1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
14 static char rcsid[] = "$FreeBSD$";
18 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
19 * double x[],y[]; int e0,nx,prec; int ipio2[];
21 * __kernel_rem_pio2 return the last three digits of N with
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.
31 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
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.
40 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
48 * y[] ouput result in an array of double precision numbers.
49 * The dimension of y[] is:
54 * The actual value is the sum of them. Thus for 113-bit
55 * precison, one may have to do something like:
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];
61 * r_tail = w - (r_head - t);
63 * e0 The exponent of x[0]
67 * prec an integer indicating the precision:
70 * 2 64 bits (extended)
74 * integer array, contains the (24*i)-th to (24*i+23)-th
75 * bit of 2/pi after binary point. The corresponding
78 * ipio2[i] * 2^(-24(i+1)).
81 * double scalbn(), floor();
84 * Here is the description of some local variables:
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.
90 * jz local integer variable indicating the number of
91 * terms of ipio2[] used.
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
99 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
100 * Hence jv = max(0,(e0-3)/24).
102 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
104 * q[] double array with integral value, representing the
105 * 24-bits chunk of the product of x and 2/pi.
107 * q0 the corresponding exponent of q[0]. Note that the
108 * exponent for q[i] would be q0-24*i.
110 * PIo2[] double precision array, obtained by cutting pi/2
111 * into 24 bits chunks.
113 * f[] ipio2[] in floating point
115 * iq[] integer array by breaking up q[] in 24-bits chunk.
117 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
119 * ih integer. If >0 it indicates q[] is >= 0.5, hence
120 * it also indicates the *sign* of the result.
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.
134 #include "math_private.h"
137 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
139 static int init_jk[] = {2,3,4,6};
143 static const double PIo2[] = {
145 static double PIo2[] = {
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 */
164 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
165 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
168 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
170 int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
171 double x[], y[]; int e0,nx,prec; int32_t ipio2[];
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];
181 /* determine jx,jv,q0, note that 3>q0 */
183 jv = (e0-3)/24; if(jv<0) jv=0;
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];
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;
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);
205 z = scalbn(z,q0); /* actual value of z */
206 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
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);
215 else if(q0==0) ih = iq[jz-1]>>23;
216 else if(z>=0.5) ih=2;
218 if(ih>0) { /* q > 0.5 */
220 for(i=0;i<jz ;i++) { /* compute 1-q */
224 carry = 1; iq[i] = 0x1000000- j;
226 } else iq[i] = 0xffffff - j;
228 if(q0>0) { /* rare case: chance is 1 in 12 */
231 iq[jz-1] &= 0x7fffff; break;
233 iq[jz-1] &= 0x3fffff; break;
238 if(carry!=0) z -= scalbn(one,q0);
242 /* check if recomputation is needed */
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 */
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];
259 /* chop off zero terms */
262 while(iq[jz]==0) { jz--; q0-=24;}
263 } else { /* break z into 24-bit if necessary */
266 fw = (double)((int32_t)(twon24*z));
267 iq[jz] = (int32_t)(z-two24*fw);
269 iq[jz] = (int32_t) fw;
270 } else iq[jz] = (int32_t) z ;
273 /* convert integer "bit" chunk to floating-point value */
276 q[i] = fw*(double)iq[i]; fw*=twon24;
279 /* compute PIo2[0,...,jp]*q[jz,...,0] */
281 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
285 /* compress fq[] into y[] */
289 for (i=jz;i>=0;i--) fw += fq[i];
290 y[0] = (ih==0)? fw: -fw;
295 for (i=jz;i>=0;i--) fw += fq[i];
296 y[0] = (ih==0)? fw: -fw;
298 for (i=1;i<=jz;i++) fw += fq[i];
299 y[1] = (ih==0)? fw: -fw;
301 case 3: /* painful */
312 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
314 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
316 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;