summaryrefslogtreecommitdiff
path: root/libm/ldouble/README.txt
diff options
context:
space:
mode:
Diffstat (limited to 'libm/ldouble/README.txt')
-rw-r--r--libm/ldouble/README.txt3502
1 files changed, 0 insertions, 3502 deletions
diff --git a/libm/ldouble/README.txt b/libm/ldouble/README.txt
deleted file mode 100644
index 30fcaad36..000000000
--- a/libm/ldouble/README.txt
+++ /dev/null
@@ -1,3502 +0,0 @@
-/* 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<k 0.0
- */
- /* 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