1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /*
13 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
14 * double x[],y[]; int e0,nx,prec; int ipio2[];
15 *
16 * __kernel_rem_pio2 return the last three digits of N with
17 * y = x - N*pi/2
18 * so that |y| < pi/2.
19 *
20 * The method is to compute the integer (mod 8) and fraction parts of
21 * (2/pi)*x without doing the full multiplication. In general we
22 * skip the part of the product that are known to be a huge integer (
23 * more accurately, = 0 mod 8 ). Thus the number of operations are
24 * independent of the exponent of the input.
25 *
26 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
27 *
28 * Input parameters:
29 * x[] The input value (must be positive) is broken into nx
30 * pieces of 24-bit integers in double precision format.
31 * x[i] will be the i-th 24 bit of x. The scaled exponent
32 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
33 * match x's up to 24 bits.
34 *
35 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
36 * e0 = ilogb(z)-23
37 * z = scalbn(z,-e0)
38 * for i = 0,1,2
39 * x[i] = floor(z)
40 * z = (z-x[i])*2**24
41 *
42 *
43 * y[] ouput result in an array of double precision numbers.
44 * The dimension of y[] is:
45 * 24-bit precision 1
46 * 53-bit precision 2
47 * 64-bit precision 2
48 * 113-bit precision 3
49 * The actual value is the sum of them. Thus for 113-bit
50 * precison, one may have to do something like:
51 *
52 * long double t,w,r_head, r_tail;
53 * t = (long double)y[2] + (long double)y[1];
54 * w = (long double)y[0];
55 * r_head = t+w;
56 * r_tail = w - (r_head - t);
57 *
58 * e0 The exponent of x[0]
59 *
60 * nx dimension of x[]
61 *
62 * prec an integer indicating the precision:
63 * 0 24 bits (single)
64 * 1 53 bits (double)
65 * 2 64 bits (extended)
66 * 3 113 bits (quad)
67 *
68 * ipio2[]
69 * integer array, contains the (24*i)-th to (24*i+23)-th
70 * bit of 2/pi after binary point. The corresponding
71 * floating value is
72 *
73 * ipio2[i] * 2^(-24(i+1)).
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 recommended value is 2,3,4,
83 * 6 for single, double, extended,and quad.
84 *
85 * jz local integer variable indicating the number of
86 * terms of ipio2[] used.
87 *
88 * jx nx - 1
89 *
90 * jv index for pointing to the suitable ipio2[] for the
91 * computation. In general, we want
92 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
93 * is an integer. Thus
94 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
95 * Hence jv = max(0,(e0-3)/24).
96 *
97 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
98 *
99 * q[] double array with integral value, representing the
100 * 24-bits chunk of the product of x and 2/pi.
101 *
102 * q0 the corresponding exponent of q[0]. Note that the
103 * exponent for q[i] would be q0-24*i.
104 *
105 * PIo2[] double precision array, obtained by cutting pi/2
106 * into 24 bits chunks.
107 *
108 * f[] ipio2[] in floating point
109 *
110 * iq[] integer array by breaking up q[] in 24-bits chunk.
111 *
112 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
113 *
114 * ih integer. If >0 it indicates q[] is >= 0.5, hence
115 * it also indicates the *sign* of the result.
116 *
117 */
118
119
120 /*
121 * Constants:
122 * The hexadecimal values are the intended ones for the following
123 * constants. The decimal values may be used, provided that the
124 * compiler will convert from decimal to binary accurately enough
125 * to produce the hexadecimal values shown.
