed
 * such that
 *
 * Ci(x) = f(x) sin(x) - g(x) cos(x)
 * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
 *
 *
 * ACCURACY:
 * Test interval = [0,50].
 * Absolute error, except relative when > 1:
 * arithmetic function # trials peak rms
 * IEEE Si 30000 2.1e-7 4.3e-8
 * IEEE Ci 30000 3.9e-7 2.2e-8
 */

/*							sindgf.c
 *
 *	Circular sine of angle in degrees
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, sindgf();
 *
 * y = sindgf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Range reduction is into intervals of 45 degrees.
 *
 * Two polynomial approximating functions are employed.
 * Between 0 and pi/4 the sine is approximated by
 * x + x**3 P(x**2).
 * Between pi/4 and pi/2 the cosine is represented as
 * 1 - x**2 Q(x**2).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-3600 100,000 1.2e-7 3.0e-8
 * 
 * ERROR MESSAGES:
 *
 * message condition value returned
 * sin total loss x > 2^24 0.0
 *
 */

/*							cosdgf.c
 *
 *	Circular cosine of angle in degrees
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, cosdgf();
 *
 * y = cosdgf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Range reduction is into intervals of 45 degrees.
 *
 * Two polynomial approximating functions are employed.
 * Between 0 and pi/4 the cosine is approximated by
 * 1 - x**2 Q(x**2).
 * Between pi/4 and pi/2 the sine is represented as
 * x + x**3 P(x**2).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
 */

/*							sinf.c
 *
 *	Circular sine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, sinf();
 *
 * y = sinf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Range reduction is into intervals of pi/4. The reduction
 * error is nearly eliminated by contriving an extended precision
 * modular arithmetic.
 *
 * Two polynomial approximating functions are employed.
 * Between 0 and pi/4 the sine is approximated by
 * x + x**3 P(x**2).
 * Between pi/4 and pi/2 the cosine is represented as
 * 1 - x**2 Q(x**2).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -4096,+4096 100,000 1.2e-7 3.0e-8
 * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
 * 
 * ERROR MESSAGES:
 *
 * message condition value returned
 * sin total loss x > 2^24 0.0
 *
 * Partial loss of accuracy begins to occur at x = 2^13
 * = 8192. Results may be meaningless for x >= 2^24
 * The routine as implemented flags a TLOSS error
 * for x >= 2^24 and returns 0.0.
 */

/*							cosf.c
 *
 *	Circular cosine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, cosf();
 *
 * y = cosf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Range reduction is into intervals of pi/4. The reduction
 * error is nearly eliminated by contriving an extended precision
 * modular arithmetic.
 *
 * Two polynomial approximating functions are employed.
 * Between 0 and pi/4 the cosine is approximated by
 * 1 - x**2 Q(x**2).
 * Between pi/4 and pi/2 the sine is represented as
 * x + x**3 P(x**2).
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
 */

/*							sinhf.c
 *
 *	Hyperbolic sine
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, sinhf();
 *
 * y = sinhf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns hyperbolic sine of argument in the range MINLOGF to
 * MAXLOGF.
 *
 * The range is partitioned into two segments. If |x| <= 1, a
 * polynomial approximation is used.
 * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-MAXLOG 100000 1.1e-7 2.9e-8
 *
 */

/*							spencef.c
 *
 *	Dilogarithm
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, spencef();
 *
 * y = spencef( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes the integral
 *
 * x
 * -
 * | | log t
 * spence(x) = - | ----- dt
 * | | t - 1
 * -
 * 1
 *
 * for x >= 0. A rational approximation gives the integral in
 * the interval (0.5, 1.5). Transformation formulas for 1/x
 * and 1-x are employed outside the basic expansion range.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,4 30000 4.4e-7 6.3e-8
 *
 *
 */

/*							sqrtf.c
 *
 *	Square root
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, sqrtf();
 *
 * y = sqrtf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the square root of x.
 *
 * Range reduction involves isolating the power of two of the
 * argument and using a polynomial approximation to obtain
 * a rough value for the square root. Then Heron's iteration
 * is used three times to converge to an accurate value.
 *
 *
 *
 * ACCURACY:
 *
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,1.e38 100000 8.7e-8 2.9e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * sqrtf domain x < 0 0.0
 *
 */

/*							stdtrf.c
 *
 *	Student's t distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * float t, stdtrf();
 * short k;
 *
 * y = stdtrf( k, t );
 *
 *
 * DESCRIPTION:
 *
 * Computes the integral from minus infinity to t of the Student
 * t distribution with integer k > 0 degrees of freedom:
 *
 * t
 * -
 * | |
 * - | 2 -(k+1)/2
 * | ( (k+1)/2 ) | ( x )
 * ---------------------- | ( 1 + --- ) dx
 * - | ( k )
 * sqrt( k pi ) | ( k/2 ) |
 * | |
 * -
 * -inf.
 * 
 * Relation to incomplete beta integral:
 *
 * 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
 * where
 * z = k/(k + t**2).
 *
 * For t < -1, this is the method of computation. For higher t,
 * a direct method is derived from integration by parts.
 * Since the function is symmetric about t=0, the area under the
 * right tail of the density is found by calling the function
 * with -t instead of t.
 * 
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +/- 100 5000 2.3e-5 2.9e-6
 */

