sysdeps/powerpc/powerpc32/power4/fpu/mpa.c

   1
   2 /*
   3  * IBM Accurate Mathematical Library
   4  * written by International Business Machines Corp.
   5  * Copyright (C) 2001, 2006 Free Software Foundation
   6  *
   7  * This program is free software; you can redistribute it and/or modify
   8  * it under the terms of the GNU Lesser General Public License as published by
   9  * the Free Software Foundation; either version 2.1 of the License, or
  10  * (at your option) any later version.
  11  *
  12  * This program is distributed in the hope that it will be useful,
  13  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  14  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  15  * GNU Lesser General Public License for more details.
  16  *
  17  * You should have received a copy of the GNU Lesser General Public License
  18  * along with this program; if not, write to the Free Software
  19  * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
  20  */
  21 /************************************************************************/
  22 /*  MODULE_NAME: mpa.c                                                  */
  23 /*                                                                      */
  24 /*  FUNCTIONS:                                                          */
  25 /*               mcr                                                    */
  26 /*               acr                                                    */
  27 /*               cr                                                     */
  28 /*               cpy                                                    */
  29 /*               cpymn                                                  */
  30 /*               norm                                                   */
  31 /*               denorm                                                 */
  32 /*               mp_dbl                                                 */
  33 /*               dbl_mp                                                 */
  34 /*               add_magnitudes                                         */
  35 /*               sub_magnitudes                                         */
  36 /*               add                                                    */
  37 /*               sub                                                    */
  38 /*               mul                                                    */
  39 /*               inv                                                    */
  40 /*               dvd                                                    */
  41 /*                                                                      */
  42 /* Arithmetic functions for multiple precision numbers.                 */
  43 /* Relative errors are bounded                                          */
  44 /************************************************************************/
  45
  46
  47 #include "endian.h"
  48 #include "mpa.h"
  49 #include "mpa2.h"
  50 #include <sys/param.h>  /* For MIN() */
  51 /* mcr() compares the sizes of the mantissas of two multiple precision  */
  52 /* numbers. Mantissas are compared regardless of the signs of the       */
  53 /* numbers, even if x->d[0] or y->d[0] are zero. Exponents are also     */
  54 /* disregarded.                                                         */
  55 static int mcr(const mp_no *x, const mp_no *y, int p) {
  56   long i;
  57   long p2 = p;
  58   for (i=1; i<=p2; i++) {
  59     if      (X[i] == Y[i])  continue;
  60     else if (X[i] >  Y[i])  return  1;
  61     else                    return -1; }
  62   return 0;
  63 }
  64
  65
  66
  67 /* acr() compares the absolute values of two multiple precision numbers */
  68 int __acr(const mp_no *x, const mp_no *y, int p) {
  69   long i;
  70
  71   if      (X[0] == ZERO) {
  72     if    (Y[0] == ZERO) i= 0;
  73     else                 i=-1;
  74   }
  75   else if (Y[0] == ZERO) i= 1;
  76   else {
  77     if      (EX >  EY)   i= 1;
  78     else if (EX <  EY)   i=-1;
  79     else                 i= mcr(x,y,p);
  80   }
  81
  82   return i;
  83 }
  84
  85
  86 /* cr90 compares the values of two multiple precision numbers           */
  87 int  __cr(const mp_no *x, const mp_no *y, int p) {
  88   int i;
  89
  90   if      (X[0] > Y[0])  i= 1;
  91   else if (X[0] < Y[0])  i=-1;
  92   else if (X[0] < ZERO ) i= __acr(y,x,p);
  93   else                   i= __acr(x,y,p);
  94
  95   return i;
  96 }
  97
  98
  99 /* Copy a multiple precision number. Set *y=*x. x=y is permissible.      */
 100 void __cpy(const mp_no *x, mp_no *y, int p) {
 101   long i;
 102
 103   EY = EX;
 104   for (i=0; i <= p; i++)    Y[i] = X[i];
 105
 106   return;
 107 }
 108
 109
 110 /* Copy a multiple precision number x of precision m into a */
 111 /* multiple precision number y of precision n. In case n>m, */
 112 /* the digits of y beyond the m'th are set to zero. In case */
 113 /* n<m, the digits of x beyond the n'th are ignored.        */
 114 /* x=y is permissible.                                      */
 115
 116 void __cpymn(const mp_no *x, int m, mp_no *y, int n) {
 117
 118   long i,k;
 119   long n2 = n;
 120   long m2 = m;
 121
 122   EY = EX;     k=MIN(m2,n2);
 123   for (i=0; i <= k; i++)    Y[i] = X[i];
 124   for (   ; i <= n2; i++)    Y[i] = ZERO;
 125
 126   return;
 127 }
 128
 129 /* Convert a multiple precision number *x into a double precision */
