uclibc-ng/libm/ldouble

/* acoshl.c
*
* Inverse hyperbolic cosine, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, acoshl();
*
* y = acoshl( x );
*
*
*
* DESCRIPTION:
*
* Returns inverse hyperbolic cosine of argument.
*
* If 1 <= x < 1.5, a rational approximation
*
* sqrt(2z) * P(z)/Q(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 30000 2.0e-19 3.9e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* acoshl domain |x| < 1 0.0
*
*/

/* asinhl.c
*
* Inverse hyperbolic sine, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, asinhl();
*
* y = asinhl( 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 30000 1.7e-19 3.5e-20
*
*/

/* asinl.c
*
* Inverse circular sine, long double precision
*
*
*
* SYNOPSIS:
*
* double x, y, asinl();
*
* y = asinl( x );
*
*
*
* DESCRIPTION:
*
* Returns radian angle between -pi/2 and +pi/2 whose sine is x.
*
* A rational function of the form x + x**3 P(x**2)/Q(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 30000 2.7e-19 4.8e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* asin domain |x| > 1 0.0
*
*/
/* acosl()
*
* Inverse circular cosine, long double precision
*
*
*
* SYNOPSIS:
*
* double x, y, acosl();
*
* y = acosl( 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 30000 1.4e-19 3.5e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* asin domain |x| > 1 0.0
*/

/* atanhl.c
*
* Inverse hyperbolic tangent, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, atanhl();
*
* y = atanhl( x );
*
*
*
* DESCRIPTION:
*
* Returns inverse hyperbolic tangent of argument in the range
* MINLOGL to MAXLOGL.
*
* If |x| < 0.5, the rational form x + x**3 P(x)/Q(x) is
* employed. Otherwise,
* atanh(x) = 0.5 * log( (1+x)/(1-x) ).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -1,1 30000 1.1e-19 3.3e-20
*
*/

/* atanl.c
*
* Inverse circular tangent, long double precision
* (arctangent)
*
*
*
* SYNOPSIS:
*
* long double x, y, atanl();
*
* y = atanl( 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 ). The approximant uses a rational
* function of degree 3/4 of the form x + x**3 P(x)/Q(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10, 10 150000 1.3e-19 3.0e-20
*
*/
/* atan2l()
*
* Quadrant correct inverse circular tangent,
* long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, z, atan2l();
*
* z = atan2l( 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 60000 1.7e-19 3.2e-20
* See atan.c.
*
*/

/* bdtrl.c
*
* Binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, bdtrl();
*
* y = bdtrl( 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:
*
* Tested at random points (k,n,p) with a and b between 0
* and 10000 and p between 0 and 1.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,10000 3000 1.6e-14 2.2e-15
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtrl domain k < 0 0.0
* n < k
* x < 0, x > 1
*
*/
/* bdtrcl()
*
* Complemented binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, bdtrcl();
*
* y = bdtrcl( 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:
*
* See incbet.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtrcl domain x<0, x>1, n */
/* bdtril()
*
* Inverse binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, bdtril();
*
* p = bdtril( 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:
*
* See incbi.c.
* Tested at random k, n between 1 and 10000. The "domain" refers to p:
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,1 3500 2.0e-15 8.2e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtril domain k < 0, n <= k 0.0
* x < 0, x > 1
*/


/* btdtrl.c
*
* Beta distribution
*
*
*
* SYNOPSIS:
*
* long double a, b, x, y, btdtrl();
*
* y = btdtrl( a, b, x );
*
*
*
* DESCRIPTION:
*
* Returns the area from zero to x under the beta density
* function:
*
*
* x
* - -
* | (a+b) | | a-1 b-1
* P(x) = ---------- | t (1-t) dt
* - - | |
* | (a) | (b) -
* 0
*
*
* The mean value of this distribution is a/(a+b). The variance
* is ab/[(a+b)^2 (a+b+1)].
*
* This function is identical to the incomplete beta integral
* function, incbetl(a, b, x).
*
* The complemented function is
*
* 1 - P(1-x) = incbetl( b, a, x );
*
*
* ACCURACY:
*
* See incbetl.c.
*
*/

/* cbrtl.c
*
* Cube root, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, cbrtl();
*
* y = cbrtl( 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 three times to converge to an accurate
* result.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE .125,8 80000 7.0e-20 2.2e-20
* IEEE exp(+-707) 100000 7.0e-20 2.4e-20
*
*/

/* chdtrl.c
*
* Chi-square distribution
*
*
*
* SYNOPSIS:
*
* long double df, x, y, chdtrl();
*
* y = chdtrl( 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:
*
* See igam().
*
* ERROR MESSAGES:
*
* message condition value returned
* chdtr domain x < 0 or v < 1 0.0
*/
/* chdtrcl()
*
* Complemented Chi-square distribution
*
*
*
* SYNOPSIS:
*
* long double v, x, y, chdtrcl();
*
* y = chdtrcl( 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:
*
* See igamc().
*
* ERROR MESSAGES:
*
* message condition value returned
* chdtrc domain x < 0 or v < 1 0.0
*/
/* chdtril()
*
* Inverse of complemented Chi-square distribution
*
*
*
* SYNOPSIS:
*
* long double df, x, y, chdtril();
*
* x = chdtril( 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:
*
* See igami.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* chdtri domain y < 0 or y > 1 0.0
* v < 1
*
*/

