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/*-
* Copyright (c) 2013 Bruce D. Evans
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include <complex.h>
#include <float.h>
#include "fpmath.h"
#include "math.h"
#include "math_private.h"
#define MANT_DIG FLT_MANT_DIG
#define MAX_EXP FLT_MAX_EXP
#define MIN_EXP FLT_MIN_EXP
static const float
ln2f_hi = 6.9314575195e-1, /* 0xb17200.0p-24 */
ln2f_lo = 1.4286067653e-6; /* 0xbfbe8e.0p-43 */
float complex
clogf(float complex z)
{
float_t ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl, sh, sl, t;
float x, y, v;
uint32_t hax, hay;
int kx, ky;
x = crealf(z);
y = cimagf(z);
v = atan2f(y, x);
ax = fabsf(x);
ay = fabsf(y);
if (ax < ay) {
t = ax;
ax = ay;
ay = t;
}
GET_FLOAT_WORD(hax, ax);
kx = (hax >> 23) - 127;
GET_FLOAT_WORD(hay, ay);
ky = (hay >> 23) - 127;
/* Handle NaNs and Infs using the general formula. */
if (kx == MAX_EXP || ky == MAX_EXP)
return (CMPLXF(logf(hypotf(x, y)), v));
/* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */
if (hax == 0x3f800000) {
if (ky < (MIN_EXP - 1) / 2)
return (CMPLXF((ay / 2) * ay, v));
return (CMPLXF(log1pf(ay * ay) / 2, v));
}
/* Avoid underflow when ax is not small. Also handle zero args. */
if (kx - ky > MANT_DIG || hay == 0)
return (CMPLXF(logf(ax), v));
/* Avoid overflow. */
if (kx >= MAX_EXP - 1)
return (CMPLXF(logf(hypotf(x * 0x1p-126F, y * 0x1p-126F)) +
(MAX_EXP - 2) * ln2f_lo + (MAX_EXP - 2) * ln2f_hi, v));
if (kx >= (MAX_EXP - 1) / 2)
return (CMPLXF(logf(hypotf(x, y)), v));
/* Reduce inaccuracies and avoid underflow when ax is denormal. */
if (kx <= MIN_EXP - 2)
return (CMPLXF(logf(hypotf(x * 0x1p127F, y * 0x1p127F)) +
(MIN_EXP - 2) * ln2f_lo + (MIN_EXP - 2) * ln2f_hi, v));
/* Avoid remaining underflows (when ax is small but not denormal). */
if (ky < (MIN_EXP - 1) / 2 + MANT_DIG)
return (CMPLXF(logf(hypotf(x, y)), v));
/* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */
t = (float)(ax * (0x1p12F + 1));
axh = (float)(ax - t) + t;
axl = ax - axh;
ax2h = ax * ax;
ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl;
t = (float)(ay * (0x1p12F + 1));
ayh = (float)(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 < 0.5F || sh >= 3)
return (CMPLXF(logf(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);
return (CMPLXF(log1pf(ay2l + t + sh) / 2, v));
}
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