126 */
127
128 #include "math_libm.h"
129 #include "math_private.h"
130
131 #include "SDL_assert.h"
132
133 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
134
135 static const double PIo2[] = {
136 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
137 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
138 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
139 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
140 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
141 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
142 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
143 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
144 };
145
146 static const double
147 zero = 0.0,
148 one = 1.0,
149 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
150 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
151
__kernel_rem_pio2(const double * x,double * y,int e0,int nx,const unsigned int prec,const int32_t * ipio2)152 int32_t attribute_hidden __kernel_rem_pio2(const double *x, double *y, int e0, int nx, const unsigned int prec, const int32_t *ipio2)
153 {
154 int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
155 double z,fw,f[20],fq[20],q[20];
156
157 if (nx < 1) {
158 return 0;
159 }
160
161 /* initialize jk*/
162 SDL_assert(prec < SDL_arraysize(init_jk));
163 jk = init_jk[prec];
164 SDL_assert(jk > 0);
165 jp = jk;
166
167 /* determine jx,jv,q0, note that 3>q0 */
168 jx = nx-1;
169 jv = (e0-3)/24; if(jv<0) jv=0;
170 q0 = e0-24*(jv+1);
171
172 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
173 j = jv-jx; m = jx+jk;
174 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
175 if ((m+1) < SDL_arraysize(f)) {
176 SDL_memset(&f[m+1], 0, sizeof (f) - ((m+1) * sizeof (f[0])));
177 }
178
179 /* compute q[0],q[1],...q[jk] */
180 for (i=0;i<=jk;i++) {
181 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
182 q[i] = fw;
183 }
184
185 jz = jk;
186 recompute:
187 /* distill q[] into iq[] reversingly */
188 for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
189 fw = (double)((int32_t)(twon24* z));
190 iq[i] = (int32_t)(z-two24*fw);
191 z = q[j-1]+fw;
192 }
193 if (jz < SDL_arraysize(iq)) {
194 SDL_memset(&iq[jz], 0, sizeof (q) - (jz * sizeof (iq[0])));
195 }
196
197 /* compute n */
198 z = scalbn(z,q0); /* actual value of z */
199 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
200 n = (int32_t) z;
201 z -= (double)n;
202 ih = 0;
203 if(q0>0) { /* need iq[jz-1] to determine n */
204 i = (iq[jz-1]>>(24-q0)); n += i;
205 iq[jz-1] -= i<<(24-q0);
206 ih = iq[jz-1]>>(23-q0);
207 }
208 else if(q0==0) ih = iq[jz-1]>>23;
209 else if(z>=0.5) ih=2;
210
211 if(ih>0) { /* q > 0.5 */
212 n += 1; carry = 0;
213 for(i=0;i<jz ;i++) { /* compute 1-q */
214 j = iq[i];
215 if(carry==0) {
216 if(j!=0) {
217 carry = 1; iq[i] = 0x1000000- j;
218 }
219 } else iq[i] = 0xffffff - j;
220 }
221 if(q0>0) { /* rare case: chance is 1 in 12 */
222 switch(q0) {
223 case 1:
224 iq[jz-1] &= 0x7fffff; break;
225 case 2:
226 iq[jz-1] &= 0x3fffff; break;
227 }
228 }
229 if(ih==2) {
230 z = one - z;
231 if(carry!=0) z -= scalbn(one,q0);
232 }
233 }
234
235 /* check if recomputation is needed */
236 if(z==zero) {
237 j = 0;
238 for (i=jz-1;i>=jk;i--) j |= iq[i];
239 if(j==0) { /* need recomputation */
240 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
241
242 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
243 f[jx+i] = (double) ipio2[jv+i];
244 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
245 q[i] = fw;
246 }
247 jz += k;
248 goto recompute;
249 }
250 }
251
252 /* chop off zero terms */
253 if(z==0.0) {
254 jz -= 1; q0 -= 24;
255 SDL_assert(jz >= 0);
256 while(iq[jz]==0) { jz--; SDL_assert(jz >= 0); q0-=24;}
257 } else { /* break z into 24-bit if necessary */
258 z = scalbn(z,-q0);
259 if(z>=two24) {
260 fw = (double)((int32_t)(twon24*z));
261 iq[jz] = (int32_t)(z-two24*fw);
262 jz += 1; q0 += 24;
263 iq[jz] = (int32_t) fw;
264 } else iq[jz] = (int32_t) z ;
265 }
266
267 /* convert integer "bit" chunk to floating-point value */
268 fw = scalbn(one,q0);
269 for(i=jz;i>=0;i--) {
270 q[i] = fw*(double)iq[i]; fw*=twon24;
271 }
272
273 /* compute PIo2[0,...,jp]*q[jz,...,0] */
274 for(i=jz;i>=0;i--) {
275 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
276 fq[jz-i] = fw;
277 }
278 if ((jz+1) < SDL_arraysize(f)) {
279 SDL_memset(&fq[jz+1], 0, sizeof (fq) - ((jz+1) * sizeof (fq[0])));
280 }
281
282 /* compress fq[] into y[] */
283 switch(prec) {
284 case 0:
285 fw = 0.0;
286 for (i=jz;i>=0;i--) fw += fq[i];
287 y[0] = (ih==0)? fw: -fw;
288 break;
289 case 1:
290 case 2:
291 fw = 0.0;
292 for (i=jz;i>=0;i--) fw += fq[i];
293 y[0] = (ih==0)? fw: -fw;
294 fw = fq[0]-fw;
295 for (i=1;i<=jz;i++) fw += fq[i];
296 y[1] = (ih==0)? fw: -fw;
297 break;
298 case 3: /* painful */
299 for (i=jz;i>0;i--) {
300 fw = fq[i-1]+fq[i];
301 fq[i] += fq[i-1]-fw;
302 fq[i-1] = fw;
303 }
304 for (i=jz;i>1;i--) {
305 fw = fq[i-1]+fq[i];
306 fq[i] += fq[i-1]-fw;
307 fq[i-1] = fw;
308 }
309 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
310 if(ih==0) {
311 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
312 } else {
313 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
314 }
315 }
316 return n&7;
317 }
318