693 Opensource replacement of sunwlibm

Reviewed by: Igor Kozhukhov ikozhukhov@gmail.com
Reviewed by: Keith M Wesolowski <keith.wesolowski@joyent.com>
Reviewed by: Richard Lowe <richlowe@richlowe.net>
Approved by: Dan McDonald <danmcd@omniti.com>

1 files changed, 266 insertions, 0 deletions

diff --git a/usr/src/lib/libm/common/C/expm1.c b/usr/src/lib/libm/common/C/expm1.c
new file mode 100644
index 0000000000..fad9d55bc7
--- /dev/null
+++ b/usr/src/lib/libm/common/C/expm1.c
@@ -0,0 +1,266 @@
+/*
+ * 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 2006 Sun Microsystems, Inc.  All rights reserved.
+ * Use is subject to license terms.
+ */
+
+#pragma weak expm1 = __expm1
+
+/* INDENT OFF */
+/*
+ * expm1(x)
+ * Returns exp(x)-1, the exponential of x minus 1.
+ *
+ * Method
+ *   1. Arugment reduction:
+ *	Given x, find r and integer k such that
+ *
+ *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
+ *
+ *      Here a correction term c will be computed to compensate
+ *	the error in r when rounded to a floating-point number.
+ *
+ *   2. Approximating expm1(r) by a special rational function on
+ *	the interval [0,0.34658]:
+ *	Since
+ *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
+ *	we define R1(r*r) by
+ *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
+ *	That is,
+ *	    R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+ *		     = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+ *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
+ *      We use a special Reme algorithm on [0,0.347] to generate
+ * 	a polynomial of degree 5 in r*r to approximate R1. The
+ *	maximum error of this polynomial approximation is bounded
+ *	by 2**-61. In other words,
+ *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+ *	where 	Q1  =  -1.6666666666666567384E-2,
+ * 		Q2  =   3.9682539681370365873E-4,
+ * 		Q3  =  -9.9206344733435987357E-6,
+ * 		Q4  =   2.5051361420808517002E-7,
+ * 		Q5  =  -6.2843505682382617102E-9;
+ *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
+ *	with error bounded by
+ *	    |                  5           |     -61
+ *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
+ *	    |                              |
+ *
+ *	expm1(r) = exp(r)-1 is then computed by the following
+ * 	specific way which minimize the accumulation rounding error:
+ *			       2     3
+ *			      r     r    [ 3 - (R1 + R1*r/2)  ]
+ *	      expm1(r) = r + --- + --- * [--------------------]
+ *		              2     2    [ 6 - r*(3 - R1*r/2) ]
+ *
+ *	To compensate the error in the argument reduction, we use
+ *		expm1(r+c) = expm1(r) + c + expm1(r)*c
+ *			   ~ expm1(r) + c + r*c
+ *	Thus c+r*c will be added in as the correction terms for
+ *	expm1(r+c). Now rearrange the term to avoid optimization
+ * 	screw up:
+ *		        (      2                                    2 )
+ *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
+ *	 expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+ *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
+ *                      (                                             )
+ *
+ *		   = r - E
+ *   3. Scale back to obtain expm1(x):
+ *	From step 1, we have
+ *	   expm1(x) = either 2^k*[expm1(r)+1] - 1
+ *		    = or     2^k*[expm1(r) + (1-2^-k)]
+ *   4. Implementation notes:
+ *	(A). To save one multiplication, we scale the coefficient Qi
+ *	     to Qi*2^i, and replace z by (x^2)/2.
+ *	(B). To achieve maximum accuracy, we compute expm1(x) by
+ *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x != inf)
+ *	  (ii)  if k=0, return r-E
+ *	  (iii) if k=-1, return 0.5*(r-E)-0.5
+ *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
+ *					else	     return  1.0+2.0*(r-E);
+ *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
+ *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
+ *	  (vii) return 2^k(1-((E+2^-k)-r))
+ *
+ * Special cases:
+ *	expm1(INF) is INF, expm1(NaN) is NaN;
+ *	expm1(-INF) is -1, and
+ *	for finite argument, only expm1(0)=0 is exact.
+ *
+ * Accuracy:
+ *	according to an error analysis, the error is always less than