/* clogl.c
*
* Complex natural logarithm
*
*
*
* SYNOPSIS:
*
* void clogl();
* cmplxl z, w;
*
* clogl( &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
* DEC -10,+10 7000 8.5e-17 1.9e-17
* IEEE -10,+10 30000 5.0e-15 1.1e-16
*
* Larger relative error can be observed for z near 1 +i0.
* In IEEE arithmetic the peak absolute error is 5.2e-16, rms
* absolute error 1.0e-16.
*/

/* cexpl()
*
* Complex exponential function
*
*
*
* SYNOPSIS:
*
* void cexpl();
* cmplxl z, w;
*
* cexpl( &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
* DEC -10,+10 8700 3.7e-17 1.1e-17
* IEEE -10,+10 30000 3.0e-16 8.7e-17
*
*/
/* csinl()
*
* Complex circular sine
*
*
*
* SYNOPSIS:
*
* void csinl();
* cmplxl z, w;
*
* csinl( &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
* DEC -10,+10 8400 5.3e-17 1.3e-17
* IEEE -10,+10 30000 3.8e-16 1.0e-16
* Also tested by csin(casin(z)) = z.
*
*/
/* ccosl()
*
* Complex circular cosine
*
*
*
* SYNOPSIS:
*
* void ccosl();
* cmplxl z, w;
*
* ccosl( &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
* DEC -10,+10 8400 4.5e-17 1.3e-17
* IEEE -10,+10 30000 3.8e-16 1.0e-16
*/
/* ctanl()
*
* Complex circular tangent
*
*
*
* SYNOPSIS:
*
* void ctanl();
* cmplxl z, w;
*
* ctanl( &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
* DEC -10,+10 5200 7.1e-17 1.6e-17
* IEEE -10,+10 30000 7.2e-16 1.2e-16
* Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
*/
/* ccotl()
*
* Complex circular cotangent
*
*
*
* SYNOPSIS:
*
* void ccotl();
* cmplxl z, w;
*
* ccotl( &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
* DEC -10,+10 3000 6.5e-17 1.6e-17
* IEEE -10,+10 30000 9.2e-16 1.2e-16
* Also tested by ctan * ccot = 1 + i0.
*/

/* casinl()
*
* Complex circular arc sine
*
*
*
* SYNOPSIS:
*
* void casinl();
* cmplxl z, w;
*
* casinl( &z, &w );
*
*
*
* DESCRIPTION:
*
* Inverse complex sine:
*
* 2
* w = -i clog( iz + csqrt( 1 - z ) ).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 10100 2.1e-15 3.4e-16
* IEEE -10,+10 30000 2.2e-14 2.7e-15
* Larger relative error can be observed for z near zero.
* Also tested by csin(casin(z)) = z.
*/
/* cacosl()
*
* Complex circular arc cosine
*
*
*
* SYNOPSIS:
*
* void cacosl();
* cmplxl z, w;
*
* cacosl( &z, &w );
*
*
*
* DESCRIPTION:
*
*
* w = arccos z = PI/2 - arcsin z.
*
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5200 1.6e-15 2.8e-16
* IEEE -10,+10 30000 1.8e-14 2.2e-15
*/

/* catanl()
*
* Complex circular arc tangent
*
*
*
* SYNOPSIS:
*
* void catanl();
* cmplxl z, w;
*
* catanl( &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
* DEC -10,+10 5900 1.3e-16 7.8e-18
* IEEE -10,+10 30000 2.3e-15 8.5e-17
* The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
* had peak relative error 1.5e-16, rms relative error
* 2.9e-17. See also clog().
*/

/* cmplxl.c
*
* Complex number arithmetic
*
*
*
* SYNOPSIS:
*
* typedef struct {
* long double r; real part
* long double i; imaginary part
* }cmplxl;
*
* cmplxl *a, *b, *c;
*
* caddl( a, b, c ); c = b + a
* csubl( a, b, c ); c = b - a
* cmull( a, b, c ); c = b * a
* cdivl( a, b, c ); c = b / a
* cnegl( c ); c = -c
* cmovl( 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
* DEC cadd 10000 1.4e-17 3.4e-18
* IEEE cadd 100000 1.1e-16 2.7e-17
* DEC csub 10000 1.4e-17 4.5e-18
* IEEE csub 100000 1.1e-16 3.4e-17
* DEC cmul 3000 2.3e-17 8.7e-18
* IEEE cmul 100000 2.1e-16 6.9e-17
* DEC cdiv 18000 4.9e-17 1.3e-17
* IEEE cdiv 100000 3.7e-16 1.1e-16
*/

