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 */
12 #include <LibConfig.h>
13 #include <sys/EfiCdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: k_rem_pio2.c,v 1.11 2003/01/04 23:43:03 wiz Exp $");
16 #endif
17
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 */
124
125
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 */
133
134 #include "math.h"
135 #include "math_private.h"
136
137 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
138
139 static const double PIo2[] = {
140 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
141 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
142 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
143 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
144 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
145 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
146 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
147 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
148 };
149
150 static const double
151 zero = 0.0,
152 one = 1.0,
153 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
154 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
155
156 int
__kernel_rem_pio2(double * x,double * y,int e0,int nx,int prec,const int32_t * ipio2)157 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
158 {
159 int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
160 double z,fw,f[20],fq[20],q[20];
161
162 /* initialize jk*/
163 jk = init_jk[prec];
164 jp = jk;
165
166 /* determine jx,jv,q0, note that 3>q0 */
167 jx = nx-1;
168 jv = (e0-3)/24; if(jv<0) jv=0;
169 q0 = e0-24*(jv+1);
170
171 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
172 j = jv-jx; m = jx+jk;
173 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
174
175 /* compute q[0],q[1],...q[jk] */
176 for (i=0;i<=jk;i++) {
177 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
178 }
179
180 jz = jk;
181 recompute:
182 /* distill q[] into iq[] reversingly */
183 for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
184 fw = (double)((int32_t)(twon24* z));
185 iq[i] = (int32_t)(z-two24*fw);
186 z = q[j-1]+fw;
187 }
188
189 /* compute n */
190 z = scalbn(z,q0); /* actual value of z */
191 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
192 n = (int32_t) z;
193 z -= (double)n;
194 ih = 0;
195 if(q0>0) { /* need iq[jz-1] to determine n */
196 i = (iq[jz-1]>>(24-q0)); n += i;
197 iq[jz-1] -= i<<(24-q0);
198 ih = iq[jz-1]>>(23-q0);
199 }
200 else if(q0==0) ih = iq[jz-1]>>23;
201 else if(z>=0.5) ih=2;
202
203 if(ih>0) { /* q > 0.5 */
204 n += 1; carry = 0;
205 for(i=0;i<jz ;i++) { /* compute 1-q */
206 j = iq[i];
207 if(carry==0) {
208 if(j!=0) {
209 carry = 1; iq[i] = 0x1000000- j;
210 }
211 } else iq[i] = 0xffffff - j;
212 }
213 if(q0>0) { /* rare case: chance is 1 in 12 */
214 switch(q0) {
215 case 1:
216 iq[jz-1] &= 0x7fffff; break;
217 case 2:
218 iq[jz-1] &= 0x3fffff; break;
219 }
220 }
221 if(ih==2) {
222 z = one - z;
223 if(carry!=0) z -= scalbn(one,q0);
224 }
225 }
226
227 /* check if recomputation is needed */
228 if(z==zero) {
229 j = 0;
230 for (i=jz-1;i>=jk;i--) j |= iq[i];
231 if(j==0) { /* need recomputation */
232 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
233
234 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
235 f[jx+i] = (double) ipio2[jv+i];
236 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
237 q[i] = fw;
238 }
239 jz += k;
240 goto recompute;
241 }
242 }
243
244 /* chop off zero terms */
245 if(z==0.0) {
246 jz -= 1; q0 -= 24;
247 while(iq[jz]==0) { jz--; q0-=24;}
248 } else { /* break z into 24-bit if necessary */
249 z = scalbn(z,-q0);
250 if(z>=two24) {
251 fw = (double)((int32_t)(twon24*z));
252 iq[jz] = (int32_t)(z-two24*fw);
253 jz += 1; q0 += 24;
254 iq[jz] = (int32_t) fw;
255 } else iq[jz] = (int32_t) z ;
256 }
257
258 /* convert integer "bit" chunk to floating-point value */
259 fw = scalbn(one,q0);
260 for(i=jz;i>=0;i--) {
261 q[i] = fw*(double)iq[i]; fw*=twon24;
262 }
263
264 /* compute PIo2[0,...,jp]*q[jz,...,0] */
265 for(i=jz;i>=0;i--) {
266 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
267 fq[jz-i] = fw;
268 }
269
270 /* compress fq[] into y[] */
271 switch(prec) {
272 case 0:
273 fw = 0.0;
274 for (i=jz;i>=0;i--) fw += fq[i];
275 y[0] = (ih==0)? fw: -fw;
276 break;
277 case 1:
278 case 2:
279 fw = 0.0;
280 for (i=jz;i>=0;i--) fw += fq[i];
281 y[0] = (ih==0)? fw: -fw;
282 fw = fq[0]-fw;
283 for (i=1;i<=jz;i++) fw += fq[i];
284 y[1] = (ih==0)? fw: -fw;
285 break;
286 case 3: /* painful */
287 for (i=jz;i>0;i--) {
288 fw = fq[i-1]+fq[i];
289 fq[i] += fq[i-1]-fw;
290 fq[i-1] = fw;
291 }
292 for (i=jz;i>1;i--) {
293 fw = fq[i-1]+fq[i];
294 fq[i] += fq[i-1]-fw;
295 fq[i-1] = fw;
296 }
297 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
298 if(ih==0) {
299 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
300 } else {
301 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
302 }
303 }
304 return n&7;
305 }
306