Inverse trigonometric sine function.
- Version
- 1.3
- Date
- 95/01/18
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.
- Copyright
- Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. Developed at SunSoft, a Sun Microsystems, Inc. business. Permission to use, copy, modify, and distribute this software is freely granted, provided that this notice is preserved.
Definition at line 104 of file asin.c.
References huge, one, pio2_hi, pio2_lo, pio4_hi, pS0, pS1, pS2, pS3, pS4, pS5, qS1, qS2, qS3, qS4, and sqrt().
Referenced by RealNumber::ArcCosecant(), and RealNumber::ArcSine().
106 double t,w,p,q,c,r,s;
110 if(ix>= 0x3ff00000) {
113 if(((ix-0x3ff00000)|lx)==0)
117 }
else if (ix<0x3fe00000) {
#define GET_HIGH_WORD(i, d)
Get the more significant 32 bit int from a double.
static const double pio2_hi
#define GET_LOW_WORD(i, d)
Get the less significant 32 bit int from a double.
signed int sword
32 bit signed integer.
double fabs(double x)
Returns the absolute value of x.
static const double pio4_hi
double sqrt(double x)
Square root function.
unsigned int uword
32 bit unsigned integer.
static const double pio2_lo
#define SET_LOW_WORD(d, v)
Set the less significant 32 bits of a double from an int.
1.8.11