/* cabsl()
*
* Complex absolute value
*
*
*
* SYNOPSIS:
*
* long double cabsl();
* cmplxl z;
* long double a;
*
* a = cabs( &z );
*
*
*
* DESCRIPTION:
*
*
* If z = x + iy
*
* then
*
* a = sqrt( x**2 + y**2 ).
*
* Overflow and underflow are avoided by testing the magnitudes
* of x and y before squaring. If either is outside half of
* the floating point full scale range, both are rescaled.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -30,+30 30000 3.2e-17 9.2e-18
* IEEE -10,+10 100000 2.7e-16 6.9e-17
*/
/* csqrtl()
*
* Complex square root
*
*
*
* SYNOPSIS:
*
* void csqrtl();
* cmplxl z, w;
*
* csqrtl( &z, &w );
*
*
*
* DESCRIPTION:
*
*
* If z = x + iy, r = |z|, then
*
* 1/2
* Im w = [ (r - x)/2 ] ,
*
* Re w = y / 2 Im w.
*
*
* Note that -w is also a square root of z. The root chosen
* is always in the upper half plane.
*
* Because of the potential for cancellation error in r - x,
* the result is sharpened by doing a Heron iteration
* (see sqrt.c) in complex arithmetic.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 25000 3.2e-17 9.6e-18
* IEEE -10,+10 100000 3.2e-16 7.7e-17
*
* 2
* Also tested by csqrt( z ) = z, and tested by arguments
* close to the real axis.
*/

/* coshl.c
*
* Hyperbolic cosine, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, coshl();
*
* y = coshl( x );
*
*
*
* DESCRIPTION:
*
* Returns hyperbolic cosine of argument in the range MINLOGL to
* MAXLOGL.
*
* cosh(x) = ( exp(x) + exp(-x) )/2.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-10000 30000 1.1e-19 2.8e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* cosh overflow |x| > MAXLOGL MAXNUML
*
*
*/

/* elliel.c
*
* Incomplete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* long double phi, m, y, elliel();
*
* y = elliel( 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 [-10, 10] and m in
* [0, 1].
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 50000 2.7e-18 2.3e-19
*
*
*/

/* ellikl.c
*
* Incomplete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* long double phi, m, y, ellikl();
*
* y = ellikl( 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 m in [0, 1] and phi as indicated.
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 30000 3.6e-18 4.1e-19
*
*
*/

/* ellpel.c
*
* Complete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* long double m1, y, ellpel();
*
* y = ellpel( 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 10000 1.1e-19 3.5e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ellpel domain x<0, x>1 0.0
*
*/

/* ellpjl.c
*
* Jacobian Elliptic Functions
*
*
*
* SYNOPSIS:
*
* long double u, m, sn, cn, dn, phi;
* int ellpjl();
*
* ellpjl( 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-12 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-18 2.3e-19
* IEEE cn 20000 1.6e-18 2.2e-19
* IEEE dn 10000 4.7e-15 2.7e-17
* IEEE phi 10000 4.0e-19* 6.6e-20*
*
* Accuracy deteriorates when u is large.
*
*/

/* ellpkl.c
*
* Complete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* long double m1, y, ellpkl();
*
* y = ellpkl( 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 10000 1.1e-19 3.3e-20
*
* ERROR MESSAGES:
*
* message condition value returned
* ellpkl domain x<0, x>1 0.0
*
*/

/* exp10l.c
*
* Base 10 exponential function, long double precision
* (Common antilogarithm)
*
*
*
* SYNOPSIS:
*
* long double x, y, exp10l()
*
* y = exp10l( 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).
* The Pade' form
*
* 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
*
* is used to approximate 10**f.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-4900 30000 1.0e-19 2.7e-20
*
* ERROR MESSAGES:
*
* message condition value returned
* exp10l underflow x < -MAXL10 0.0
* exp10l overflow x > MAXL10 MAXNUM
*
* IEEE arithmetic: MAXL10 = 4932.0754489586679023819
*
*/

/* exp2l.c
*
* Base 2 exponential function, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, exp2l();
*
* y = exp2l( 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 Pade' form
*
* 1 + 2x P(x**2) / (Q(x**2) - x P(x**2) )
*
* approximates 2**x in the basic range [-0.5, 0.5].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-16300 300000 9.1e-20 2.6e-20
*
*
* See exp.c for comments on error amplification.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp2l underflow x < -16382 0.0
* exp2l overflow x >= 16384 MAXNUM
*
*/

/* expl.c
*
* Exponential function, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, expl();
*
* y = expl( 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 Pade' form of degree 2/3 is used to approximate exp(f) - 1
* in the basic range [-0.5 ln 2, 0.5 ln 2].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-10000 50000 1.12e-19 2.81e-20
*
*
* 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 long double precision
* computer number, the result contains amplified roundoff
* error for large arguments not exactly represented.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp underflow x < MINLOG 0.0
* exp overflow x > MAXLOG MAXNUM
*
*/

/* fabsl.c
*
* Absolute value
*
*
*
* SYNOPSIS:
*
* long double x, y;
*
* y = fabsl( x );
*
*
*
* DESCRIPTION:
*
* Returns the absolute value of the argument.
*
*/

