2 /* @(#)k_rem_pio2.c 1.3 95/01/18 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
11 * ====================================================
15 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
16 * double x[],y[]; int e0,nx,prec; int ipio2[];
18 * __kernel_rem_pio2 return the last three digits of N with
22 * The method is to compute the integer (mod 8) and fraction parts of
23 * (2/pi)*x without doing the full multiplication. In general we
24 * skip the part of the product that are known to be a huge integer (
25 * more accurately, = 0 mod 8 ). Thus the number of operations are
26 * independent of the exponent of the input.
28 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
31 * x[] The input value (must be positive) is broken into nx
32 * pieces of 24-bit integers in double precision format.
33 * x[i] will be the i-th 24 bit of x. The scaled exponent
34 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
35 * match x's up to 24 bits.
37 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
45 * y[] ouput result in an array of double precision numbers.
46 * The dimension of y[] is:
51 * The actual value is the sum of them. Thus for 113-bit
52 * precison, one may have to do something like:
54 * long double t,w,r_head, r_tail;
55 * t = (long double)y[2] + (long double)y[1];
56 * w = (long double)y[0];
58 * r_tail = w - (r_head - t);
60 * e0 The exponent of x[0]
64 * prec an integer indicating the precision:
67 * 2 64 bits (extended)
71 * integer array, contains the (24*i)-th to (24*i+23)-th
72 * bit of 2/pi after binary point. The corresponding
75 * ipio2[i] * 2^(-24(i+1)).
78 * double scalbn(), floor();
81 * Here is the description of some local variables:
83 * jk jk+1 is the initial number of terms of ipio2[] needed
84 * in the computation. The recommended value is 2,3,4,
85 * 6 for single, double, extended,and quad.
87 * jz local integer variable indicating the number of
88 * terms of ipio2[] used.
92 * jv index for pointing to the suitable ipio2[] for the
93 * computation. In general, we want
94 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
96 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
97 * Hence jv = max(0,(e0-3)/24).
99 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
101 * q[] double array with integral value, representing the
102 * 24-bits chunk of the product of x and 2/pi.
104 * q0 the corresponding exponent of q[0]. Note that the
105 * exponent for q[i] would be q0-24*i.
107 * PIo2[] double precision array, obtained by cutting pi/2
108 * into 24 bits chunks.
110 * f[] ipio2[] in floating point
112 * iq[] integer array by breaking up q[] in 24-bits chunk.
114 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
116 * ih integer. If >0 it indicates q[] is >= 0.5, hence
117 * it also indicates the *sign* of the result.
124 * The hexadecimal values are the intended ones for the following
125 * constants. The decimal values may be used, provided that the
126 * compiler will convert from decimal to binary accurately enough
127 * to produce the hexadecimal values shown.
131 #include "math_private.h"
133 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
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 */
149 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
150 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
152 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
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];
161 /* determine jx,jv,q0, note that 3>q0 */
163 jv = (e0-3)/24; if(jv<0) jv=0;
166 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
167 j = jv-jx; m = jx+jk;
168 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
170 /* compute q[0],q[1],...q[jk] */
171 for (i=0;i<=jk;i++) {
172 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
177 /* distill q[] into iq[] reversingly */
178 for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
179 fw = (double)((int32_t)(twon24* z));
180 iq[i] = (int32_t)(z-two24*fw);
185 z = scalbn(z,q0); /* actual value of z */
186 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
190 if(q0>0) { /* need iq[jz-1] to determine n */
191 i = (iq[jz-1]>>(24-q0)); n += i;
192 iq[jz-1] -= i<<(24-q0);
193 ih = iq[jz-1]>>(23-q0);
195 else if(q0==0) ih = iq[jz-1]>>23;
196 else if(z>=0.5) ih=2;
198 if(ih>0) { /* q > 0.5 */
200 for(i=0;i<jz ;i++) { /* compute 1-q */
204 carry = 1; iq[i] = 0x1000000- j;
206 } else iq[i] = 0xffffff - j;
208 if(q0>0) { /* rare case: chance is 1 in 12 */
211 iq[jz-1] &= 0x7fffff; break;
213 iq[jz-1] &= 0x3fffff; break;
218 if(carry!=0) z -= scalbn(one,q0);
222 /* check if recomputation is needed */
225 for (i=jz-1;i>=jk;i--) j |= iq[i];
226 if(j==0) { /* need recomputation */
227 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
229 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
230 f[jx+i] = (double) ipio2[jv+i];
231 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
239 /* chop off zero terms */
242 while(iq[jz]==0) { jz--; q0-=24;}
243 } else { /* break z into 24-bit if necessary */
246 fw = (double)((int32_t)(twon24*z));
247 iq[jz] = (int32_t)(z-two24*fw);
249 iq[jz] = (int32_t) fw;
250 } else iq[jz] = (int32_t) z ;
253 /* convert integer "bit" chunk to floating-point value */
256 q[i] = fw*(double)iq[i]; fw*=twon24;
259 /* compute PIo2[0,...,jp]*q[jz,...,0] */
261 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
265 /* compress fq[] into y[] */
269 for (i=jz;i>=0;i--) fw += fq[i];
270 y[0] = (ih==0)? fw: -fw;
275 for (i=jz;i>=0;i--) fw += fq[i];
276 y[0] = (ih==0)? fw: -fw;
278 for (i=1;i<=jz;i++) fw += fq[i];
279 y[1] = (ih==0)? fw: -fw;
281 case 3: /* painful */
292 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
294 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
296 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;