catan.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
/* INDENT OFF */
/*
* dcomplex catan(dcomplex z);
*
* If
* z = x + iy,
*
* then
* 1 ( 2x ) 1 2 2
* Re w = - arctan(-----------) = - ATAN2(2x, 1 - x - y )
* 2 ( 2 2) 2
* (1 - x - y )
*
* ( 2 2)
* 1 (x + (y+1) ) 1 4y
* Im w = - log(------------) .= --- log [ 1 + ------------- ]
* 4 ( 2 2) 4 2 2
* (x + (y-1) ) x + (y-1)
*
* 2 16 3 y
* = t - 2t + -- t - ..., where t = -----------------
* 3 x*x + (y-1)*(y-1)
*
* Note that: if catan( x, y) = ( u, v), then
* catan(-x, y) = (-u, v)
* catan( x,-y) = ( u,-v)
*
* Also, catan(x,y) = -i*catanh(-y,x), or
* catanh(x,y) = i*catan(-y,x)
* So, if catanh(y,x) = (v,u), then catan(x,y) = -i*(-v,u) = (u,v), i.e.,
* catan(x,y) = (u,v)
*
* catan( 0 , 0 ) = (0 , 0 )
* catan( NaN, 0 ) = (NaN , 0 )
* catan( 0 , 1 ) = (0 , +inf) with divide-by-zero
* catan( inf, y ) = (pi/2 , 0 ) for finite +y
* catan( NaN, y ) = (NaN , NaN ) with invalid for finite y != 0
* catan( x , inf ) = (pi/2 , 0 ) for finite +x
* catan( inf, inf ) = (pi/2 , 0 )
* catan( NaN, inf ) = (NaN , 0 )
* catan( x , NaN ) = (NaN , NaN ) with invalid for finite x
* catan( inf, NaN ) = (pi/2 , +-0 )
*/
/* INDENT ON */
#include "complex_wrapper.h"
/* INDENT OFF */
static const double
pi_2 = 1.570796326794896558e+00,
zero = 0.0,
half = 0.5,
two = 2.0,
ln2 = 6.931471805599453094172321214581765680755e-0001,
one = 1.0;
/* INDENT ON */
x = D_RE(z);
y = D_IM(z);
/* x is inf or NaN */
if (ix >= 0x7ff00000) {
} else {
else
}
} else if (iy >= 0x7ff00000) {
/* y is inf or NaN */
} else {
}
/* INDENT OFF */
/*
* x = 0
* 1 1
* A = --- * atan2(2x, 1-x*x-y*y) = --- atan2(0,1-|y|)
* 2 2
*
* 1 [ (y+1)*(y+1) ] 1 2 1 2y
* B = - log [ ------------ ] = - log (1+ ---) or - log(1+ ----)
* 4 [ (y-1)*(y-1) ] 2 y-1 2 1-y
*/
/* INDENT ON */
/* y=1: catan(0,1)=(0,+inf) with 1/0 signal */
} else { /* y<1 */
}
/* INDENT OFF */
/*
* Tiny y (relative to 1+|x|)
* |y| < E*(1+|x|)
* where E=2**-29, -35, -60 for double, double extended, quad precision
*
* 1 [ x<=1: atan(x)
* A = --- * atan2(2x, 1-x*x-y*y) ~ [ 1 1+x
* 2 [ x>=1: - atan2(2,(1-x)*(-----))
* 2 x
*
* y/x
* B ~ t*(1-2t), where t = ----------------- is tiny
* x + (y-1)*(y-1)/x
*/
/* INDENT ON */
if (ix < 0x3ff00000)
else
} else {
if (ix < 0x3e200000)
else if (ix > 0x41c00000)
else
}
/* INDENT OFF */
/*
* Huge y relative to 1+|x|
* |y| > Einv*(1+|x|), where Einv~2**(prec/2+3),
* 1
* A ~ --- * atan2(2x, -y*y) ~ pi/2
* 2
* y
* B ~ t*(1-2t), where t = --------------- is tiny
* (y-1)*(y-1)
*/
/* INDENT ON */
/* INDENT OFF */
/*
* y = 1
* 1 1
* A = --- * atan2(2x, -x*x) = --- atan2(2,-x)
* 2 2
*
* 1 [x*x + 4] 1 4 [ 0.5(log2-logx) if
* B = - log [-------] = - log (1+ ---) = [ |x|<E, else 0.25*
* 4 [ x*x ] 4 x*x [ log1p((2/x)*(2/x))
*/
/* INDENT ON */
if (ix < 0x3e200000)
else {
}
} else if (ix >= 0x43900000) {
/* INDENT OFF */
/*
* Huge x:
* when |x| > 1/E^2,
* 1 pi
* A ~ --- * atan2(2x, -x*x-y*y) ~ ---
* 2 2
* y y/x
* B ~ t*(1-2t), where t = --------------- = (-------------- )/x
* x*x+(y-1)*(y-1) 1+((y-1)/x)^2
*/
/* INDENT ON */
} else if (ix < 0x38b00000) {
/* INDENT OFF */
/*
* Tiny x:
* when |x| < E^4, (note that y != 1)
* 1 1
* A = --- * atan2(2x, 1-x*x-y*y) ~ --- * atan2(2x,(1-y)*(1+y))
* 2 2
*
* 1 [(y+1)*(y+1)] 1 2 1 2y
* B = - log [-----------] = - log (1+ ---) or - log(1+ ----)
* 4 [(y-1)*(y-1)] 2 y-1 2 1-y
*/
/* INDENT ON */
if (iy >= 0x3ff00000)
else
} else {
/* INDENT OFF */
/*
* normal x,y
* 1
* A = --- * atan2(2x, 1-x*x-y*y)
* 2
*
* 1 [x*x+(y+1)*(y+1)] 1 4y
* B = - log [---------------] = - log (1+ -----------------)
* 4 [x*x+(y-1)*(y-1)] 4 x*x + (y-1)*(y-1)
*/
/* INDENT ON */
/* y close to 1 */
/* x close to 1 */
} else
}
if (hx < 0)
if (hy < 0)
return (ans);
}