xref: /illumos-gate/usr/src/lib/libm/common/complex/k_clog_r.c (revision 25c28e83)

/*
 * CDDL HEADER START
 *
 * The contents of this file are subject to the terms of the
 * Common Development and Distribution License (the "License").
 * You may not use this file except in compliance with the License.
 *
 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
 * or http://www.opensolaris.org/os/licensing.
 * See the License for the specific language governing permissions
 * and limitations under the License.
 *
 * When distributing Covered Code, include this CDDL HEADER in each
 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
 * If applicable, add the following below this CDDL HEADER, with the
 * fields enclosed by brackets "[]" replaced with your own identifying
 * information: Portions Copyright [yyyy] [name of copyright owner]
 *
 * CDDL HEADER END
 */
/*
 * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
 */
/*
 * Copyright 2005 Sun Microsystems, Inc.  All rights reserved.
 * Use is subject to license terms.
 */

#include "libm.h"		/* __k_clog_r */
#include "complex_wrapper.h"

/* INDENT OFF */
/*
 * double __k_clog_r(double x, double y, double *e);
 *
 * Compute real part of complex natural logarithm of x+iy in extra precision
 *
 * __k_clog_r returns log(hypot(x, y)) with a correction term e.
 *
 * Accuracy: 70 bits
 *
 * Method.
 * Let Z = x*x + y*y.  Z can be normalized as Z = 2^N * z,  1 <= z < 2.
 * We further break down z into 1 + zk + zh + zt, where
 *	zk = K*(2^-7) matches z to 7.5 significant bits, 0 <= K <= 2^(-7)-1
 *	zh = (z-zk) rounded to 24 bits
 *	zt = (z-zk-zh) rounded.
 *
 *          z - (1+zk)        (zh+zt)
 * Let s = ------------ = ---------------, then
 *          z + (1+zk)     2(1+zk)+zh+zt
 *                                                       z
 * log(Z) = N*log2 + log(z) = N*log2 + log(1+zk) + log(------)
 *                                                      1+zk
 *                                    1+s
 *         = N*log2 + log(1+zk) + log(---)
 *                                    1-s
 *
 *                                     1     3    1     5
 *        = N*log2 + log(1+zk) + 2s + -- (2s)  + -- (2s)  + ...
 *                                    12         80
 *
 * Note 1. For IEEE double precision,  a seven degree odd polynomial
 *		2s + P1*(2s)^3 + P2*(2s)^5 + P3*(2s)^7
 *         is generated by a special remez algorithm to
 *         approx log((1+s)/(1-s)) accurte to 72 bits.
 * Note 2. 2s can be computed accurately as s2h+s2t by
 *	   r = 2/((zh+zt)+2(1+zk))
 *	   s2 = r*(zh+zt)
 *	   s2h = s2 rounded to float;  v = 0.5*s2h;
 *	   s2t = r*((((zh-s2h*(1+zk))-v*zh)+zt)-v*zt)
 */
/* INDENT ON */

static const double
zero  = 0.0,
half  = 0.5,
two   = 2.0,
two120 = 1.32922799578491587290e+36,  /* 2^120 */
ln2_h  = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
ln2_t  = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
P1 =  .083333333333333351554108717377986202224765262191125,
P2 =  .01249999999819227552330700574633767185896464873834375,
P3 =  .0022321938458645656605471559987512516234702284287265625;

/*
* T[2k, 2k+1] = log(1+k*2^-7) for k = 0, ..., 2^7 - 1,
* with T[2k] * 2^40 is an int
*/