/* fdtrl.c
*
* F distribution, long double precision
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* long double x, y, fdtrl();
*
* y = fdtrl( 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) = incbetl( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
*
*
* The arguments a and b are greater than zero, and x
* x is nonnegative.
*
* ACCURACY:
*
* Tested at random points (a,b,x) in the indicated intervals.
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 1,100 10000 9.3e-18 2.9e-19
* IEEE 0,1 1,10000 10000 1.9e-14 2.9e-15
* IEEE 1,5 1,10000 10000 5.8e-15 1.4e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtrl domain a<0, b<0, x<0 0.0
*
*/
/* fdtrcl()
*
* Complemented F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* long double x, y, fdtrcl();
*
* y = fdtrcl( 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:
*
* See incbet.c.
* Tested at random points (a,b,x).
*
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 0,100 10000 4.2e-18 3.3e-19
* IEEE 0,1 1,10000 10000 7.2e-15 2.6e-16
* IEEE 1,5 1,10000 10000 1.7e-14 3.0e-15
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtrcl domain a<0, b<0, x<0 0.0
*
*/
/* fdtril()
*
* Inverse of complemented F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* long double x, p, fdtril();
*
* x = fdtril( df1, df2, p );
*
* 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 p.
*
* This is accomplished using the inverse beta integral
* function and the relations
*
* z = incbi( df2/2, df1/2, p )
* 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, p )
* x = df2 z / (df1 (1-z)).
*
* ACCURACY:
*
* See incbi.c.
* Tested at random points (a,b,p).
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between .001 and 1:
* IEEE 1,100 40000 4.6e-18 2.7e-19
* IEEE 1,10000 30000 1.7e-14 1.4e-16
* For p between 10^-6 and .001:
* IEEE 1,100 20000 1.9e-15 3.9e-17
* IEEE 1,10000 30000 2.7e-15 4.0e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtril domain p <= 0 or p > 1 0.0
* v < 1
*/

/* ceill()
* floorl()
* frexpl()
* ldexpl()
* fabsl()
*
* Floating point numeric utilities
*
*
*
* SYNOPSIS:
*
* long double x, y;
* long double ceill(), floorl(), frexpl(), ldexpl(), fabsl();
* int expnt, n;
*
* y = floorl(x);
* y = ceill(x);
* y = frexpl( x, &expnt );
* y = ldexpl( x, n );
* y = fabsl( x );
*
*
*
* DESCRIPTION:
*
* All four routines return a long double precision floating point
* result.
*
* floorl() returns the largest integer less than or equal to x.
* It truncates toward minus infinity.
*
* ceill() returns the smallest integer greater than or equal
* to x. It truncates toward plus infinity.
*
* frexpl() 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.
*
* ldexpl() multiplies x by 2**n.
*
* fabsl() returns the absolute value of its argument.
*
* These functions are part of the standard C run time library
* for some but not all C compilers. The ones supplied are
* written in C for 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.
*/

/* gammal.c
*
* Gamma function
*
*
*
* SYNOPSIS:
*
* long double x, y, gammal();
* extern int sgngam;
*
* y = gammal( 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 sgngam.
* This variable is also filled in by the logarithmic gamma
* function lgam().
*
* Arguments |x| <= 13 are reduced by recurrence and the function
* approximated by a rational function of degree 7/8 in the
* interval (2,3). Large arguments are handled by Stirling's
* formula. Large negative arguments are made positive using
* a reflection formula.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -40,+40 10000 3.6e-19 7.9e-20
* IEEE -1755,+1755 10000 4.8e-18 6.5e-19
*
* Accuracy for large arguments is dominated by error in powl().
*
*/
/* lgaml()
*
* Natural logarithm of gamma function
*
*
*
* SYNOPSIS:
*
* long double x, y, lgaml();
* extern int sgngam;
*
* y = lgaml( 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 sgngam.
*
* For arguments greater than 33, the logarithm of the gamma
* function is approximated by the logarithmic version of
* Stirling's formula using a polynomial approximation of
* degree 4. Arguments between -33 and +33 are reduced by
* recurrence to the interval [2,3] of a rational approximation.
* The cosecant reflection formula is employed for arguments
* less than -33.
*
* Arguments greater than MAXLGML (10^4928) return MAXNUML.
*
*
*
* ACCURACY:
*
*
* arithmetic domain # trials peak rms
* IEEE -40, 40 100000 2.2e-19 4.6e-20
* IEEE 10^-2000,10^+2000 20000 1.6e-19 3.3e-20
* The error criterion was relative when the function magnitude
* was greater than one but absolute when it was less than one.
*
*/

/* gdtrl.c
*
* Gamma distribution function
*
*
*
* SYNOPSIS:
*
* long double a, b, x, y, gdtrl();
*
* y = gdtrl( 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:
*
* See igam().
*
* ERROR MESSAGES:
*
* message condition value returned
* gdtrl domain x < 0 0.0
*
*/
/* gdtrcl.c
*
* Complemented gamma distribution function
*
*
*
* SYNOPSIS:
*
* long double a, b, x, y, gdtrcl();
*
* y = gdtrcl( 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:
*
* See igamc().
*
* ERROR MESSAGES:
*
* message condition value returned
* gdtrcl domain x < 0 0.0
*
*/