+ *	1 ulp (unit in the last place).
+ *
+ * Misc. info.
+ *	For IEEE double
+ *	    if x >  7.09782712893383973096e+02 then expm1(x) overflow
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+/* INDENT ON */
+
+#include "libm_synonyms.h"	/* __expm1 */
+#include "libm_macros.h"
+#include <math.h>
+
+static const double xxx[] = {
+/* one */		 1.0,
+/* huge */		 1.0e+300,
+/* tiny */		 1.0e-300,
+/* o_threshold */	 7.09782712893383973096e+02,	/* 40862E42 FEFA39EF */
+/* ln2_hi */		 6.93147180369123816490e-01,	/* 3FE62E42 FEE00000 */
+/* ln2_lo */		 1.90821492927058770002e-10,	/* 3DEA39EF 35793C76 */
+/* invln2 */		 1.44269504088896338700e+00,	/* 3FF71547 652B82FE */
+/* scaled coefficients related to expm1 */
+/* Q1 */		-3.33333333333331316428e-02,	/* BFA11111 111110F4 */
+/* Q2 */		 1.58730158725481460165e-03,	/* 3F5A01A0 19FE5585 */
+/* Q3 */		-7.93650757867487942473e-05,	/* BF14CE19 9EAADBB7 */
+/* Q4 */		 4.00821782732936239552e-06,	/* 3ED0CFCA 86E65239 */
+/* Q5 */		-2.01099218183624371326e-07	/* BE8AFDB7 6E09C32D */
+};
+#define	one		xxx[0]
+#define	huge		xxx[1]
+#define	tiny		xxx[2]
+#define	o_threshold	xxx[3]
+#define	ln2_hi		xxx[4]
+#define	ln2_lo		xxx[5]
+#define	invln2		xxx[6]
+#define	Q1		xxx[7]
+#define	Q2		xxx[8]
+#define	Q3		xxx[9]
+#define	Q4		xxx[10]
+#define	Q5		xxx[11]
+
+double
+expm1(double x) {
+	double y, hi, lo, c = 0.0L, t, e, hxs, hfx, r1;
+	int k, xsb;
+	unsigned hx;
+
+	hx = ((unsigned *) &x)[HIWORD];		/* high word of x */
+	xsb = hx & 0x80000000;			/* sign bit of x */
+	if (xsb == 0)
+		y = x;
+	else
+		y = -x;				/* y = |x| */
+	hx &= 0x7fffffff;			/* high word of |x| */
+
+	/* filter out huge and non-finite argument */
+	/* for example exp(38)-1 is approximately 3.1855932e+16 */
+	if (hx >= 0x4043687A) {
+		/* if |x|>=56*ln2 (~38.8162...) */
+		if (hx >= 0x40862E42) {		/* if |x|>=709.78... -> inf */
+			if (hx >= 0x7ff00000) {
+				if (((hx & 0xfffff) | ((int *) &x)[LOWORD])
+					!= 0)
+					return (x * x);	/* + -> * for Cheetah */
+				else
+					/* exp(+-inf)={inf,-1} */
+					return (xsb == 0 ? x : -1.0);
+			}
+			if (x > o_threshold)
+				return (huge * huge);	/* overflow */
+		}
+		if (xsb != 0) {		/* x < -56*ln2, return -1.0 w/inexact */
+			if (x + tiny < 0.0)		/* raise inexact */
+				return (tiny - one);	/* return -1 */
+		}
+	}
+
+	/* argument reduction */
+	if (hx > 0x3fd62e42) {			/* if  |x| > 0.5 ln2 */
+		if (hx < 0x3FF0A2B2) {		/* and |x| < 1.5 ln2 */
+			if (xsb == 0) {		/* positive number */
+				hi = x - ln2_hi;
+				lo = ln2_lo;
+				k = 1;
+			} else {
+				/* negative number */
+				hi = x + ln2_hi;
+				lo = -ln2_lo;
+				k = -1;
+			}
+		} else {
+			/* |x| > 1.5 ln2 */
+			k = (int) (invln2 * x + (xsb == 0 ? 0.5 : -0.5));
+			t = k;
+			hi = x - t * ln2_hi;	/* t*ln2_hi is exact here */
+			lo = t * ln2_lo;
+		}
+		x = hi - lo;
+		c = (hi - x) - lo; /* still at |x| > 0.5 ln2 */
+	} else if (hx < 0x3c900000) {
+		/* when |x|<2**-54, return x */
+		t = huge + x;		/* return x w/inexact when x != 0 */
+		return (x - (t - (huge + x)));
+	} else
+		/* |x| <= 0.5 ln2 */
+		k = 0;
+
+	/* x is now in primary range */
+	hfx = 0.5 * x;
+	hxs = x * hfx;
+	r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
+	t = 3.0 - r1 * hfx;
+	e = hxs * ((r1 - t) / (6.0 - x * t));
+	if (k == 0) /* |x| <= 0.5 ln2 */
+		return (x - (x * e - hxs));
+	else {		/* |x| > 0.5 ln2 */
+		e = (x * (e - c) - c);
+		e -= hxs;
+		if (k == -1)
+			return (0.5 * (x - e) - 0.5);
+		if (k == 1) {
+			if (x < -0.25)
+				return (-2.0 * (e - (x + 0.5)));
+			else
+				return (one + 2.0 * (x - e));
+		}
+		if (k <= -2 || k > 56) {	/* suffice to return exp(x)-1 */
+			y = one - (e - x);
+			((int *) &y)[HIWORD] += k << 20;
+			return (y - one);
+		}
+		t = one;
+		if (k < 20) {
+			((int *) &t)[HIWORD] = 0x3ff00000 - (0x200000 >> k);
+							/* t = 1 - 2^-k */
+			y = t - (e - x);
+			((int *) &y)[HIWORD] += k << 20;
+		} else {
+			((int *) &t)[HIWORD] = (0x3ff - k) << 20; /* 2^-k */
+			y = x - (e + t);
+			y += one;
+			((int *) &y)[HIWORD] += k << 20;
+		}
+	}
+	return (y);
+}