Annotation of src/lib/libm/src/k_rem_pio2.c, Revision 1.11
1.1 jtc 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
1.9 simonb 8: * software is freely granted, provided that this notice
1.1 jtc 9: * is preserved.
10: * ====================================================
11: */
1.3 jtc 12:
1.8 lukem 13: #include <sys/cdefs.h>
1.6 jtc 14: #if defined(LIBM_SCCS) && !defined(lint)
1.11 ! wiz 15: __RCSID("$NetBSD: k_rem_pio2.c,v 1.10 2002/05/26 22:01:53 wiz Exp $");
1.3 jtc 16: #endif
1.1 jtc 17:
18: /*
19: * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
20: * double x[],y[]; int e0,nx,prec; int ipio2[];
1.9 simonb 21: *
22: * __kernel_rem_pio2 return the last three digits of N with
1.1 jtc 23: * y = x - N*pi/2
24: * so that |y| < pi/2.
25: *
1.9 simonb 26: * The method is to compute the integer (mod 8) and fraction parts of
1.1 jtc 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:
1.9 simonb 35: * x[] The input value (must be positive) is broken into nx
1.1 jtc 36: * pieces of 24-bit integers in double precision format.
1.9 simonb 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
1.1 jtc 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: *
1.11 ! wiz 49: * y[] output result in an array of double precision numbers.
1.1 jtc 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[]
1.9 simonb 75: * integer array, contains the (24*i)-th to (24*i+23)-th
76: * bit of 2/pi after binary point. The corresponding
1.1 jtc 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: *
1.9 simonb 91: * jz local integer variable indicating the number of
92: * terms of ipio2[] used.
1.1 jtc 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
1.9 simonb 112: * into 24 bits chunks.
1.1 jtc 113: *
1.9 simonb 114: * f[] ipio2[] in floating point
1.1 jtc 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:
1.9 simonb 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
1.1 jtc 131: * to produce the hexadecimal values shown.
132: */
133:
1.4 jtc 134: #include "math.h"
135: #include "math_private.h"
1.1 jtc 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:
1.9 simonb 150: static const double
1.1 jtc 151: zero = 0.0,
152: one = 1.0,
153: two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
154: twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
155:
1.10 wiz 156: int
157: __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
1.1 jtc 158: {
1.5 jtc 159: int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
1.1 jtc 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--) {
1.5 jtc 184: fw = (double)((int32_t)(twon24* z));
185: iq[i] = (int32_t)(z-two24*fw);
1.1 jtc 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 */
1.5 jtc 192: n = (int32_t) z;
1.1 jtc 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);
1.9 simonb 199: }
1.1 jtc 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);
1.9 simonb 250: if(z>=two24) {
1.5 jtc 251: fw = (double)((int32_t)(twon24*z));
252: iq[jz] = (int32_t)(z-two24*fw);
1.1 jtc 253: jz += 1; q0 += 24;
1.5 jtc 254: iq[jz] = (int32_t) fw;
255: } else iq[jz] = (int32_t) z ;
1.1 jtc 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];
1.9 simonb 275: y[0] = (ih==0)? fw: -fw;
1.1 jtc 276: break;
277: case 1:
278: case 2:
279: fw = 0.0;
1.9 simonb 280: for (i=jz;i>=0;i--) fw += fq[i];
281: y[0] = (ih==0)? fw: -fw;
1.1 jtc 282: fw = fq[0]-fw;
283: for (i=1;i<=jz;i++) fw += fq[i];
1.9 simonb 284: y[1] = (ih==0)? fw: -fw;
1.1 jtc 285: break;
286: case 3: /* painful */
287: for (i=jz;i>0;i--) {
1.9 simonb 288: fw = fq[i-1]+fq[i];
1.1 jtc 289: fq[i] += fq[i-1]-fw;
290: fq[i-1] = fw;
291: }
292: for (i=jz;i>1;i--) {
1.9 simonb 293: fw = fq[i-1]+fq[i];
1.1 jtc 294: fq[i] += fq[i-1]-fw;
295: fq[i-1] = fw;
296: }
1.9 simonb 297: for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
1.1 jtc 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: }
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