/*
C
C ..................................................................
C
C SUBROUTINE GELS
C
C PURPOSE
C TO SOLVE A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH
C SYMMETRIC COEFFICIENT MATRIX UPPER TRIANGULAR PART OF WHICH
C IS ASSUMED TO BE STORED COLUMNWISE.
C
C USAGE
C CALL GELS(R,A,M,N,EPS,IER,AUX)
C
C DESCRIPTION OF PARAMETERS
C R - M BY N RIGHT HAND SIDE MATRIX. (DESTROYED)
C ON RETURN R CONTAINS THE SOLUTION OF THE EQUATIONS.
C A - UPPER TRIANGULAR PART OF THE SYMMETRIC
C M BY M COEFFICIENT MATRIX. (DESTROYED)
C M - THE NUMBER OF EQUATIONS IN THE SYSTEM.
C N - THE NUMBER OF RIGHT HAND SIDE VECTORS.
C EPS - AN INPUT CONSTANT WHICH IS USED AS RELATIVE
C TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE.
C IER - RESULTING ERROR PARAMETER CODED AS FOLLOWS
C IER=0 - NO ERROR,
C IER=-1 - NO RESULT BECAUSE OF M LESS THAN 1 OR
C PIVOT ELEMENT AT ANY ELIMINATION STEP
C EQUAL TO 0,
C IER=K - WARNING DUE TO POSSIBLE LOSS OF SIGNIFI-
C CANCE INDICATED AT ELIMINATION STEP K+1,
C WHERE PIVOT ELEMENT WAS LESS THAN OR
C EQUAL TO THE INTERNAL TOLERANCE EPS TIMES
C ABSOLUTELY GREATEST MAIN DIAGONAL
C ELEMENT OF MATRIX A.
C AUX - AN AUXILIARY STORAGE ARRAY WITH DIMENSION M-1.
C
C REMARKS
C UPPER TRIANGULAR PART OF MATRIX A IS ASSUMED TO BE STORED
C COLUMNWISE IN M*(M+1)/2 SUCCESSIVE STORAGE LOCATIONS, RIGHT
C HAND SIDE MATRIX R COLUMNWISE IN N*M SUCCESSIVE STORAGE
C LOCATIONS. ON RETURN SOLUTION MATRIX R IS STORED COLUMNWISE
C TOO.
C THE PROCEDURE GIVES RESULTS IF THE NUMBER OF EQUATIONS M IS
C GREATER THAN 0 AND PIVOT ELEMENTS AT ALL ELIMINATION STEPS
C ARE DIFFERENT FROM 0. HOWEVER WARNING IER=K - IF GIVEN -
C INDICATES POSSIBLE LOSS OF SIGNIFICANCE. IN CASE OF A WELL
C SCALED MATRIX A AND APPROPRIATE TOLERANCE EPS, IER=K MAY BE
C INTERPRETED THAT MATRIX A HAS THE RANK K. NO WARNING IS
C GIVEN IN CASE M=1.
C ERROR PARAMETER IER=-1 DOES NOT NECESSARILY MEAN THAT
C MATRIX A IS SINGULAR, AS ONLY MAIN DIAGONAL ELEMENTS
C ARE USED AS PIVOT ELEMENTS. POSSIBLY SUBROUTINE GELG (WHICH
C WORKS WITH TOTAL PIVOTING) WOULD BE ABLE TO FIND A SOLUTION.
C
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
C NONE
C
C METHOD
C SOLUTION IS DONE BY MEANS OF GAUSS-ELIMINATION WITH
C PIVOTING IN MAIN DIAGONAL, IN ORDER TO PRESERVE
C SYMMETRY IN REMAINING COEFFICIENT MATRICES.
C
C ..................................................................
C
*/

/* igamil()
*
* Inverse of complemented imcomplete gamma integral
*
*
*
* SYNOPSIS:
*
* long double a, x, y, igamil();
*
* x = igamil( a, y );
*
*
*
* DESCRIPTION:
*
* Given y, the function finds x such that
*
* igamc( a, x ) = y.
*
* 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 0.5 to 30 and x from 0 to 0.5.
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,0.5 3400 8.8e-16 1.3e-16
* IEEE 0,0.5 10000 1.1e-14 1.0e-15
*
*/

/* igaml.c
*
* Incomplete gamma integral
*
*
*
* SYNOPSIS:
*
* long double a, x, y, igaml();
*
* y = igaml( 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
* DEC 0,30 4000 4.4e-15 6.3e-16
* IEEE 0,30 10000 3.6e-14 5.1e-15
*
*/
/* igamcl()
*
* Complemented incomplete gamma integral
*
*
*
* SYNOPSIS:
*
* long double a, x, y, igamcl();
*
* y = igamcl( 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
* DEC 0,30 2000 2.7e-15 4.0e-16
* IEEE 0,30 60000 1.4e-12 6.3e-15
*
*/

/* incbetl.c
*
* Incomplete beta integral
*
*
* SYNOPSIS:
*
* long double a, b, x, y, incbetl();
*
* y = incbetl( 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
* or, when b*x is small, by a power series.
*
* ACCURACY:
*
* Tested at random points (a,b,x) with x between 0 and 1.
* arithmetic domain # trials peak rms
* IEEE 0,5 20000 4.5e-18 2.4e-19
* IEEE 0,100 100000 3.9e-17 1.0e-17
* Half-integer a, b:
* IEEE .5,10000 100000 3.9e-14 4.4e-15
* Outputs smaller than the IEEE gradual underflow threshold
* were excluded from these statistics.
*
* ERROR MESSAGES:
*
* message condition value returned
* incbetl domain x<0, x>1 0.0
*/

/* incbil()
*
* Inverse of imcomplete beta integral
*
*
*
* SYNOPSIS:
*
* long double a, b, x, y, incbil();
*
* x = incbil( 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 .5,10000 10000 1.1e-14 1.4e-16
*/

