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[/] [openrisc/] [trunk/] [gnu-old/] [newlib-1.17.0/] [newlib/] [libm/] [math/] [k_rem_pio2.c] - Blame information for rev 855

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1 148 jeremybenn
 
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/* @(#)k_rem_pio2.c 5.1 93/09/24 */
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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/*
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 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
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 * double x[],y[]; int e0,nx,prec; int ipio2[];
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 *
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 * __kernel_rem_pio2 return the last three digits of N with
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 *              y = x - N*pi/2
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 * so that |y| < pi/2.
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 *
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 * The method is to compute the integer (mod 8) and fraction parts of
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 * (2/pi)*x without doing the full multiplication. In general we
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 * skip the part of the product that are known to be a huge integer (
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 * more accurately, = 0 mod 8 ). Thus the number of operations are
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 * independent of the exponent of the input.
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 *
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 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
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 *
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 * Input parameters:
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 *      x[]     The input value (must be positive) is broken into nx
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 *              pieces of 24-bit integers in double precision format.
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 *              x[i] will be the i-th 24 bit of x. The scaled exponent
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 *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
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 *              match x's up to 24 bits.
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 *
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 *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
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 *                      e0 = ilogb(z)-23
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 *                      z  = scalbn(z,-e0)
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 *              for i = 0,1,2
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 *                      x[i] = floor(z)
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 *                      z    = (z-x[i])*2**24
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 *
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 *
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 *      y[]     ouput result in an array of double precision numbers.
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 *              The dimension of y[] is:
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 *                      24-bit  precision       1
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 *                      53-bit  precision       2
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 *                      64-bit  precision       2
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 *                      113-bit precision       3
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 *              The actual value is the sum of them. Thus for 113-bit
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 *              precison, one may have to do something like:
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 *
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 *              long double t,w,r_head, r_tail;
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 *              t = (long double)y[2] + (long double)y[1];
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 *              w = (long double)y[0];
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 *              r_head = t+w;
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 *              r_tail = w - (r_head - t);
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 *
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 *      e0      The exponent of x[0]
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 *
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 *      nx      dimension of x[]
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 *
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 *      prec    an integer indicating the precision:
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 *                      0        24  bits (single)
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 *                      1       53  bits (double)
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 *                      2       64  bits (extended)
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 *                      3       113 bits (quad)
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 *
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 *      ipio2[]
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 *              integer array, contains the (24*i)-th to (24*i+23)-th
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 *              bit of 2/pi after binary point. The corresponding
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 *              floating value is
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 *
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 *                      ipio2[i] * 2^(-24(i+1)).
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 *
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 * External function:
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 *      double scalbn(), floor();
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 *
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 *
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 * Here is the description of some local variables:
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 *
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 *      jk      jk+1 is the initial number of terms of ipio2[] needed
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 *              in the computation. The recommended value is 2,3,4,
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 *              6 for single, double, extended,and quad.
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 *
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 *      jz      local integer variable indicating the number of
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 *              terms of ipio2[] used.
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 *
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 *      jx      nx - 1
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 *
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 *      jv      index for pointing to the suitable ipio2[] for the
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 *              computation. In general, we want
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 *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
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 *              is an integer. Thus
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 *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
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 *              Hence jv = max(0,(e0-3)/24).
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 *
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 *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
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 *
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 *      q[]     double array with integral value, representing the
102
 *              24-bits chunk of the product of x and 2/pi.
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 *
104
 *      q0      the corresponding exponent of q[0]. Note that the
105
 *              exponent for q[i] would be q0-24*i.
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 *
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 *      PIo2[]  double precision array, obtained by cutting pi/2
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 *              into 24 bits chunks.
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 *
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 *      f[]     ipio2[] in floating point
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 *
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 *      iq[]    integer array by breaking up q[] in 24-bits chunk.
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 *
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 *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
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 *
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 *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
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 *              it also indicates the *sign* of the result.
118
 *
119
 */
120
 
121
 
122
/*
123
 * Constants:
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.
128
 */
129
 
130
#include "fdlibm.h"
131
 
132
#ifndef _DOUBLE_IS_32BITS
133
 
134
#ifdef __STDC__
135
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
136
#else
137
static int init_jk[] = {2,3,4,6};
138
#endif
139
 
140
#ifdef __STDC__
141
static const double PIo2[] = {
142
#else
143
static double PIo2[] = {
144
#endif
145
  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
146
  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
147
  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
148
  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
149
  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
150
  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
151
  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
152
  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
153
};
154
 
155
#ifdef __STDC__
156
static const double
157
#else
158
static double
159
#endif
160
zero   = 0.0,
161
one    = 1.0,
162
two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
163
twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
164
 
165
#ifdef __STDC__
166
        int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const __int32_t *ipio2)
167
#else
168
        int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
169
        double x[], y[]; int e0,nx,prec; __int32_t ipio2[];
170
#endif
171
{
172
        __int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
173
        double z,fw,f[20],fq[20],q[20];
174
 
175
    /* initialize jk*/
176
        jk = init_jk[prec];
177
        jp = jk;
178
 
