long double
cbrtl(long double x) {
long double s, t, r, w, y;
double dx, dy;
int *py = (int *) &dy;
int n, m, m3, sx;
if (!finitel(x))
return (x + x);
if (iszerol(x))
return (x);
sx = signbitl(x);
x = fabsl(x);
n = ilogbl(x);
m = n / 3;
m3 = m + m + m;
y = scalbnl(x, -m3);
dx = (double) y;
dy = cbrt(dx);
py[1 - n0] += 2;
if (py[1 - n0] == 0)
py[n0] += 1;
/* one step newton iteration to 113 bits with error < 0.667ulps */
t = (long double) dy;
t = scalbnl(t, m);
s = t * t;
r = x / s;
w = t + t;
r = (r - t) / (w + r);
t += t * r;
return (sx == 0 ? t : -t);
}
ldcomplex
catanl(ldcomplex z) {
ldcomplex ans;
long double x, y, t1, ax, ay, t;
int hx, hy, ix, iy;
x = LD_RE(z);
y = LD_IM(z);
ax = fabsl(x);
ay = fabsl(y);
hx = HI_XWORD(x);
hy = HI_XWORD(y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* x is inf or NaN */
if (ix >= 0x7fff0000) {
if (isinfl(x)) {
LD_RE(ans) = pi_2;
LD_IM(ans) = zero;
} else {
LD_RE(ans) = x + x;
if (iszerol(y) || (isinfl(y)))
LD_IM(ans) = zero;
else
LD_IM(ans) = (fabsl(y) - ay) / (fabsl(y) - ay);
}
} else if (iy >= 0x7fff0000) {
/* y is inf or NaN */
if (isinfl(y)) {
LD_RE(ans) = pi_2;
LD_IM(ans) = zero;
} else {
LD_RE(ans) = (fabsl(x) - ax) / (fabsl(x) - ax);
LD_IM(ans) = y;
}
} else if (iszerol(x)) {
/* 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 */
t = one - ay;
if (ay == one) {
/* y=1: catan(0,1)=(0,+inf) with 1/0 signal */
LD_IM(ans) = ay / ax;
LD_RE(ans) = zero;
} else if (ay > one) { /* y>1 */
LD_IM(ans) = half * log1pl(two / (-t));
LD_RE(ans) = pi_2;
} else { /* y<1 */
LD_IM(ans) = half * log1pl((ay + ay) / t);
LD_RE(ans) = zero;
}
} else if (ay < E * (one + ax)) {
/* INDENT OFF */
/*
* Tiny y (relative to 1+|x|)
* |y| < E*(1+|x|)
* where E=2**-29, -35, -60 for 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
*
* y
* (when x < 2**-60, t = ----------- )
* (y-1)*(y-1)
*/
/* INDENT ON */
if (ay == zero)
LD_IM(ans) = ay;
else {
t1 = ay - one;
if (ix < 0x3fc30000)
t = ay / (t1 * t1);
else if (ix > 0x403b0000)
t = (ay / ax) / ax;
else
t = ay / (ax * ax + t1 * t1);
LD_IM(ans) = t * (one - two * t);
}
if (ix < 0x3fff0000)
LD_RE(ans) = atanl(ax);
else
LD_RE(ans) = half * atan2l(two, (one - ax) * (one +
one / ax));
} else if (ay > Einv * (one + ax)) {
/* 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 */
LD_RE(ans) = pi_2;
t = (ay / (ay - one)) / (ay - one);
LD_IM(ans) = t * (one - (t + t));
} else if (ay == one) {
/* 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 */
LD_RE(ans) = half * atan2l(two, -ax);
if (ax < E)
LD_IM(ans) = half * (ln2 - logl(ax));
else {
t = two / ax;
LD_IM(ans) = 0.25L * log1pl(t * t);
}
} else if (ax > Einv * Einv) {
/* 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 */
LD_RE(ans) = pi_2;
t = ((ay / ax) / (one + ((ay - one) / ax) * ((ay - one) /
ax))) / ax;
LD_IM(ans) = t * (one - (t + t));
} else if (ax < E * E * E * E) {
/* 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*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 */
LD_RE(ans) = half * atan2l(ax + ax, (one - ay) * (one + ay));
if (ay > one) /* y>1 */
LD_IM(ans) = half * log1pl(two / (ay - one));
else /* y<1 */
LD_IM(ans) = half * log1pl((ay + ay) / (one - ay));
} 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 */
t = one - ay;
if (iy >= 0x3ffe0000 && iy < 0x40000000) {
/* y close to 1 */
LD_RE(ans) = half * (atan2l((ax + ax), (t * (one +
ay) - ax * ax)));
} else if (ix >= 0x3ffe0000 && ix < 0x40000000) {
/* x close to 1 */
LD_RE(ans) = half * atan2l((ax + ax), ((one - ax) *
(one + ax) - ay * ay));
} else
LD_RE(ans) = half * atan2l((ax + ax), ((one - ax *
ax) - ay * ay));
LD_IM(ans) = 0.25L * log1pl((4.0L * ay) / (ax * ax + t * t));
}
if (hx < 0)
LD_RE(ans) = -LD_RE(ans);
if (hy < 0)
LD_IM(ans) = -LD_IM(ans);
return (ans);
}
