1 /*
2  * CDDL HEADER START
3  *
4  * The contents of this file are subject to the terms of the
5  * Common Development and Distribution License (the "License").
6  * You may not use this file except in compliance with the License.
7  *
8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9  * or http://www.opensolaris.org/os/licensing.
10  * See the License for the specific language governing permissions
11  * and limitations under the License.
12  *
13  * When distributing Covered Code, include this CDDL HEADER in each
14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
17  * information: Portions Copyright [yyyy] [name of copyright owner]
18  *
19  * CDDL HEADER END
20  */
21 
22 /*
23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
24  */
25 /*
26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
27  * Use is subject to license terms.
28  */
29 
30 /* INDENT OFF */
31 /*
32  * double __k_cexp(double x, int *n);
33  * Returns the exponential of x in the form of 2**n * y, y=__k_cexp(x,&n).
34  *
35  * Method
36  *   1. Argument reduction:
37  *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
38  *	Given x, find r and integer k such that
39  *
40  *               x = k*ln2 + r,  |r| <= 0.5*ln2.
41  *
42  *      Here r will be represented as r = hi-lo for better
43  *	accuracy.
44  *
45  *   2. Approximation of exp(r) by a special rational function on
46  *	the interval [0,0.34658]:
47  *	Write
48  *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
49  *      We use a special Remez algorithm on [0,0.34658] to generate
50  * 	a polynomial of degree 5 to approximate R. The maximum error
51  *	of this polynomial approximation is bounded by 2**-59. In
52  *	other words,
53  *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
54  *  	(where z=r*r, and the values of P1 to P5 are listed below)
55  *	and
56  *	    |                  5          |     -59
57  *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
58  *	    |                             |
59  *	The computation of exp(r) thus becomes
60  *                             2*r
61  *		exp(r) = 1 + -------
62  *		              R - r
63  *                                 r*R1(r)
64  *		       = 1 + r + ----------- (for better accuracy)
65  *		                  2 - R1(r)
66  *	where
67  *			         2       4             10
68  *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
69  *
70  *   3. Return n = k and __k_cexp = exp(r).
71  *
72  * Special cases:
73  *	exp(INF) is INF, exp(NaN) is NaN;
74  *	exp(-INF) is 0, and
75  *	for finite argument, only exp(0)=1 is exact.
76  *
77  * Range and Accuracy:
78  *      When |x| is really big, say |x| > 50000, the accuracy
79  *      is not important because the ultimate result will over or under
80  *      flow. So we will simply replace n = 50000 and r = 0.0. For
81  *      moderate size x, according to an error analysis, the error is
82  *      always less than 1 ulp (unit in the last place).
83  *
84  * Constants:
85  * The hexadecimal values are the intended ones for the following
86  * constants. The decimal values may be used, provided that the
87  * compiler will convert from decimal to binary accurately enough
88  * to produce the hexadecimal values shown.
89  */
90 /* INDENT ON */
91 
92 #include "libm.h"		/* __k_cexp */
93 #include "complex_wrapper.h"	/* HI_WORD/LO_WORD */
94 
95 /* INDENT OFF */
96 static const double
97 one = 1.0,
98 two128 = 3.40282366920938463463e+38,
99 halF[2]	= {
100 	0.5, -0.5,
101 },
102 ln2HI[2] = {
103 	6.93147180369123816490e-01,	/* 0x3fe62e42, 0xfee00000 */
104 	-6.93147180369123816490e-01,	/* 0xbfe62e42, 0xfee00000 */
105 },
106 ln2LO[2] = {
107 	1.90821492927058770002e-10,	/* 0x3dea39ef, 0x35793c76 */
108 	-1.90821492927058770002e-10,	/* 0xbdea39ef, 0x35793c76 */
109 },
110 invln2 = 1.44269504088896338700e+00,	/* 0x3ff71547, 0x652b82fe */
111 P1 = 1.66666666666666019037e-01,	/* 0x3FC55555, 0x5555553E */
112 P2 = -2.77777777770155933842e-03,	/* 0xBF66C16C, 0x16BEBD93 */
113 P3 = 6.61375632143793436117e-05,	/* 0x3F11566A, 0xAF25DE2C */
114 P4 = -1.65339022054652515390e-06,	/* 0xBEBBBD41, 0xC5D26BF1 */
115 P5 = 4.13813679705723846039e-08;	/* 0x3E663769, 0x72BEA4D0 */
116 /* INDENT ON */
117 
118 double
__k_cexp(double x,int * n)119 __k_cexp(double x, int *n) {
120 	double hi = 0.0L, lo = 0.0L, c, t;
121 	int k, xsb;
122 	unsigned hx, lx;
123 
124 	hx = HI_WORD(x);	/* high word of x */
125 	lx = LO_WORD(x);	/* low word of x */
126 	xsb = (hx >> 31) & 1;	/* sign bit of x */
127 	hx &= 0x7fffffff;	/* high word of |x| */
128 
129 	/* filter out non-finite argument */
130 	if (hx >= 0x40e86a00) {	/* if |x| > 50000 */
131 		if (hx >= 0x7ff00000) {
132 			*n = 1;
133 			if (((hx & 0xfffff) | lx) != 0)
134 				return (x + x);	/* NaN */
135 			else
136 				return ((xsb == 0) ? x : 0.0);
137 							/* exp(+-inf)={inf,0} */
138 		}
139 		*n = (xsb == 0) ? 50000 : -50000;
140 		return (one + ln2LO[1] * ln2LO[1]);	/* generate inexact */
141 	}
142 
143 	*n = 0;
144 	/* argument reduction */
145 	if (hx > 0x3fd62e42) {	/* if  |x| > 0.5 ln2 */
146 		if (hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
147 			hi = x - ln2HI[xsb];
148 			lo = ln2LO[xsb];
149 			k = 1 - xsb - xsb;
150 		} else {
151 			k = (int) (invln2 * x + halF[xsb]);
152 			t = k;
153 			hi = x - t * ln2HI[0];
154 					/* t*ln2HI is exact for t<2**20 */
155 			lo = t * ln2LO[0];
156 		}
157 		x = hi - lo;
158 		*n = k;
159 	} else if (hx < 0x3e300000) {	/* when |x|<2**-28 */
160 		return (one + x);
161 	} else
162 		k = 0;
163 
164 	/* x is now in primary range */
165 	t = x * x;
166 	c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
167 	if (k == 0)
168 		return (one - ((x * c) / (c - 2.0) - x));
169 	else {
170 		t = one - ((lo - (x * c) / (2.0 - c)) - hi);
171 		if (k > 128) {
172 			t *= two128;
173 			*n = k - 128;
174 		} else if (k > 0) {
175 			HI_WORD(t) += (k << 20);
176 			*n = 0;
177 		}
178 		return (t);
179 	}
180 }
181