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2 /* @(#)k_rem_pio2.c 1.3 95/01/18 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
14 #ifndef lint
15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_rem_pio2.c,v 1.11 2008/02/25 11:43:20 bde Exp $";
16 #endif
18 /*
19 * __kernel_rem_pio2(x,y,e0,nx,prec)
20 * double x[],y[]; int e0,nx,prec;
21 *
22 * __kernel_rem_pio2 return the last three digits of N with
23 * y = x - N*pi/2
24 * so that |y| < pi/2.
25 *
26 * The method is to compute the integer (mod 8) and fraction parts of
27 * (2/pi)*x without doing the full multiplication. In general we
28 * skip the part of the product that are known to be a huge integer (
29 * more accurately, = 0 mod 8 ). Thus the number of operations are
30 * independent of the exponent of the input.
31 *
32 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
33 *
34 * Input parameters:
35 * x[] The input value (must be positive) is broken into nx
36 * pieces of 24-bit integers in double precision format.
37 * x[i] will be the i-th 24 bit of x. The scaled exponent
38 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
39 * match x's up to 24 bits.
40 *
41 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
42 * e0 = ilogb(z)-23
43 * z = scalbn(z,-e0)
44 * for i = 0,1,2
45 * x[i] = floor(z)
46 * z = (z-x[i])*2**24
47 *
48 *
49 * y[] output result in an array of double precision numbers.
50 * The dimension of y[] is:
51 * 24-bit precision 1
52 * 53-bit precision 2
53 * 64-bit precision 2
54 * 113-bit precision 3
55 * The actual value is the sum of them. Thus for 113-bit
56 * precison, one may have to do something like:
57 *
58 * long double t,w,r_head, r_tail;
59 * t = (long double)y[2] + (long double)y[1];
60 * w = (long double)y[0];
61 * r_head = t+w;
62 * r_tail = w - (r_head - t);
63 *
64 * e0 The exponent of x[0]. Must be <= 16360 or you need to
65 * expand the ipio2 table.
66 *
67 * nx dimension of x[]
68 *
69 * prec an integer indicating the precision:
70 * 0 24 bits (single)
71 * 1 53 bits (double)
72 * 2 64 bits (extended)
73 * 3 113 bits (quad)
74 *
75 * External function:
76 * double scalbn(), floor();
77 *
78 *
79 * Here is the description of some local variables:
80 *
81 * jk jk+1 is the initial number of terms of ipio2[] needed
82 * in the computation. The minimum and recommended value
83 * for jk is 3,4,4,6 for single, double, extended, and quad.
84 * jk+1 must be 2 larger than you might expect so that our
85 * recomputation test works. (Up to 24 bits in the integer
86 * part (the 24 bits of it that we compute) and 23 bits in
87 * the fraction part may be lost to cancellation before we
88 * recompute.)
89 *
90 * jz local integer variable indicating the number of
91 * terms of ipio2[] used.
92 *
93 * jx nx - 1
94 *
95 * jv index for pointing to the suitable ipio2[] for the
96 * computation. In general, we want
97 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
98 * is an integer. Thus
99 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
100 * Hence jv = max(0,(e0-3)/24).
101 *
102 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
103 *
104 * q[] double array with integral value, representing the
105 * 24-bits chunk of the product of x and 2/pi.
106 *
107 * q0 the corresponding exponent of q[0]. Note that the
108 * exponent for q[i] would be q0-24*i.
109 *
110 * PIo2[] double precision array, obtained by cutting pi/2
111 * into 24 bits chunks.
112 *
113 * f[] ipio2[] in floating point
114 *
115 * iq[] integer array by breaking up q[] in 24-bits chunk.
116 *
117 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
118 *
119 * ih integer. If >0 it indicates q[] is >= 0.5, hence
120 * it also indicates the *sign* of the result.
121 *
122 */
125 /*
126 * Constants:
127 * The hexadecimal values are the intended ones for the following
128 * constants. The decimal values may be used, provided that the
129 * compiler will convert from decimal to binary accurately enough
130 * to produce the hexadecimal values shown.
131 */
133 #include <float.h>
140 /*
141 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
142 *
143 * integer array, contains the (24*i)-th to (24*i+23)-th
144 * bit of 2/pi after binary point. The corresponding
145 * floating value is
146 *
147 * ipio2[i] * 2^(-24(i+1)).
148 *
149 * NB: This table must have at least (e0-3)/24 + jk terms.
150 * For quad precision (e0 <= 16360, jk = 6), this is 686.
151 */
165 #if LDBL_MAX_EXP > 1024
166 #if LDBL_MAX_EXP > 16384
168 #endif
273 #endif
275 };
286 };
288 static const double
294 int
296 {
300 /* initialize jk*/
304 /* determine jx,jv,q0, note that 3>q0 */
309 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
313 /* compute q[0],q[1],...q[jk] */
317 }
320 recompute:
321 /* distill q[] into iq[] reversingly */
326 }
328 /* compute n */
338 }
349 }
351 }
358 }
359 }
363 }
364 }
366 /* check if recomputation is needed */
377 }
380 }
381 }
383 /* chop off zero terms */
395 }
397 /* convert integer "bit" chunk to floating-point value */
401 }
403 /* compute PIo2[0,...,jp]*q[jz,...,0] */
407 }
409 /* compress fq[] into y[] */
431 }
436 }
442 }
443 }
445 }