cc68a136 |
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 | |
13 | #if defined(LIBM_SCCS) && !defined(lint) |
14 | static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $"; |
15 | #endif |
16 | |
17 | /* |
18 | * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
19 | * double x[],y[]; int e0,nx,prec; int ipio2[]; |
20 | * |
21 | * __kernel_rem_pio2 return the last three digits of N with |
22 | * y = x - N*pi/2 |
23 | * so that |y| < pi/2. |
24 | * |
25 | * The method is to compute the integer (mod 8) and fraction parts of |
26 | * (2/pi)*x without doing the full multiplication. In general we |
27 | * skip the part of the product that are known to be a huge integer ( |
28 | * more accurately, = 0 mod 8 ). Thus the number of operations are |
29 | * independent of the exponent of the input. |
30 | * |
31 | * (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
32 | * |
33 | * Input parameters: |
34 | * x[] The input value (must be positive) is broken into nx |
35 | * pieces of 24-bit integers in double precision format. |
36 | * x[i] will be the i-th 24 bit of x. The scaled exponent |
37 | * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
38 | * match x's up to 24 bits. |
39 | * |
40 | * Example of breaking a double positive z into x[0]+x[1]+x[2]: |
41 | * e0 = ilogb(z)-23 |
42 | * z = scalbn(z,-e0) |
43 | * for i = 0,1,2 |
44 | * x[i] = floor(z) |
45 | * z = (z-x[i])*2**24 |
46 | * |
47 | * |
48 | * y[] ouput result in an array of double precision numbers. |
49 | * The dimension of y[] is: |
50 | * 24-bit precision 1 |
51 | * 53-bit precision 2 |
52 | * 64-bit precision 2 |
53 | * 113-bit precision 3 |
54 | * The actual value is the sum of them. Thus for 113-bit |
55 | * precison, one may have to do something like: |
56 | * |
57 | * long double t,w,r_head, r_tail; |
58 | * t = (long double)y[2] + (long double)y[1]; |
59 | * w = (long double)y[0]; |
60 | * r_head = t+w; |
61 | * r_tail = w - (r_head - t); |
62 | * |
63 | * e0 The exponent of x[0] |
64 | * |
65 | * nx dimension of x[] |
66 | * |
67 | * prec an integer indicating the precision: |
68 | * 0 24 bits (single) |
69 | * 1 53 bits (double) |
70 | * 2 64 bits (extended) |
71 | * 3 113 bits (quad) |
72 | * |
73 | * ipio2[] |
74 | * integer array, contains the (24*i)-th to (24*i+23)-th |
75 | * bit of 2/pi after binary point. The corresponding |
76 | * floating value is |
77 | * |
78 | * ipio2[i] * 2^(-24(i+1)). |
79 | * |
80 | * External function: |
81 | * double scalbn(), floor(); |
82 | * |
83 | * |
84 | * Here is the description of some local variables: |
85 | * |
86 | * jk jk+1 is the initial number of terms of ipio2[] needed |
87 | * in the computation. The recommended value is 2,3,4, |
88 | * 6 for single, double, extended,and quad. |
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 | */ |
123 | |
124 | |
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 | */ |
132 | |
133 | #include "math.h" |
134 | #include "math_private.h" |
135 | |
136 | #ifdef __STDC__ |
137 | static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ |
138 | #else |
139 | static int init_jk[] = {2,3,4,6}; |
140 | #endif |
141 | |
142 | #ifdef __STDC__ |
143 | static const double PIo2[] = { |
144 | #else |
145 | static double PIo2[] = { |
146 | #endif |
147 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
148 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
149 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
150 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
151 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
152 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
153 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
154 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
155 | }; |
156 | |
157 | #ifdef __STDC__ |
158 | static const double |
159 | #else |
160 | static double |
161 | #endif |
162 | zero = 0.0, |
163 | one = 1.0, |
164 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
165 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
166 | |
167 | #ifdef __STDC__ |
168 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) |
169 | #else |
170 | int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
171 | double x[], y[]; int e0,nx,prec; int32_t ipio2[]; |