/*							struvef.c
 *
 * Struve function
 *
 *
 *
 * SYNOPSIS:
 *
 * float v, x, y, struvef();
 *
 * y = struvef( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes the Struve function Hv(x) of order v, argument x.
 * Negative x is rejected unless v is an integer.
 *
 * This module also contains the hypergeometric functions 1F2
 * and 3F0 and a routine for the Bessel function Yv(x) with
 * noninteger v.
 *
 *
 *
 * ACCURACY:
 *
 * v varies from 0 to 10.
 * Absolute error (relative error when |Hv(x)| > 1):
 * arithmetic domain # trials peak rms
 * IEEE -10,10 100000 9.0e-5 4.0e-6
 *
 */

/*							tandgf.c
 *
 *	Circular tangent of angle in degrees
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, tandgf();
 *
 * y = tandgf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the circular tangent of the radian argument x.
 *
 * Range reduction is into intervals of 45 degrees.
 *
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-2^24 50000 2.4e-7 4.8e-8
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * tanf total loss x > 2^24 0.0
 *
 */

/*							cotdgf.c
 *
 *	Circular cotangent of angle in degrees
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, cotdgf();
 *
 * y = cotdgf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Range reduction is into intervals of 45 degrees.
 * A common routine computes either the tangent or cotangent.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-2^24 50000 2.4e-7 4.8e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * cot total loss x > 2^24 0.0
 * cot singularity x = 0 MAXNUMF
 *
 */

/*							tanf.c
 *
 *	Circular tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, tanf();
 *
 * y = tanf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the circular tangent of the radian argument x.
 *
 * Range reduction is modulo pi/4. A polynomial approximation
 * is employed in the basic interval [0, pi/4].
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-4096 100000 3.3e-7 4.5e-8
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * tanf total loss x > 2^24 0.0
 *
 */

/*							cotf.c
 *
 *	Circular cotangent
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, cotf();
 *
 * y = cotf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the circular cotangent of the radian argument x.
 * A common routine computes either the tangent or cotangent.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE +-4096 100000 3.0e-7 4.5e-8
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * cot total loss x > 2^24 0.0
 * cot singularity x = 0 MAXNUMF
 *
 */

/*							tanhf.c
 *
 *	Hyperbolic tangent
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, tanhf();
 *
 * y = tanhf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns hyperbolic tangent of argument in the range MINLOG to
 * MAXLOG.
 *
 * A polynomial approximation is used for |x| < 0.625.
 * Otherwise,
 *
 * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE -2,2 100000 1.3e-7 2.6e-8
 *
 */

/*							ynf.c
 *
 *	Bessel function of second kind of integer order
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, ynf();
 * int n;
 *
 * y = ynf( n, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Bessel function of order n, where n is a
 * (possibly negative) integer.
 *
 * The function is evaluated by forward recurrence on
 * n, starting with values computed by the routines
 * y0() and y1().
 *
 * If n = 0 or 1 the routine for y0 or y1 is called
 * directly.
 *
 *
 *
 * ACCURACY:
 *
 *
 * Absolute error, except relative when y > 1:
 * 
 * arithmetic domain # trials peak rms
 * IEEE 0, 30 10000 2.3e-6 3.4e-7
 *
 *
 * ERROR MESSAGES:
 *
 * message condition value returned
 * yn singularity x = 0 MAXNUMF
 * yn overflow MAXNUMF
 *
 * Spot checked against tables for x, n between 0 and 100.
 *
 */

/*							zetacf.c
 *
 *	Riemann zeta function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, zetacf();
 *
 * y = zetacf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 *
 * inf.
 * - -x
 * zetac(x) = > k , x > 1,
 * -
 * k=2
 *
 * is related to the Riemann zeta function by
 *
 *	Riemann zeta(x) = zetac(x) + 1.
 *
 * Extension of the function definition for x < 1 is implemented.
 * Zero is returned for x > log2(MAXNUM).
 *
 * An overflow error may occur for large negative x, due to the
 * gamma function in the reflection formula.
 *
 * ACCURACY:
 *
 * Tabulated values have full machine accuracy.
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 1,50 30000 5.5e-7 7.5e-8
 *
 *
 */

/*							zetaf.c
 *
 *	Riemann zeta function of two arguments
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, q, y, zetaf();
 *
 * y = zetaf( x, q );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 *
 * inf.
 * - -x
 * zeta(x,q) = > (k+q) 
 * -
 * k=0
 *
 * where x > 1 and q is not a negative integer or zero.
 * The Euler-Maclaurin summation formula is used to obtain
 * the expansion
 *
 * n 
 * - -x
 * zeta(x,q) = > (k+q) 
 * - 
 * k=1 
 *
 * 1-x inf. B x(x+1)...(x+2j)
 * (n+q) 1 - 2j
 * + --------- - ------- + > --------------------
 * x-1 x - x+2j+1
 * 2(n+q) j=1 (2j)! (n+q)
 *
 * where the B2j are Bernoulli numbers. Note that (see zetac.c)
 * zeta(x,1) = zetac(x) + 1.
 *
 *
 *
 * ACCURACY:
 *
 * Relative error:
 * arithmetic domain # trials peak rms
 * IEEE 0,25 10000 6.9e-7 1.0e-7
 *
 * Large arguments may produce underflow in powf(), in which
 * case the results are inaccurate.
 *
 * REFERENCE:
 *
 * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
 * Series, and Products, p. 1073; Academic Press, 1980.
 *
 */

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Last update: 5 October 2014