static const double TBL_log1k[] = {
0.00000000000000000000e+00,  0.00000000000000000000e+00,
7.78214044203195953742e-03,  2.29894100462035112076e-14,
1.55041865355087793432e-02,  4.56474807636434698847e-13,
2.31670592811497044750e-02,  3.84673753843363762372e-13,
3.07716586667083902285e-02,  4.52981425779092882775e-14,
3.83188643018002039753e-02,  3.36395218465265063278e-13,
4.58095360309016541578e-02,  3.92549008891706208826e-13,
5.32445145181554835290e-02,  6.56799336898521766515e-13,
6.06246218158048577607e-02,  6.29984819938331143924e-13,
6.79506619080711971037e-02,  4.36552290856295281946e-13,
7.52234212368421140127e-02,  7.45411685916941618656e-13,
8.24436692109884461388e-02,  8.61451293608781447223e-14,
8.96121586893059429713e-02,  3.81189648692113819551e-13,
9.67296264579999842681e-02,  5.51128027471986918274e-13,
1.03796793680885457434e-01,  7.58107392301637643358e-13,
1.10814366339582193177e-01,  7.07921017612766061755e-13,
1.17783035655520507134e-01,  8.62947404296943765415e-13,
1.24703478500123310369e-01,  8.33925494898414856118e-13,
1.31576357788617315236e-01,  1.01957352237084734958e-13,
1.38402322858382831328e-01,  7.36304357708705134617e-13,
1.45182009843665582594e-01,  8.32314688404647202319e-13,
1.51916042025732167531e-01,  1.09807540998552379211e-13,
1.58605030175749561749e-01,  8.89022343972466269900e-13,
1.65249572894936136436e-01,  3.71026439894104998399e-13,
1.71850256926518341061e-01,  1.40881279371111350341e-13,
1.78407657472234859597e-01,  5.83437522462346671423e-13,
1.84922338493379356805e-01,  6.32635858668445232946e-13,
1.91394852999110298697e-01,  5.19155912393432989209e-13,
1.97825743329303804785e-01,  6.16075577558872326221e-13,
2.04215541428311553318e-01,  3.79338185766902218086e-13,
2.10564769106895255391e-01,  4.54382278998146218219e-13,
2.16873938300523150247e-01,  9.12093724991498410553e-14,
2.23143551314024080057e-01,  1.85675709597960106615e-13,
2.29374101064422575291e-01,  4.23254700234549300166e-13,
2.35566071311950508971e-01,  8.16400106820959292914e-13,
2.41719936886511277407e-01,  6.33890736899755317832e-13,
2.47836163904139539227e-01,  4.41717553713155466566e-13,
2.53915209980732470285e-01,  2.30973852175869394892e-13,
2.59957524436686071567e-01,  2.39995404842117353465e-13,
2.65963548496984003577e-01,  1.53937761744554075681e-13,
2.71933715483100968413e-01,  5.40790418614551497411e-13,
2.77868451003087102436e-01,  3.69203750820800887027e-13,
2.83768173129828937817e-01,  8.15660529536291275782e-13,
2.89633292582948342897e-01,  9.43339818951269030846e-14,
2.95464212893421063200e-01,  4.14813187042585679830e-13,
3.01261330577290209476e-01,  8.71571536970835103739e-13,
3.07025035294827830512e-01,  8.40315630479242455758e-14,
3.12755710003330023028e-01,  5.66865358290073900922e-13,
3.18453731118097493891e-01,  4.37121919574291444278e-13,
3.24119468653407238889e-01,  8.04737201185162774515e-13,
3.29753286371669673827e-01,  7.98307987877335024112e-13,
3.35355541920762334485e-01,  3.75495772572598557174e-13,
3.40926586970454081893e-01,  1.39128412121975659358e-13,
3.46466767346100823488e-01,  1.07757430375726404546e-13,
3.51976423156884266064e-01,  2.93918591876480007730e-13,
3.57455888921322184615e-01,  4.81589611172320539489e-13,
3.62905493689140712377e-01,  2.27740761140395561986e-13,
3.68325561158599157352e-01,  1.08495696229679121506e-13,
3.73716409792905324139e-01,  6.78756682315870616582e-13,
3.79078352934811846353e-01,  1.57612037739694350287e-13,
3.84411698910298582632e-01,  3.34571026954408237380e-14,
3.89716751139530970249e-01,  4.94243121138567024911e-13,
3.94993808240542421117e-01,  3.26556988969071456956e-13,
4.00243164126550254878e-01,  4.62452051668403792833e-13,
