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/* @(#)e_asin.c 5.1 93/09/24 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: e_asin.c,v 1.3.2.1 1997/02/23 11:03:00 joerg Exp $";
#endif
/* __ieee754_asin(x)
* Method :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
* where
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
* and its remez error is bounded by
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const double
#else
static double
#endif
one
= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
huge
= 1.000e+300,
pio2_hi
= 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo
= 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
pio4_hi
= 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
/* coefficient for R(x^2) */
pS0
= 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
pS1
= -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
pS2
= 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
pS3
= -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
pS4
= 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
pS5
= 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
qS1
= -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
qS2
= 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
qS3
= -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
qS4
= 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
#ifdef __STDC__
double __generic___ieee754_asin
(double x
)
#else
double __generic___ieee754_asin
(x
)
double x
;
#endif
{
double t
=0.0,w
,p
,q
,c
,r
,s
;
int32_t hx
,ix
;
GET_HIGH_WORD
(hx
,x
);
ix
= hx
&0x7fffffff;
if(ix
>= 0x3ff00000) { /* |x|>= 1 */
u_int32_t lx
;
GET_LOW_WORD
(lx
,x
);
if(((ix
-0x3ff00000)|lx
)==0)
/* asin(1)=+-pi/2 with inexact */
return x
*pio2_hi
+x
*pio2_lo
;
return (x
-x
)/(x
-x
); /* asin(|x|>1) is NaN */
} else if (ix
<0x3fe00000) { /* |x|<0.5 */
if(ix
<0x3e400000) { /* if |x| < 2**-27 */
if(huge
+x
>one
) return x
;/* return x with inexact if x!=0*/
} else
t
= x
*x
;
p
= t
*(pS0
+t
*(pS1
+t
*(pS2
+t
*(pS3
+t
*(pS4
+t
*pS5
)))));
q
= one
+t
*(qS1
+t
*(qS2
+t
*(qS3
+t
*qS4
)));
w
= p
/q
;
return x
+x
*w
;
}
/* 1> |x|>= 0.5 */
w
= one
-fabs(x
);
t
= w
*0.5;
p
= t
*(pS0
+t
*(pS1
+t
*(pS2
+t
*(pS3
+t
*(pS4
+t
*pS5
)))));
q
= one
+t
*(qS1
+t
*(qS2
+t
*(qS3
+t
*qS4
)));
s
= sqrt(t
);
if(ix
>=0x3FEF3333) { /* if |x| > 0.975 */
w
= p
/q
;
t
= pio2_hi
-(2.0*(s
+s
*w
)-pio2_lo
);
} else {
w
= s
;
SET_LOW_WORD
(w
,0);
c
= (t
-w
*w
)/(s
+w
);
r
= p
/q
;
p
= 2.0*s
*r
-(pio2_lo
-2.0*c
);
q
= pio4_hi
-2.0*w
;
t
= pio4_hi
-(p
-q
);
}
if(hx
>0) return t
; else return -t
;
}