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
119 __k_cexp(double x, int *n) {
120 double hi, lo, 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 }