/* $NetBSD: n_asincos.c,v 1.11 2024/06/09 13:35:38 riastradh Exp $ */
/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/* ASIN(X)
* RETURNS ARC SINE OF X
* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
*
* Required system supported functions:
* copysign(x,y)
* sqrt(x)
*
* Required kernel function:
* atan2(y,x)
*
* Method :
* asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
* computed as follows
* 1-x*x if x < 0.5,
* 2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN.
*
* Accuracy:
* 1) If atan2() uses machine PI, then
*
* asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
* and PI is the exact pi rounded to machine precision (see atan2 for
* details):
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in hexadecimal:
* pi = 3.243F6A8885A308D313198A2E....
* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
*
* In a test run with more than 200,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 2.06 ulps. (comparing against (PI/pi)*(exact asin(x)));
*
* 2) If atan2() uses true pi, then
*
* asin(x) returns the exact asin(x) with error below about 2 ulps.
*
* In a test run with more than 1,024,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 1.99 ulps.
*/
double
asin(double x)
{
double s,t,one=1.0;
#if !defined(__vax__)&&!defined(tahoe)
if(x!=x) return(x); /* x is NaN */
#endif /* !defined(__vax__)&&!defined(tahoe) */
s=copysign(x,one);
if(s <= 0.5)
return(atan2(x,sqrt(one-x*x)));
else
{ t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
}
float
asinf(float x)
{
return (float)asin(x);
}
/* ACOS(X)
* RETURNS ARC COS OF X
* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
*
* Required system supported functions:
* copysign(x,y)
* sqrt(x)
*
* Required kernel function:
* atan2(y,x)
*
* Method :
* ________
* / 1 - x
* acos(x) = 2*atan2( / -------- , 1 ) .
* \/ 1 + x
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN.
*
* Accuracy:
* 1) If atan2() uses machine PI, then
*
* acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
* and PI is the exact pi rounded to machine precision (see atan2 for
* details):
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in hexadecimal:
* pi = 3.243F6A8885A308D313198A2E....
* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
*
* In a test run with more than 200,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 2.07 ulps. (comparing against (PI/pi)*(exact acos(x)));
*
* 2) If atan2() uses true pi, then
*
* acos(x) returns the exact acos(x) with error below about 2 ulps.
*
* In a test run with more than 1,024,000 random arguments on a VAX, the
* maximum observed error in ulps (units in the last place) was
* 2.15 ulps.
*/