4.05465108107819105498e-01,  3.45276479520397708744e-13,
4.10659924984429380856e-01,  8.39005077851830734139e-13,
4.15827895143593195826e-01,  1.17769787513692141889e-13,
4.20969294643327884842e-01,  8.01751287156832458079e-13,
4.26084395310681429692e-01,  2.18633432932159103190e-13,
4.31173464818130014464e-01,  2.41326394913331314894e-13,
4.36236766774527495727e-01,  3.90574622098307022265e-13,
4.41274560804231441580e-01,  6.43787909737320689684e-13,
4.46287102628048160113e-01,  3.71351419195920213229e-13,
4.51274644138720759656e-01,  7.37825488412103968058e-13,
4.56237433480964682531e-01,  6.22911850193784704748e-13,
4.61175715121498797089e-01,  6.71369279138460114513e-13,
4.66089729924533457961e-01,  6.57665976858006147528e-14,
4.70979715218163619284e-01,  6.27393263311115598424e-13,
4.75845904869856894948e-01,  1.07019317621142549209e-13,
4.80688529345570714213e-01,  1.81193463664411114729e-13,
4.85507815781602403149e-01,  9.84046527823262695501e-14,
4.90303988044615834951e-01,  5.78003198945402769376e-13,
4.95077266797125048470e-01,  7.26466128212511528295e-13,
4.99827869555701909121e-01,  7.47420700205478712293e-13,
5.04556010751912253909e-01,  4.83033149495532022300e-13,
5.09261901789614057634e-01,  1.93889170049107088943e-13,
5.13945751101346104406e-01,  8.88212395185718544720e-13,
5.18607764207445143256e-01,  6.00488896640545761201e-13,
5.23248143764249107335e-01,  2.98729182044413286731e-13,
5.27867089620485785417e-01,  3.56599696633478298092e-13,
5.32464798869114019908e-01,  3.57823965912763837621e-13,
5.37041465896436420735e-01,  4.47233831757482468946e-13,
5.41597282432121573947e-01,  6.22797629172251525649e-13,
5.46132437597407260910e-01,  7.28389472720657362987e-13,
5.50647117952394182794e-01,  2.68096466152116723636e-13,
5.55141507539701706264e-01,  7.99886451312335479470e-13,
5.59615787935399566777e-01,  2.31194938380053776320e-14,
5.64070138284478161950e-01,  3.24804121719935740729e-13,
5.68504735351780254859e-01,  8.88457219261483317716e-13,
5.72919753561109246220e-01,  6.76262872317054154667e-13,
5.77315365034337446559e-01,  4.86157758891509033842e-13,
5.81691739634152327199e-01,  4.70155322075549811780e-13,
5.86049045003164792433e-01,  4.13416470738355643357e-13,
5.90387446602107957006e-01,  6.84176364159146659095e-14,
5.94707107746216934174e-01,  4.75855340044306376333e-13,
5.99008189645246602595e-01,  8.36796786747576938145e-13,
6.03290851438032404985e-01,  5.18573553063418286042e-14,
6.07555250224322662689e-01,  2.19132812293400917731e-13,
6.11801541105705837253e-01,  2.87066276408616768331e-13,
6.16029877214714360889e-01,  7.99658758518543977451e-13,
6.20240409751204424538e-01,  6.53104313776336534177e-13,
6.24433288011459808331e-01,  4.33692711555820529733e-13,
6.28608659421843185555e-01,  5.30952189118357790115e-13,
6.32766669570628437214e-01,  4.09392332186786656392e-13,
6.36907462236194987781e-01,  8.74243839148582888557e-13,
6.41031179420679109171e-01,  2.52181884568428814231e-13,
6.45137961372711288277e-01,  8.73413388168702670246e-13,
6.49227946624705509748e-01,  4.04309142530119209805e-13,
6.53301272011958644725e-01,  7.86994033233553225797e-13,
6.57358072708120744210e-01,  2.39285932153437645135e-13,
6.61398482245203922503e-01,  1.61085757539324585156e-13,
6.65422632544505177066e-01,  5.85271884362515112697e-13,
6.69430653942072240170e-01,  5.57027128793880294600e-13,
6.73422675211440946441e-01,  7.25773856816637653180e-13,
6.77398823590920073912e-01,  8.86066898134949155668e-13,
6.81359224807238206267e-01,  6.64862680714687006264e-13,
6.85304003098281100392e-01,  6.38316151706465171657e-13,
6.89233281238557538018e-01,  2.51442307283760746611e-13,
};

