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glibc  2.9
k_rem_pio2.c
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00001 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
00002 /*
00003  * ====================================================
00004  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00005  *
00006  * Developed at SunPro, a Sun Microsystems, Inc. business.
00007  * Permission to use, copy, modify, and distribute this
00008  * software is freely granted, provided that this notice
00009  * is preserved.
00010  * ====================================================
00011  */
00012 
00013 #if defined(LIBM_SCCS) && !defined(lint)
00014 static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
00015 #endif
00016 
00017 /*
00018  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
00019  * double x[],y[]; int e0,nx,prec; int ipio2[];
00020  *
00021  * __kernel_rem_pio2 return the last three digits of N with
00022  *            y = x - N*pi/2
00023  * so that |y| < pi/2.
00024  *
00025  * The method is to compute the integer (mod 8) and fraction parts of
00026  * (2/pi)*x without doing the full multiplication. In general we
00027  * skip the part of the product that are known to be a huge integer (
00028  * more accurately, = 0 mod 8 ). Thus the number of operations are
00029  * independent of the exponent of the input.
00030  *
00031  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
00032  *
00033  * Input parameters:
00034  *     x[]    The input value (must be positive) is broken into nx
00035  *            pieces of 24-bit integers in double precision format.
00036  *            x[i] will be the i-th 24 bit of x. The scaled exponent
00037  *            of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
00038  *            match x's up to 24 bits.
00039  *
00040  *            Example of breaking a double positive z into x[0]+x[1]+x[2]:
00041  *                   e0 = ilogb(z)-23
00042  *                   z  = scalbn(z,-e0)
00043  *            for i = 0,1,2
00044  *                   x[i] = floor(z)
00045  *                   z    = (z-x[i])*2**24
00046  *
00047  *
00048  *     y[]    ouput result in an array of double precision numbers.
00049  *            The dimension of y[] is:
00050  *                   24-bit  precision    1
00051  *                   53-bit  precision    2
00052  *                   64-bit  precision    2
00053  *                   113-bit precision    3
00054  *            The actual value is the sum of them. Thus for 113-bit
00055  *            precision, one may have to do something like:
00056  *
00057  *            long double t,w,r_head, r_tail;
00058  *            t = (long double)y[2] + (long double)y[1];
00059  *            w = (long double)y[0];
00060  *            r_head = t+w;
00061  *            r_tail = w - (r_head - t);
00062  *
00063  *     e0     The exponent of x[0]
00064  *
00065  *     nx     dimension of x[]
00066  *
00067  *     prec   an integer indicating the precision:
00068  *                   0      24  bits (single)
00069  *                   1      53  bits (double)
00070  *                   2      64  bits (extended)
00071  *                   3      113 bits (quad)
00072  *
00073  *     ipio2[]
00074  *            integer array, contains the (24*i)-th to (24*i+23)-th
00075  *            bit of 2/pi after binary point. The corresponding
00076  *            floating value is
00077  *
00078  *                   ipio2[i] * 2^(-24(i+1)).
00079  *
00080  * External function:
00081  *     double scalbn(), floor();
00082  *
00083  *
00084  * Here is the description of some local variables:
00085  *
00086  *     jk     jk+1 is the initial number of terms of ipio2[] needed
00087  *            in the computation. The recommended value is 2,3,4,
00088  *            6 for single, double, extended,and quad.
00089  *
00090  *     jz     local integer variable indicating the number of
00091  *            terms of ipio2[] used.
00092  *
00093  *     jx     nx - 1
00094  *
00095  *     jv     index for pointing to the suitable ipio2[] for the
00096  *            computation. In general, we want
00097  *                   ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
00098  *            is an integer. Thus
00099  *                   e0-3-24*jv >= 0 or (e0-3)/24 >= jv
00100  *            Hence jv = max(0,(e0-3)/24).
00101  *
00102  *     jp     jp+1 is the number of terms in PIo2[] needed, jp = jk.
00103  *
00104  *     q[]    double array with integral value, representing the
00105  *            24-bits chunk of the product of x and 2/pi.
00106  *
00107  *     q0     the corresponding exponent of q[0]. Note that the
00108  *            exponent for q[i] would be q0-24*i.
00109  *
00110  *     PIo2[] double precision array, obtained by cutting pi/2
00111  *            into 24 bits chunks.
00112  *
00113  *     f[]    ipio2[] in floating point
00114  *
00115  *     iq[]   integer array by breaking up q[] in 24-bits chunk.
00116  *
00117  *     fq[]   final product of x*(2/pi) in fq[0],..,fq[jk]
00118  *
00119  *     ih     integer. If >0 it indicates q[] is >= 0.5, hence
00120  *            it also indicates the *sign* of the result.
00121  *
00122  */
00123 
00124 
00125 /*
00126  * Constants:
00127  * The hexadecimal values are the intended ones for the following
00128  * constants. The decimal values may be used, provided that the
00129  * compiler will convert from decimal to binary accurately enough
00130  * to produce the hexadecimal values shown.
