static inline long double
divl(long double a, long double b, long double c, long double d,
long double e, long double f)
{
long double inv, r;
float fr, fw;
_2sumF(a, c);
b = b + c;
_2sumF(d, f);
e = e + f;
inv = 1 / (d + e);
r = (a + b) * inv;
fr = r;
r = fr;
fw = d + e;
e = d - fw + e;
d = fw;
r = r + (a - d * r + b - e * r) * inv;
return r;
}
long double complex
clogl(long double complex z)
{
long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl;
long double sh, sl, t;
long double x, y, v;
uint16_t hax, hay;
int kx, ky;
ENTERIT(long double complex);
x = creall(z);
y = cimagl(z);
v = atan2l(y, x);
ax = fabsl(x);
ay = fabsl(y);
if (ax < ay) {
t = ax;
ax = ay;
ay = t;
}
GET_LDBL_EXPSIGN(hax, ax);
kx = hax - 16383;
GET_LDBL_EXPSIGN(hay, ay);
ky = hay - 16383;
/* Handle NaNs and Infs using the general formula. */
if (kx == MAX_EXP || ky == MAX_EXP)
RETURNI(CMPLXL(logl(hypotl(x, y)), v));
/* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */
if (ax == 1) {
if (ky < (MIN_EXP - 1) / 2)
RETURNI(CMPLXL((ay / 2) * ay, v));
RETURNI(CMPLXL(log1pl(ay * ay) / 2, v));
}
/* Avoid underflow when ax is not small. Also handle zero args. */
if (kx - ky > MANT_DIG || ay == 0)
RETURNI(CMPLXL(logl(ax), v));
/* Avoid overflow. */
if (kx >= MAX_EXP - 1)
RETURNI(CMPLXL(logl(hypotl(x * 0x1p-16382L, y * 0x1p-16382L)) +
(MAX_EXP - 2) * ln2l_lo + (MAX_EXP - 2) * ln2_hi, v));
if (kx >= (MAX_EXP - 1) / 2)
RETURNI(CMPLXL(logl(hypotl(x, y)), v));
/* Reduce inaccuracies and avoid underflow when ax is denormal. */
if (kx <= MIN_EXP - 2)
RETURNI(CMPLXL(logl(hypotl(x * 0x1p16383L, y * 0x1p16383L)) +
(MIN_EXP - 2) * ln2l_lo + (MIN_EXP - 2) * ln2_hi, v));
/* Avoid remaining underflows (when ax is small but not denormal). */
if (ky < (MIN_EXP - 1) / 2 + MANT_DIG)
RETURNI(CMPLXL(logl(hypotl(x, y)), v));
/* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */
t = (long double)(ax * (MULT_REDUX + 1));
axh = (long double)(ax - t) + t;
axl = ax - axh;
ax2h = ax * ax;
ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl;
t = (long double)(ay * (MULT_REDUX + 1));
ayh = (long double)(ay - t) + t;
ayl = ay - ayh;
ay2h = ay * ay;
ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl;
/*
* When log(|z|) is far from 1, accuracy in calculating the sum
* of the squares is not very important since log() reduces
* inaccuracies. We depended on this to use the general
* formula when log(|z|) is very far from 1. When log(|z|) is
* moderately far from 1, we go through the extra-precision
* calculations to reduce branches and gain a little accuracy.
*
* When |z| is near 1, we subtract 1 and use log1p() and don't
* leave it to log() to subtract 1, since we gain at least 1 bit
* of accuracy in this way.
*
* When |z| is very near 1, subtracting 1 can cancel almost
* 3*MANT_DIG bits. We arrange that subtracting 1 is exact in
* doubled precision, and then do the rest of the calculation
* in sloppy doubled precision. Although large cancellations
* often lose lots of accuracy, here the final result is exact
* in doubled precision if the large calculation occurs (because
* then it is exact in tripled precision and the cancellation
* removes enough bits to fit in doubled precision). Thus the
* result is accurate in sloppy doubled precision, and the only
* significant loss of accuracy is when it is summed and passed
* to log1p().
*/
sh = ax2h;
sl = ay2h;
_2sumF(sh, sl);
if (sh < 0.5 || sh >= 3)
RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v));
sh -= 1;
_2sum(sh, sl);
_2sum(ax2l, ay2l);
/* Briggs-Kahan algorithm (except we discard the final low term): */
_2sum(sh, ax2l);
_2sum(sl, ay2l);
t = ax2l + sl;
_2sumF(sh, t);
RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v));
}