/*
 * Compute N*log2 + log(1+zk+zh+zt) in extra precision
 */
static double k_log_NKz(int N, int K, double zh, double *zt)
{
	double y, r, w, s2, s2h, s2t, t, zk, v, P;

	((int *)&zk)[HIWORD] = 0x3ff00000 + (K << 13);
	((int *)&zk)[LOWORD] = 0;
	t  = zh + (*zt);
	r = two / (t + two * zk);
	s2h = s2 = r * t;
	((int *)&s2h)[LOWORD] &= 0xe0000000;
	v = half * s2h;
	w = s2 * s2;
	s2t = r * ((((zh - s2h * zk) - v * zh) + (*zt)) - v * (*zt));
	P = s2t + (w * s2) * ((P1 + w * P2) + (w * w) * P3);
	P += N * ln2_t + TBL_log1k[K + K + 1];
	t = N*ln2_h + TBL_log1k[K+K];
	y = t + (P + s2h);
	P -= ((y - t) - s2h);
	*zt = P;
	return (y);
}

double
__k_clog_r(double x, double y, double *er)
{
	double t1, t2, t3, t4, tk, z, wh, w, zh, zk;
	int n, k, ix, iy, iz, nx, ny, nz, i, j;
	unsigned lx, ly;

	ix = (((int *)&x)[HIWORD]) & ~0x80000000;
	lx = ((unsigned *)&x)[LOWORD];
	iy = (((int *)&y)[HIWORD]) & ~0x80000000;
	ly = ((unsigned *)&y)[LOWORD];
	y = fabs(y); x = fabs(x);
	if (ix < iy || (ix == iy && lx < ly)) {		/* force x >= y */
		tk = x;  x = y;   y = tk;
		n = ix, ix = iy; iy = n;
		n = lx, lx = ly; ly = n;
	}
	*er = zero;
	nx = ix >> 20; ny = iy >> 20;
	if (nx >= 0x7ff) {   	/* x or y is Inf or NaN */
		if (ISINF(ix, lx))
			return (x);
		else if (ISINF(iy, ly))
			return (y);
		else
			return (x+y);
	}
/*
 * for tiny y (double y < 2^-35, extended y < 2^-46, quad y < 2^-70):
 * 	log(sqrt(1+y^2)) = (y^2)/2 - (y^4)/8 + ... ~= (y^2)/2
 */
	if ((((ix - 0x3ff00000) | lx) == 0) && ny < (0x3ff - 35))  {
		t2 = y * y;
		if (ny >= 565) {	/* compute er = tail of t2 */
			((int *)&wh)[HIWORD] =  iy;
			((unsigned *)&wh)[LOWORD] = ly & 0xf8000000;
			*er = half * ((y - wh) * (y + wh) - (t2 - wh * wh));
		}
		return (half * t2);
	}
/*
 * x or y is subnormal or zero
 */
	if (nx == 0) {
		if ((ix | lx) == 0)
			return (-1.0 / x);
		else {
			x *= two120;
			y *= two120;
			ix = ((int *)&x)[HIWORD];
			lx = ((unsigned *)&x)[LOWORD];
			iy = ((int *)&y)[HIWORD];
			ly = ((unsigned *)&y)[LOWORD];
			nx = (ix >> 20) - 120;
			ny = (iy >> 20) - 120;
			/* guard subnormal flush to 0 */
			if ((ix | lx) == 0)
				return (-1.0 / x);
		}
	} else if (ny == 0) {	/* y subnormal, scale it */
		y *= two120;
		iy = ((int *)&y)[HIWORD];
		ly = ((unsigned *)&y)[LOWORD];
		ny = (iy >> 20) - 120;
	}
	n = nx - ny;
/*
 * return log(x) when y is zero or x >> y so that
 * log(x) ~ log(sqrt(x*x+y*y)) to 27 extra bits
