atanl.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
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/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak atanl = __atanl
/*
* atanl(x)
* Table look-up algorithm
* By K.C. Ng, March 9, 1989
*
* Algorithm.
*
* The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)).
* We use poly1(x) to approximate atan(x) for x in [0,1/8] with
* error (relative)
* |(atan(x)-poly1(x))/x|<= 2^-115.94 long double
* |(atan(x)-poly1(x))/x|<= 2^-58.85 double
* |(atan(x)-poly1(x))/x|<= 2^-25.53 float
* and use poly2(x) to approximate atan(x) for x in [0,1/65] with
* error (absolute)
* |atan(x)-poly2(x)|<= 2^-122.15 long double
* |atan(x)-poly2(x)|<= 2^-64.79 double
* |atan(x)-poly2(x)|<= 2^-35.36 float
* Here poly1 and poly2 are odd polynomial with the following form:
* x + x^3*(a1+x^2*(a2+...))
*
* (0). Purge off Inf and NaN and 0
* (1). Reduce x to positive by atan(x) = -atan(-x).
* (2). For x <= 1/8, use
* (2.1) if x < 2^(-prec/2-2), atan(x) = x with inexact
* (2.2) Otherwise
* atan(x) = poly1(x)
* (3). For x >= 8 then
* (3.1) if x >= 2^(prec+2), atan(x) = atan(inf) - pio2lo
* (3.2) if x >= 2^(prec/3+2), atan(x) = atan(inf) - 1/x
* (3.3) if x > 65, atan(x) = atan(inf) - poly2(1/x)
* (3.4) Otherwise, atan(x) = atan(inf) - poly1(1/x)
*
* (4). Now x is in (0.125, 8)
* Find y that match x to 4.5 bit after binary (easy).
* If iy is the high word of y, then
* single : j = (iy - 0x3e000000) >> 19
* double : j = (iy - 0x3fc00000) >> 16
* quad : j = (iy - 0x3ffc0000) >> 12
*
* Let s = (x-y)/(1+x*y). Then
* atan(x) = atan(y) + poly1(s)
* = _TBL_atanl_hi[j] + (_TBL_atanl_lo[j] + poly2(s) )
*
* Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125
*
*/
#include "libm.h"
extern const long double _TBL_atanl_hi[], _TBL_atanl_lo[];
static const long double
one = 1.0L,
p1 = -3.333333333333333333333333333331344526118e-0001L,
p2 = 1.999999999999999999999999989931277668570e-0001L,
p3 = -1.428571428571428571428553606221309530901e-0001L,
p4 = 1.111111111111111111095219842737139747418e-0001L,
p5 = -9.090909090909090825503603835248061123323e-0002L,
p6 = 7.692307692307664052130743214708925258904e-0002L,
p7 = -6.666666666660213835187713228363717388266e-0002L,
p8 = 5.882352940152439399097283359608661949504e-0002L,
p9 = -5.263157780447533993046614040509529668487e-0002L,
p10 = 4.761895816878184933175855990886788439447e-0002L,
p11 = -4.347345005832274022681019724553538135922e-0002L,
p12 = 3.983031914579635037502589204647752042736e-0002L,
p13 = -3.348206704469830575196657749413894897554e-0002L,
q1 = -3.333333333333333333333333333195273650186e-0001L,
q2 = 1.999999999999999999999988146114392615808e-0001L,
q3 = -1.428571428571428571057630319435467111253e-0001L,
q4 = 1.111111111111105373263048208994541544098e-0001L,
q5 = -9.090909090421834209167373258681021816441e-0002L,
q6 = 7.692305377813692706850171767150701644539e-0002L,
q7 = -6.660896644393861499914731734305717901330e-0002L,
pio2hi = 1.570796326794896619231321691639751398740e+0000L,
pio2lo = 4.335905065061890512398522013021675984381e-0035L;
#define i0 0
#define i1 3
long double
atanl(long double x) {
long double y, z, r, p, s;
int *px = (int *) &x, *py = (int *) &y;
int ix, iy, sign, j;
ix = px[i0];
sign = ix & 0x80000000;
ix ^= sign;
/* for |x| < 1/8 */
if (ix < 0x3ffc0000) {
if (ix < 0x3feb0000) { /* when |x| < 2**(-prec/6-2) */
if (ix < 0x3fc50000) { /* if |x| < 2**(-prec/2-2) */
s = one;
*(3 - i0 + (int *) &s) = -1; /* s = 1-ulp */
*(1 + (int *) &s) = -1;
*(2 + (int *) &s) = -1;
*(i0 + (int *) &s) -= 1;
if ((int) (s * x) < 1)
return (x); /* raise inexact */
}
z = x * x;
if (ix < 0x3fe20000) { /* if |x| < 2**(-prec/4-1) */
return (x + (x * z) * p1);
} else { /* if |x| < 2**(-prec/6-2) */
return (x + (x * z) * (p1 + z * p2));
}
}
z = x * x;
return (x + (x * z) * (p1 + z * (p2 + z * (p3 + z * (p4 +
z * (p5 + z * (p6 + z * (p7 + z * (p8 + z * (p9 +
z * (p10 + z * (p11 + z * (p12 + z * p13)))))))))))));
}
/* for |x| >= 8.0 */
if (ix >= 0x40020000) {
px[i0] = ix;
if (ix < 0x40050400) { /* x < 65 */
r = one / x;
z = r * r;
/*
* poly1
*/
y = r * (one + z * (p1 + z * (p2 + z * (p3 +
z * (p4 + z * (p5 + z * (p6 + z * (p7 +
z * (p8 + z * (p9 + z * (p10 + z * (p11 +
z * (p12 + z * p13)))))))))))));
y -= pio2lo;
} else if (ix < 0x40260000) { /* x < 2**(prec/3+2) */
r = one / x;
z = r * r;
/*
* poly2
*/
y = r * (one + z * (q1 + z * (q2 + z * (q3 + z * (q4 +
z * (q5 + z * (q6 + z * q7)))))));
y -= pio2lo;
} else if (ix < 0x40720000) { /* x < 2**(prec+2) */
y = one / x - pio2lo;
} else if (ix < 0x7fff0000) { /* x < inf */
y = -pio2lo;
} else { /* x is inf or NaN */
if (((ix - 0x7fff0000) | px[1] | px[2] | px[i1]) != 0)
return (x - x);
y = -pio2lo;
}
if (sign == 0)
return (pio2hi - y);
else
return (y - pio2hi);
}
/* now x is between 1/8 and 8 */
px[i0] = ix;
iy = (ix + 0x00000800) & 0x7ffff000;
py[i0] = iy;
py[1] = py[2] = py[i1] = 0;
j = (iy - 0x3ffc0000) >> 12;
if (sign == 0)
s = (x - y) / (one + x * y);
else
s = (y - x) / (one + x * y);
z = s * s;
if (ix == iy)
p = s * (one + z * (q1 + z * (q2 + z * (q3 + z * q4))));
else
p = s * (one + z * (q1 + z * (q2 + z * (q3 + z * (q4 +
z * (q5 + z * (q6 + z * q7)))))));
if (sign == 0) {
r = p + _TBL_atanl_lo[j];
return (r + _TBL_atanl_hi[j]);
} else {
r = p - _TBL_atanl_lo[j];
return (r - _TBL_atanl_hi[j]);
}
}