179
    /* determine jx,jv,q0, note that 3>q0 */
180
        jx =  nx-1;
181
        jv = (e0-3)/24; if(jv<0) jv=0;
182
        q0 =  e0-24*(jv+1);
183
 
184
    /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
185
        j = jv-jx; m = jx+jk;
186
        for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
187
 
188
    /* compute q[0],q[1],...q[jk] */
189
        for (i=0;i<=jk;i++) {
190
            for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
191
        }
192
 
193
        jz = jk;
194
recompute:
195
    /* distill q[] into iq[] reversingly */
196
        for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
197
            fw    =  (double)((__int32_t)(twon24* z));
198
            iq[i] =  (__int32_t)(z-two24*fw);
199
            z     =  q[j-1]+fw;
200
        }
201
 
202
    /* compute n */
203
        z  = scalbn(z,(int)q0);         /* actual value of z */
204
        z -= 8.0*floor(z*0.125);                /* trim off integer >= 8 */
205
        n  = (__int32_t) z;
206
        z -= (double)n;
207
        ih = 0;
208
        if(q0>0) {       /* need iq[jz-1] to determine n */
209
            i  = (iq[jz-1]>>(24-q0)); n += i;
210
            iq[jz-1] -= i<<(24-q0);
211
            ih = iq[jz-1]>>(23-q0);
212
        }
213
        else if(q0==0) ih = iq[jz-1]>>23;
214
        else if(z>=0.5) ih=2;
215
 
216
        if(ih>0) {       /* q > 0.5 */
217
            n += 1; carry = 0;
218
            for(i=0;i<jz ;i++) { /* compute 1-q */
219
                j = iq[i];
220
                if(carry==0) {
221
                    if(j!=0) {
222
                        carry = 1; iq[i] = 0x1000000- j;
223
                    }
224
                } else  iq[i] = 0xffffff - j;
225
            }
226
            if(q0>0) {           /* rare case: chance is 1 in 12 */
227
                switch(q0) {
228
                case 1:
229
                   iq[jz-1] &= 0x7fffff; break;
230
                case 2:
231
                   iq[jz-1] &= 0x3fffff; break;
232
                }
233
            }
234
            if(ih==2) {
235
                z = one - z;
236
                if(carry!=0) z -= scalbn(one,(int)q0);
237
            }
238
        }
239
 
240
    /* check if recomputation is needed */
241
        if(z==zero) {
242
            j = 0;
243
            for (i=jz-1;i>=jk;i--) j |= iq[i];
244
            if(j==0) { /* need recomputation */
245
                for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
246
 
247
                for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
248
                    f[jx+i] = (double) ipio2[jv+i];
249
                    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
250
                    q[i] = fw;
251
                }
252
                jz += k;
253
                goto recompute;
254
            }
255
        }
256
 
257
    /* chop off zero terms */
258
        if(z==0.0) {
259
            jz -= 1; q0 -= 24;
260
            while(iq[jz]==0) { jz--; q0-=24;}
261
        } else { /* break z into 24-bit if necessary */
262
            z = scalbn(z,-(int)q0);
263
            if(z>=two24) {
264
                fw = (double)((__int32_t)(twon24*z));
265
                iq[jz] = (__int32_t)(z-two24*fw);
266
                jz += 1; q0 += 24;
267
                iq[jz] = (__int32_t) fw;
268
            } else iq[jz] = (__int32_t) z ;
269
        }
270
 
271
    /* convert integer "bit" chunk to floating-point value */
272
        fw = scalbn(one,(int)q0);
273
        for(i=jz;i>=0;i--) {
274
            q[i] = fw*(double)iq[i]; fw*=twon24;
275
        }
276
 
277
    /* compute PIo2[0,...,jp]*q[jz,...,0] */
278
        for(i=jz;i>=0;i--) {
279
            for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
280
            fq[jz-i] = fw;
281
        }
282
 
283
    /* compress fq[] into y[] */
284
        switch(prec) {
285
            case 0:
286
                fw = 0.0;
287
                for (i=jz;i>=0;i--) fw += fq[i];
288
                y[0] = (ih==0)? fw: -fw;
289
                break;
290
            case 1:
291
            case 2:
292
                fw = 0.0;
293
                for (i=jz;i>=0;i--) fw += fq[i];
294
                y[0] = (ih==0)? fw: -fw;
295
                fw = fq[0]-fw;
296
                for (i=1;i<=jz;i++) fw += fq[i];
297
                y[1] = (ih==0)? fw: -fw;
298
                break;
299
            case 3:     /* painful */
300
                for (i=jz;i>0;i--) {
301
                    fw      = fq[i-1]+fq[i];
302
                    fq[i]  += fq[i-1]-fw;
303
                    fq[i-1] = fw;
304
                }
305
                for (i=jz;i>1;i--) {
306
                    fw      = fq[i-1]+fq[i];
307
                    fq[i]  += fq[i-1]-fw;
308
                    fq[i-1] = fw;
309
                }
310
                for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
311
                if(ih==0) {
312
                    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
313
                } else {
314
                    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
315
                }
316
        }
317
        return n&7;
318
}
319
 
320
#endif /* defined(_DOUBLE_IS_32BITS) */

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