 * (n > 62 for double, 78 for i386 extended, 122 for quad)
 */
	if (n > 62 || (iy | ly) == 0) {
		i = (0x000fffff & ix) | 0x3ff00000;	/* normalize x */
		((int *)&x)[HIWORD] = i;
		i += 0x1000;
		((int *)&zk)[HIWORD] = i & 0xffffe000;
		((int *)&zk)[LOWORD] = 0;  /* zk matches 7.5 bits of x */
		z = x - zk;
		zh = (double)((float)z);
		i >>= 13;
		k = i & 0x7f;	/* index of zk */
		n = nx - 0x3ff;
		*er = z - zh;
		if (i >> 17) {	/* if zk = 2.0, adjust scaling */
			n += 1;
			zh *= 0.5;  *er *= 0.5;
		}
		w = k_log_NKz(n, k, zh, er);
	} else {
/*
 * compute z = x*x + y*y
 */
		ix = (ix & 0xfffff) | 0x3ff00000;
		iy = (iy & 0xfffff) | (0x3ff00000 - (n << 20));
		((int *)&x)[HIWORD] = ix; ((int *)&y)[HIWORD] = iy;
		t1 = x * x; t2 = y * y;
		j = ((lx >> 26) + 1) >> 1;
		((int *)&wh)[HIWORD] = ix + (j >> 5);
		((unsigned *)&wh)[LOWORD] = (j << 27);
		z = t1+t2;
/*
 * higher precision simulation x*x = t1 + t3, y*y = t2 + t4
 */
		tk = wh - x;
		t3 = tk * tk - (two * wh * tk - (wh * wh - t1));
		j = ((ly >> 26) + 1) >> 1;
		((int *)&wh)[HIWORD] = iy + (j >> 5);
		((unsigned *)&wh)[LOWORD] = (j << 27);
		tk = wh - y;
		t4 = tk * tk - (two * wh * tk - (wh * wh - t2));
/*
 * find zk matches z to 7.5 bits
 */
		nx -= 0x3ff;
		iz = ((int *)&z)[HIWORD] + 0x1000;
		k = (iz >> 13) & 0x7f;
		nz = (iz >> 20) - 0x3ff;
		((int *)&zk)[HIWORD] = iz & 0xffffe000;
		((int *)&zk)[LOWORD] = 0;
/*
 * order t1,t2,t3,t4 according to their size
 */
		if (t2 >= fabs(t3)) {
			if (fabs(t3) < fabs(t4)) {
				wh = t3;  t3 = t4; t4 = wh;
			}
		} else {
			wh = t2; t2 = t3; t3 = wh;
		}
/*
 * higher precision simulation: x * x + y * y = t1 + t2 + t3 + t4
 * = zk (7 bits) + zh (24 bits) + *er (tail) and call k_log_NKz
 */
		tk = t1 - zk;
		zh = ((tk + t2) + t3) + t4;
		((int *)&zh)[LOWORD] &= 0xe0000000;
		w = fabs(zh);
		if (w >= fabs(t2))
			*er = (((tk - zh) + t2) + t3) + t4;
		else {
			if (n == 0) {
				wh = half * zk;
				wh = (t1 - wh) - (wh - t2);
			} else
				wh = tk + t2;
			if (w >= fabs(t3))
				*er = ((wh - zh) + t3) + t4;
			else {
				z = t3;
				t3 += t4;
				t4 -= t3 - z;
				if (w >= fabs(t3))
					*er = ((wh - zh) + t3) + t4;
				else
					*er = ((wh + t3) - zh) + t4;
			}
		}
		if (nz == 3) {zh *= 0.125; *er *= 0.125; }
		if (nz == 2) {zh *= 0.25; *er *= 0.25; }
		if (nz == 1) {zh *= half; *er *= half; }
		nz += nx + nx;
		w = half * k_log_NKz(nz, k, zh, er);
		*er *= half;
	}
	return (w);
}