| blob | f19370d8d0a7bb92c5d17362b31c99381925a376 |
1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
13 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
16 #endif
18 /*
19 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
20 * double x[],y[]; int e0,nx,prec; int ipio2[];
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]
65 *
66 * nx dimension of x[]
67 *
68 * prec an integer indicating the precision:
69 * 0 24 bits (single)
70 * 1 53 bits (double)
71 * 2 64 bits (extended)
72 * 3 113 bits (quad)
73 *
74 * ipio2[]
75 * integer array, contains the (24*i)-th to (24*i+23)-th
76 * bit of 2/pi after binary point. The corresponding
77 * floating value is
78 *
79 * ipio2[i] * 2^(-24(i+1)).
80 *
81 * External function:
82 * double scalbn(), floor();
83 *
84 *
85 * Here is the description of some local variables:
86 *
87 * jk jk+1 is the initial number of terms of ipio2[] needed
88 * in the computation. The recommended value is 2,3,4,
89 * 6 for single, double, extended,and quad.
90 *
91 * jz local integer variable indicating the number of
92 * terms of ipio2[] used.
93 *
94 * jx nx - 1
95 *
96 * jv index for pointing to the suitable ipio2[] for the
97 * computation. In general, we want
98 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
99 * is an integer. Thus
100 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
101 * Hence jv = max(0,(e0-3)/24).
102 *
103 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
104 *
105 * q[] double array with integral value, representing the
106 * 24-bits chunk of the product of x and 2/pi.
107 *
108 * q0 the corresponding exponent of q[0]. Note that the
109 * exponent for q[i] would be q0-24*i.
110 *
111 * PIo2[] double precision array, obtained by cutting pi/2
112 * into 24 bits chunks.
113 *
114 * f[] ipio2[] in floating point
115 *
116 * iq[] integer array by breaking up q[] in 24-bits chunk.
117 *
118 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
119 *
120 * ih integer. If >0 it indicates q[] is >= 0.5, hence
121 * it also indicates the *sign* of the result.
122 *
123 */
126 /*
127 * Constants:
128 * The hexadecimal values are the intended ones for the following
129 * constants. The decimal values may be used, provided that the
130 * compiler will convert from decimal to binary accurately enough
131 * to produce the hexadecimal values shown.
132 */
149 };
151 static const double
157 int
159 {
163 /* initialize jk*/
167 /* determine jx,jv,q0, note that 3>q0 */
172 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
176 /* compute q[0],q[1],...q[jk] */
179 }
182 recompute:
183 /* distill q[] into iq[] reversingly */
188 }
190 /* compute n */
200 }
211 }
213 }
220 }
221 }
225 }
226 }
228 /* check if recomputation is needed */
239 }
242 }
243 }
245 /* chop off zero terms */
257 }
259 /* convert integer "bit" chunk to floating-point value */
263 }
265 /* compute PIo2[0,...,jp]*q[jz,...,0] */
269 }
271 /* compress fq[] into y[] */
292 }
297 }
303 }
304 }
306 }