/* j0l.c
*
* Bessel function of order zero
*
*
*
* SYNOPSIS:
*
* long double x, y, j0l();
*
* y = j0l( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of first kind, order zero of the argument.
*
* The domain is divided into the intervals [0, 9] and
* (9, infinity). In the first interval the rational approximation
* is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) P7(x^2) / Q8(x^2),
* where r, s, t are the first three zeros of the function.
* In the second interval the expansion is in terms of the
* modulus M0(x) = sqrt(J0(x)^2 + Y0(x)^2) and phase P0(x)
* = atan(Y0(x)/J0(x)). M0 is approximated by sqrt(1/x)P7(1/x)/Q7(1/x).
* The approximation to J0 is M0 * cos(x - pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 100000 2.8e-19 7.4e-20
*
*
*/
/* y0l.c
*
* Bessel function of the second kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, y0l();
*
* y = y0l( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind, of order
* zero, of the argument.
*
* The domain is divided into the intervals [0, 5>, [5,9> and
* [9, infinity). In the first interval a rational approximation
* R(x) is employed to compute y0(x) = R(x) + 2/pi * log(x) * j0(x).
*
* In the second interval, the approximation is
* (x - p)(x - q)(x - r)(x - s)P7(x)/Q7(x)
* where p, q, r, s are zeros of y0(x).
*
* The third interval uses the same approximations to modulus
* and phase as j0(x), whence y0(x) = modulus * sin(phase).
*
* ACCURACY:
*
* Absolute error, when y0(x) < 1; else relative error:
*
* arithmetic domain # trials peak rms
* IEEE 0, 30 100000 3.4e-19 7.6e-20
*
*/

/* j1l.c
*
* Bessel function of order one
*
*
*
* SYNOPSIS:
*
* long double x, y, j1l();
*
* y = j1l( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order one of the argument.
*
* The domain is divided into the intervals [0, 9] and
* (9, infinity). In the first interval the rational approximation
* is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) x P8(x^2) / Q8(x^2),
* where r, s, t are the first three zeros of the function.
* In the second interval the expansion is in terms of the
* modulus M1(x) = sqrt(J1(x)^2 + Y1(x)^2) and phase P1(x)
* = atan(Y1(x)/J1(x)). M1 is approximated by sqrt(1/x)P7(1/x)/Q8(1/x).
* The approximation to j1 is M1 * cos(x - 3 pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 40000 1.8e-19 5.0e-20
*
*
*/
/* y1l.c
*
* Bessel function of the second kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, y1l();
*
* y = y1l( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind, of order
* zero, of the argument.
*
* The domain is divided into the intervals [0, 4.5>, [4.5,9> and
* [9, infinity). In the first interval a rational approximation
* R(x) is employed to compute y0(x) = R(x) + 2/pi * log(x) * j0(x).
*
* In the second interval, the approximation is
* (x - p)(x - q)(x - r)(x - s)P9(x)/Q10(x)
* where p, q, r, s are zeros of y1(x).
*
* The third interval uses the same approximations to modulus
* and phase as j1(x), whence y1(x) = modulus * sin(phase).
*
* ACCURACY:
*
* Absolute error, when y0(x) < 1; else relative error:
*
* arithmetic domain # trials peak rms
* IEEE 0, 30 36000 2.7e-19 5.3e-20
*
*/

/* jnl.c
*
* Bessel function of integer order
*
*
*
* SYNOPSIS:
*
* int n;
* long double x, y, jnl();
*
* y = jnl( 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 domain # trials peak rms
* IEEE -30, 30 5000 3.3e-19 4.7e-20
*
*
* Not suitable for large n or x.
*
*/

/* ldrand.c
*
* Pseudorandom number generator
*
*
*
* SYNOPSIS:
*
* double y;
* int ldrand();
*
* ldrand( &y );
*
*
*
* DESCRIPTION:
*
* Yields a random number 1.0 <= y < 2.0.
*
* The three-generator congruential algorithm by Brian
* Wichmann and David Hill (BYTE magazine, March, 1987,
* pp 127-8) is used.
*
* Versions invoked by the different arithmetic compile
* time options IBMPC, and MIEEE, produce the same sequences.
*
*/

/* log10l.c
*
* Common logarithm, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, log10l();
*
* y = log10l( x );
*
*
*
* DESCRIPTION:
*
* Returns the base 10 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)/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 0.5, 2.0 30000 9.0e-20 2.6e-20
* IEEE exp(+-10000) 30000 6.0e-20 2.3e-20
*
* In the tests over the interval exp(+-10000), the logarithms
* of the random arguments were uniformly distributed over
* [-10000, +10000].
*
* ERROR MESSAGES:
*
* log singularity: x = 0; returns MINLOG
* log domain: x < 0; returns MINLOG
*/

/* log2l.c
*
* Base 2 logarithm, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, log2l();
*
* y = log2l( 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 (natural)
* 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 0.5, 2.0 30000 9.8e-20 2.7e-20
* IEEE exp(+-10000) 70000 5.4e-20 2.3e-20
*
* In the tests over the interval exp(+-10000), the logarithms
* of the random arguments were uniformly distributed over
* [-10000, +10000].
*
* ERROR MESSAGES:
*
* log singularity: x = 0; returns MINLOG
* log domain: x < 0; returns MINLOG
*/