00131  */
00132 
00133 #include "math.h"
00134 #include "math_private.h"
00135 
00136 #ifdef __STDC__
00137 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
00138 #else
00139 static int init_jk[] = {2,3,4,6};
00140 #endif
00141 
00142 #ifdef __STDC__
00143 static const double PIo2[] = {
00144 #else
00145 static double PIo2[] = {
00146 #endif
00147   1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
00148   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
00149   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
00150   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
00151   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
00152   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
00153   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
00154   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
00155 };
00156 
00157 #ifdef __STDC__
00158 static const double
00159 #else
00160 static double
00161 #endif
00162 zero   = 0.0,
00163 one    = 1.0,
00164 two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
00165 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
00166 
00167 #ifdef __STDC__
00168        int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
00169 #else
00170        int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
00171        double x[], y[]; int e0,nx,prec; int32_t ipio2[];
00172 #endif
00173 {
00174        int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
00175        double z,fw,f[20],fq[20],q[20];
00176 
00177     /* initialize jk*/
00178        jk = init_jk[prec];
00179        jp = jk;
00180 
00181     /* determine jx,jv,q0, note that 3>q0 */
00182        jx =  nx-1;
00183        jv = (e0-3)/24; if(jv<0) jv=0;
00184        q0 =  e0-24*(jv+1);
00185 
00186     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
00187        j = jv-jx; m = jx+jk;
00188        for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
00189 
00190     /* compute q[0],q[1],...q[jk] */
00191        for (i=0;i<=jk;i++) {
00192            for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
00193        }
00194 
00195        jz = jk;
00196 recompute:
00197     /* distill q[] into iq[] reversingly */
00198        for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
00199            fw    =  (double)((int32_t)(twon24* z));
00200            iq[i] =  (int32_t)(z-two24*fw);
00201            z     =  q[j-1]+fw;
00202        }
00203 
00204     /* compute n */
00205        z  = __scalbn(z,q0);        /* actual value of z */
00206        z -= 8.0*__floor(z*0.125);         /* trim off integer >= 8 */
00207        n  = (int32_t) z;
00208        z -= (double)n;
00209        ih = 0;
00210        if(q0>0) {    /* need iq[jz-1] to determine n */
00211            i  = (iq[jz-1]>>(24-q0)); n += i;
00212            iq[jz-1] -= i<<(24-q0);
00213            ih = iq[jz-1]>>(23-q0);
00214        }
00215        else if(q0==0) ih = iq[jz-1]>>23;
00216        else if(z>=0.5) ih=2;
00217 
00218        if(ih>0) {    /* q > 0.5 */
00219            n += 1; carry = 0;
00220            for(i=0;i<jz ;i++) {    /* compute 1-q */
00221               j = iq[i];
00222               if(carry==0) {
00223                   if(j!=0) {
00224                      carry = 1; iq[i] = 0x1000000- j;
00225                   }
00226               } else  iq[i] = 0xffffff - j;
00227            }
00228            if(q0>0) {              /* rare case: chance is 1 in 12 */
00229                switch(q0) {
00230                case 1:
00231                  iq[jz-1] &= 0x7fffff; break;
00232               case 2:
00233                  iq[jz-1] &= 0x3fffff; break;
00234                }
00235            }
00236            if(ih==2) {
00237               z = one - z;
00238               if(carry!=0) z -= __scalbn(one,q0);
00239            }
00240        }
00241 
00242     /* check if recomputation is needed */
00243        if(z==zero) {
00244            j = 0;
00245            for (i=jz-1;i>=jk;i--) j |= iq[i];
00246            if(j==0) { /* need recomputation */
00247               for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
00248 
00249               for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
00250                   f[jx+i] = (double) ipio2[jv+i];
00251                   for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
00252                   q[i] = fw;
00253               }
00254               jz += k;
00255               goto recompute;
00256            }
00257        }
00258 
00259     /* chop off zero terms */
00260        if(z==0.0) {
00261            jz -= 1; q0 -= 24;
00262            while(iq[jz]==0) { jz--; q0-=24;}
00263        } else { /* break z into 24-bit if necessary */
00264            z = __scalbn(z,-q0);
00265            if(z>=two24) {
00266               fw = (double)((int32_t)(twon24*z));
00267               iq[jz] = (int32_t)(z-two24*fw);
00268               jz += 1; q0 += 24;
00269               iq[jz] = (int32_t) fw;
00270            } else iq[jz] = (int32_t) z ;
00271        }
00272 
00273     /* convert integer "bit" chunk to floating-point value */
00274        fw = __scalbn(one,q0);
00275        for(i=jz;i>=0;i--) {
00276            q[i] = fw*(double)iq[i]; fw*=twon24;
00277        }
00278 
00279     /* compute PIo2[0,...,jp]*q[jz,...,0] */
00280        for(i=jz;i>=0;i--) {
00281            for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
00282            fq[jz-i] = fw;
00283        }
00284 
00285     /* compress fq[] into y[] */
00286        switch(prec) {
00287            case 0:
00288               fw = 0.0;
00289               for (i=jz;i>=0;i--) fw += fq[i];
00290               y[0] = (ih==0)? fw: -fw;
00291               break;
00292            case 1:
00293            case 2:
00294               fw = 0.0;
00295               for (i=jz;i>=0;i--) fw += fq[i];
00296               y[0] = (ih==0)? fw: -fw;
00297               fw = fq[0]-fw;
00298               for (i=1;i<=jz;i++) fw += fq[i];
00299               y[1] = (ih==0)? fw: -fw;
00300               break;
00301            case 3:   /* painful */
00302               for (i=jz;i>0;i--) {
00303                   fw      = fq[i-1]+fq[i];
00304                   fq[i]  += fq[i-1]-fw;
00305                   fq[i-1] = fw;
00306               }
00307               for (i=jz;i>1;i--) {
00308                   fw      = fq[i-1]+fq[i];
00309                   fq[i]  += fq[i-1]-fw;
00310                   fq[i-1] = fw;
00311               }
00312               for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
00313               if(ih==0) {
00314                   y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
00315               } else {
00316                   y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
00317               }
00318        }
00319        return n&7;
00320 }