172 | #endif |
173 | { |
174 | int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
175 | double z,fw,f[20],fq[20],q[20]; |
176 | |
177 | /* initialize jk*/ |
178 | jk = init_jk[prec]; |
179 | jp = jk; |
180 | |
181 | /* determine jx,jv,q0, note that 3>q0 */ |
182 | jx = nx-1; |
183 | jv = (e0-3)/24; if(jv<0) jv=0; |
184 | q0 = e0-24*(jv+1); |
185 | |
186 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
187 | j = jv-jx; m = jx+jk; |
188 | for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; |
189 | |
190 | /* compute q[0],q[1],...q[jk] */ |
191 | for (i=0;i<=jk;i++) { |
192 | for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; |
193 | } |
194 | |
195 | jz = jk; |
196 | recompute: |
197 | /* distill q[] into iq[] reversingly */ |
198 | for(i=0,j=jz,z=q[jz];j>0;i++,j--) { |
199 | fw = (double)((int32_t)(twon24* z)); |
200 | iq[i] = (int32_t)(z-two24*fw); |
201 | z = q[j-1]+fw; |
202 | } |
203 | |
204 | /* compute n */ |
205 | z = scalbn(z,q0); /* actual value of z */ |
206 | z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ |
207 | n = (int32_t) z; |
208 | z -= (double)n; |
209 | ih = 0; |
210 | if(q0>0) { /* need iq[jz-1] to determine n */ |
211 | i = (iq[jz-1]>>(24-q0)); n += i; |
212 | iq[jz-1] -= i<<(24-q0); |
213 | ih = iq[jz-1]>>(23-q0); |
214 | } |
215 | else if(q0==0) ih = iq[jz-1]>>23; |
216 | else if(z>=0.5) ih=2; |
217 | |
218 | if(ih>0) { /* q > 0.5 */ |
219 | n += 1; carry = 0; |
220 | for(i=0;i<jz ;i++) { /* compute 1-q */ |
221 | j = iq[i]; |
222 | if(carry==0) { |
223 | if(j!=0) { |
224 | carry = 1; iq[i] = 0x1000000- j; |
225 | } |
226 | } else iq[i] = 0xffffff - j; |
227 | } |
228 | if(q0>0) { /* rare case: chance is 1 in 12 */ |
229 | switch(q0) { |
230 | case 1: |
231 | iq[jz-1] &= 0x7fffff; break; |
232 | case 2: |
233 | iq[jz-1] &= 0x3fffff; break; |
234 | } |
235 | } |
236 | if(ih==2) { |
237 | z = one - z; |
238 | if(carry!=0) z -= scalbn(one,q0); |
239 | } |
240 | } |
241 | |
242 | /* check if recomputation is needed */ |
243 | if(z==zero) { |
244 | j = 0; |
245 | for (i=jz-1;i>=jk;i--) j |= iq[i]; |
246 | if(j==0) { /* need recomputation */ |
247 | for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ |
248 | |
249 | for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ |
250 | f[jx+i] = (double) ipio2[jv+i]; |
251 | for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; |
252 | q[i] = fw; |
253 | } |
254 | jz += k; |
255 | goto recompute; |
256 | } |
257 | } |
258 | |
259 | /* chop off zero terms */ |
260 | if(z==0.0) { |
261 | jz -= 1; q0 -= 24; |
262 | while(iq[jz]==0) { jz--; q0-=24;} |
263 | } else { /* break z into 24-bit if necessary */ |
264 | z = scalbn(z,-q0); |
265 | if(z>=two24) { |
266 | fw = (double)((int32_t)(twon24*z)); |
267 | iq[jz] = (int32_t)(z-two24*fw); |
268 | jz += 1; q0 += 24; |
269 | iq[jz] = (int32_t) fw; |
270 | } else iq[jz] = (int32_t) z ; |
271 | } |
272 | |
273 | /* convert integer "bit" chunk to floating-point value */ |
274 | fw = scalbn(one,q0); |
275 | for(i=jz;i>=0;i--) { |
276 | q[i] = fw*(double)iq[i]; fw*=twon24; |
277 | } |
278 | |
279 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
280 | for(i=jz;i>=0;i--) { |
281 | for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; |
282 | fq[jz-i] = fw; |
283 | } |
284 | |
285 | /* compress fq[] into y[] */ |
286 | switch(prec) { |
287 | case 0: |
288 | fw = 0.0; |
289 | for (i=jz;i>=0;i--) fw += fq[i]; |
290 | y[0] = (ih==0)? fw: -fw; |
291 | break; |
292 | case 1: |
293 | case 2: |
294 | fw = 0.0; |
295 | for (i=jz;i>=0;i--) fw += fq[i]; |
296 | y[0] = (ih==0)? fw: -fw; |
297 | fw = fq[0]-fw; |
298 | for (i=1;i<=jz;i++) fw += fq[i]; |
299 | y[1] = (ih==0)? fw: -fw; |
300 | break; |
301 | case 3: /* painful */ |
302 | for (i=jz;i>0;i--) { |
303 | fw = fq[i-1]+fq[i]; |
304 | fq[i] += fq[i-1]-fw; |
305 | fq[i-1] = fw; |
306 | } |
307 | for (i=jz;i>1;i--) { |
308 | fw = fq[i-1]+fq[i]; |
309 | fq[i] += fq[i-1]-fw; |
310 | fq[i-1] = fw; |
311 | } |
312 | for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; |
313 | if(ih==0) { |
314 | y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
315 | } else { |
316 | y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
317 | } |
318 | } |
319 | return n&7; |
320 | } |