/* logl.c
*
* Natural logarithm, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, logl();
*
* y = logl( 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)/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 0.5, 2.0 150000 8.71e-20 2.75e-20
* IEEE exp(+-10000) 100000 5.39e-20 2.34e-20
*
* In the tests over the interval exp(+-10000), the logarithms
* of the random arguments were uniformly distributed over
* [-10000, +10000].
*
* ERROR MESSAGES:
*
* log singularity: x = 0; returns MINLOG
* log domain: x < 0; returns MINLOG
*/

/* mtherr.c
*
* Library common error handling routine
*
*
*
* SYNOPSIS:
*
* char *fctnam;
* int code;
* int 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
*
*/

/* nbdtrl.c
*
* Negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, nbdtrl();
*
* y = nbdtrl( 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:
*
* Tested at random points (k,n,p) with k and n between 1 and 10,000
* and p between 0 and 1.
*
* arithmetic domain # trials peak rms
* Absolute error:
* IEEE 0,10000 10000 9.8e-15 2.1e-16
*
*/
/* nbdtrcl.c
*
* Complemented negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, nbdtrcl();
*
* y = nbdtrcl( 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:
*
* See incbetl.c.
*
*/
/* nbdtril
*
* Functional inverse of negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* long double p, y, nbdtril();
*
* p = nbdtril( k, n, y );
*
*
*
* DESCRIPTION:
*
* Finds the argument p such that nbdtr(k,n,p) is equal to y.
*
* ACCURACY:
*
* Tested at random points (a,b,y), with y between 0 and 1.
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,100
* See also incbil.c.
*/

/* ndtril.c
*
* Inverse of Normal distribution function
*
*
*
* SYNOPSIS:
*
* long double x, y, ndtril();
*
* x = ndtril( 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 log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) .
* For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
* where w = y - 0.5 .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* Arguments uniformly distributed:
* IEEE 0, 1 5000 7.8e-19 9.9e-20
* Arguments exponentially distributed:
* IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtril domain x <= 0 -MAXNUML
* ndtril domain x >= 1 MAXNUML
*
*/

/* ndtril.c
*
* Inverse of Normal distribution function
*
*
*
* SYNOPSIS:
*
* long double x, y, ndtril();
*
* x = ndtril( 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 log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) .
* For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
* where w = y - 0.5 .
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* Arguments uniformly distributed:
* IEEE 0, 1 5000 7.8e-19 9.9e-20
* Arguments exponentially distributed:
* IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtril domain x <= 0 -MAXNUML
* ndtril domain x >= 1 MAXNUML
*
*/

/* pdtrl.c
*
* Poisson distribution
*
*
*
* SYNOPSIS:
*
* int k;
* long double m, y, pdtrl();
*
* y = pdtrl( 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:
*
* See igamc().
*
*/
/* pdtrcl()
*
* Complemented poisson distribution
*
*
*
* SYNOPSIS:
*
* int k;
* long double m, y, pdtrcl();
*
* y = pdtrcl( 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:
*
* See igam.c.
*
*/
/* pdtril()
*
* Inverse Poisson distribution
*
*
*
* SYNOPSIS:
*
* int k;
* long double m, y, pdtrl();
*
* m = pdtril( 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:
*
* See igami.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* pdtri domain y < 0 or y >= 1 0.0
* k < 0
*
*/

/* polevll.c
* p1evll.c
*
* Evaluate polynomial
*
*
*
* SYNOPSIS:
*
* int N;
* long double x, y, coef[N+1], polevl[];
*
* y = polevll( 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 p1evll() assumes that coef[N] = 1.0 and is
* omitted from the array. Its calling arguments are
* otherwise the same as polevll().
*
* This module also contains the following globally declared constants:
* MAXNUML = 1.189731495357231765021263853E4932L;
* MACHEPL = 5.42101086242752217003726400434970855712890625E-20L;
* MAXLOGL = 1.1356523406294143949492E4L;
* MINLOGL = -1.1355137111933024058873E4L;
* LOGE2L = 6.9314718055994530941723E-1L;
* LOG2EL = 1.4426950408889634073599E0L;
* PIL = 3.1415926535897932384626L;
* PIO2L = 1.5707963267948966192313L;
* PIO4L = 7.8539816339744830961566E-1L;
*
* 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.
*
*/

/* powil.c
*
* Real raised to integer power, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, powil();
* int n;
*
* y = powil( 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 .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
* IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
* IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
*
* Returns MAXNUM on overflow, zero on underflow.
*
*/

/* powl.c
*
* Power function, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, z, powl();
*
* z = powl( 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/32 and pseudo extended precision arithmetic to
* obtain several extra bits of accuracy in both the logarithm
* and the exponential.
*
*
*
* ACCURACY:
*
* The relative error of pow(x,y) can be estimated
* by y dl ln(2), where dl is the absolute error of
* the internally computed base 2 logarithm. At the ends
* of the approximation interval the logarithm equal 1/32
* and its relative error is about 1 lsb = 1.1e-19. Hence
* the predicted relative error in the result is 2.3e-21 y .
*
* Relative error:
* arithmetic domain # trials peak rms
*
* IEEE +-1000 40000 2.8e-18 3.7e-19
* .001 < x < 1000, with log(x) uniformly distributed.
* -1000 < y < 1000, y uniformly distributed.
*
* IEEE 0,8700 60000 6.5e-18 1.0e-18
* 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* pow overflow x**y > MAXNUM MAXNUM
* pow underflow x**y < 1/MAXNUM 0.0
* pow domain x<0 and y noninteger 0.0
*
*/

/* sinhl.c
*
* Hyperbolic sine, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, sinhl();
*
* y = sinhl( x );
*
*
*
* DESCRIPTION:
*
* Returns hyperbolic sine of argument in the range MINLOGL to
* MAXLOGL.
*
* The range is partitioned into two segments. If |x| <= 1, a
* rational function of the form x + x**3 P(x)/Q(x) is employed.
* Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -2,2 10000 1.5e-19 3.9e-20
* IEEE +-10000 30000 1.1e-19 2.8e-20
*
*/

/* sinl.c
*
* Circular sine, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, sinl();
*
* y = sinl( 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 the Cody
* and Waite polynomial form
* x + x**3 P(x**2) .
* Between pi/4 and pi/2 the cosine is represented as
* 1 - .5 x**2 + x**4 Q(x**2) .
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-5.5e11 200,000 1.2e-19 2.9e-20
*
* ERROR MESSAGES:
*
* message condition value returned
* sin total loss x > 2**39 0.0
*
* Loss of precision occurs for x > 2**39 = 5.49755813888e11.
* The routine as implemented flags a TLOSS error for
* x > 2**39 and returns 0.0.
*/
/* cosl.c
*
* Circular cosine, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, cosl();
*
* y = cosl( 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 - .5 x**2 + x**4 Q(x**2) .
* Between pi/4 and pi/2 the sine is represented by the Cody
* and Waite polynomial form
* x + x**3 P(x**2) .
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-5.5e11 50000 1.2e-19 2.9e-20
*/

/* sqrtl.c
*
* Square root, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, sqrtl();
*
* y = sqrtl( 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.
*
* Note, some arithmetic coprocessors such as the 8087 and
* 68881 produce correctly rounded square roots, which this
* routine will not.
*
* ACCURACY:
*
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,10 30000 8.1e-20 3.1e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* sqrt domain x < 0 0.0
*
*/

/* stdtrl.c
*
* Student's t distribution
*
*
*
* SYNOPSIS:
*
* long double p, t, stdtrl();
* int k;
*
* p = stdtrl( 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 < -1.6, 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:
*
* Tested at random 1 <= k <= 100. The "domain" refers to t.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -100,-1.6 10000 5.7e-18 9.8e-19
* IEEE -1.6,100 10000 3.8e-18 1.0e-19
*/

/* stdtril.c
*
* Functional inverse of Student's t distribution
*
*
*
* SYNOPSIS:
*
* long double p, t, stdtril();
* int k;
*
* t = stdtril( k, p );
*
*
* DESCRIPTION:
*
* Given probability p, finds the argument t such that stdtrl(k,t)
* is equal to p.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 100. The "domain" refers to p:
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,1 3500 4.2e-17 4.1e-18
*/

/* tanhl.c
*
* Hyperbolic tangent, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, tanhl();
*
* y = tanhl( x );
*
*
*
* DESCRIPTION:
*
* Returns hyperbolic tangent of argument in the range MINLOGL to
* MAXLOGL.
*
* A rational function is used for |x| < 0.625. The form
* x + x**3 P(x)/Q(x) of Cody _& Waite is employed.
* Otherwise,
* tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -2,2 30000 1.3e-19 2.4e-20
*
*/

/* tanl.c
*
* Circular tangent, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, tanl();
*
* y = tanl( x );
*
*
*
* DESCRIPTION:
*
* Returns the circular tangent of the radian argument x.
*
* Range reduction is modulo pi/4. A rational function
* x + x**3 P(x**2)/Q(x**2)
* is employed in the basic interval [0, pi/4].
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-1.07e9 30000 1.9e-19 4.8e-20
*
* ERROR MESSAGES:
*
* message condition value returned
* tan total loss x > 2^39 0.0
*
*/
/* cotl.c
*
* Circular cotangent, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, cotl();
*
* y = cotl( x );
*
*
*
* DESCRIPTION:
*
* Returns the circular cotangent of the radian argument x.
*
* Range reduction is modulo pi/4. A rational function
* x + x**3 P(x**2)/Q(x**2)
* is employed in the basic interval [0, pi/4].
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-1.07e9 30000 1.9e-19 5.1e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* cot total loss x > 2^39 0.0
* cot singularity x = 0 MAXNUM
*
*/

/* unityl.c
*
* Relative error approximations for function arguments near
* unity.
*
* log1p(x) = log(1+x)
* expm1(x) = exp(x) - 1
* cos1m(x) = cos(x) - 1
*
*/

/* ynl.c
*
* Bessel function of second kind of integer order
*
*
*
* SYNOPSIS:
*
* long double x, y, ynl();
* int n;
*
* y = ynl( 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
* y0l() and y1l().
*
* If n = 0 or 1 the routine for y0l or y1l is called
* directly.
*
*
*
* ACCURACY:
*
*
* Absolute error, except relative error when y > 1.
* x >= 0, -30 <= n <= +30.
* arithmetic domain # trials peak rms
* IEEE -30, 30 10000 1.3e-18 1.8e-19
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ynl singularity x = 0 MAXNUML
* ynl overflow MAXNUML
*
* Spot checked against tables for x, n between 0 and 100.
*
*/