 130 /* number *y, normalized case  (|x| >= 2**(-1022))) */
 131 static void norm(const mp_no *x, double *y, int p)
 132 {
 133   #define R  radixi.d
 134   long i;
 135 #if 0
 136   int k;
 137 #endif
 138   double a,c,u,v,z[5];
 139   if (p<5) {
 140     if      (p==1) c = X[1];
 141     else if (p==2) c = X[1] + R* X[2];
 142     else if (p==3) c = X[1] + R*(X[2]  +   R* X[3]);
 143     else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
 144   }
 145   else {
 146     for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
 147         {a *= TWO;   z[1] *= TWO; }
 148
 149     for (i=2; i<5; i++) {
 150       z[i] = X[i]*a;
 151       u = (z[i] + CUTTER)-CUTTER;
 152       if  (u > z[i])  u -= RADIX;
 153       z[i] -= u;
 154       z[i-1] += u*RADIXI;
 155     }
 156
 157     u = (z[3] + TWO71) - TWO71;
 158     if (u > z[3])   u -= TWO19;
 159     v = z[3]-u;
 160
 161     if (v == TWO18) {
 162       if (z[4] == ZERO) {
 163         for (i=5; i <= p; i++) {
 164           if (X[i] == ZERO)   continue;
 165           else                {z[3] += ONE;   break; }
 166         }
 167       }
 168       else              z[3] += ONE;
 169     }
 170
 171     c = (z[1] + R *(z[2] + R * z[3]))/a;
 172   }
 173
 174   c *= X[0];
 175
 176   for (i=1; i<EX; i++)   c *= RADIX;
 177   for (i=1; i>EX; i--)   c *= RADIXI;
 178
 179   *y = c;
 180   return;
 181 #undef R
 182 }
 183
 184 /* Convert a multiple precision number *x into a double precision */
 185 /* number *y, denormalized case  (|x| < 2**(-1022))) */
 186 static void denorm(const mp_no *x, double *y, int p)
 187 {
 188   long i,k;
 189   long p2 = p;
 190   double c,u,z[5];
 191 #if 0
 192   double a,v;
 193 #endif
 194
 195 #define R  radixi.d
 196   if (EX<-44 || (EX==-44 && X[1]<TWO5))
 197      { *y=ZERO; return; }
 198
 199   if      (p2==1) {
 200     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=ZERO;  z[3]=ZERO;  k=3;}
 201     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=ZERO;  k=2;}
 202     else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
 203   }
 204   else if (p2==2) {
 205     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  z[3]=ZERO;  k=3;}
 206     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=X[2];  k=2;}
 207     else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
 208   }
 209   else {
 210     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  k=3;}
 211     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  k=2;}
 212     else              {z[1]=     TWO10;  z[2]=ZERO;  k=1;}
 213     z[3] = X[k];
 214   }
 215
 216   u = (z[3] + TWO57) - TWO57;
 217   if  (u > z[3])   u -= TWO5;
 218
 219   if (u==z[3]) {
 220     for (i=k+1; i <= p2; i++) {
 221       if (X[i] == ZERO)   continue;
 222       else {z[3] += ONE;   break; }
 223     }
 224   }
 225
 226   c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);
 227
 228   *y = c*TWOM1032;
 229   return;
 230
 231 #undef R
 232 }
 233
 234 /* Convert a multiple precision number *x into a double precision number *y. */
 235 /* The result is correctly rounded to the nearest/even. *x is left unchanged */
 236
 237 void __mp_dbl(const mp_no *x, double *y, int p) {
 238 #if 0
 239   int i,k;
 240   double a,c,u,v,z[5];
 241 #endif
 242
 243   if (X[0] == ZERO)  {*y = ZERO;  return; }
 244
 245   if      (EX> -42)                 norm(x,y,p);
 246   else if (EX==-42 && X[1]>=TWO10)  norm(x,y,p);
 247   else                              denorm(x,y,p);
 248 }
 249
 250
 251 /* dbl_mp() converts a double precision number x into a multiple precision  */
 252 /* number *y. If the precision p is too small the result is truncated. x is */
 253 /* left unchanged.                                                          */
 254
 255 void __dbl_mp(double x, mp_no *y, int p) {
 256
 257   long i,n;
 258   long p2 = p;
 259   double u;
 260
 261   /* Sign */
 262   if      (x == ZERO)  {Y[0] = ZERO;  return; }
 263   else if (x >  ZERO)   Y[0] = ONE;
 264   else                 {Y[0] = MONE;  x=-x;   }
 265
 266   /* Exponent */
 267   for (EY=ONE; x >= RADIX; EY += ONE)   x *= RADIXI;
 268   for (      ; x <  ONE;   EY -= ONE)   x *= RADIX;
 269
 270   /* Digits */
 271   n=MIN(p2,4);
 272   for (i=1; i<=n; i++) {
 273     u = (x + TWO52) - TWO52;
 274     if (u>x)   u -= ONE;
 275     Y[i] = u;     x -= u;    x *= RADIX; }
 276   for (   ; i<=p2; i++)     Y[i] = ZERO;
 277   return;
 278 }
 279
 280
 281 /*  add_magnitudes() adds the magnitudes of *x & *y assuming that           */
 282 /*  abs(*x) >= abs(*y) > 0.                                                 */
 283 /* The sign of the sum *z is undefined. x&y may overlap but not x&z or y&z. */
 284 /* No guard digit is used. The result equals the exact sum, truncated.      */
 285 /* *x & *y are left unchanged.                                              */
 286
 287 static void add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 288
 289   long i,j,k;
 290   long p2 = p;
 291
 292   EZ = EX;
 293
 294   i=p2;    j=p2+ EY - EX;    k=p2+1;
 295
 296   if (j<1)
 297      {__cpy(x,z,p);  return; }
 298   else   Z[k] = ZERO;
 299
 300   for (; j>0; i--,j--) {
 301     Z[k] += X[i] + Y[j];
 302     if (Z[k] >= RADIX) {
 303       Z[k]  -= RADIX;
 304       Z[--k] = ONE; }
 305     else
 306       Z[--k] = ZERO;
 307   }
 308
 309   for (; i>0; i--) {
 310     Z[k] += X[i];
 311     if (Z[k] >= RADIX) {
 312       Z[k]  -= RADIX;
 313       Z[--k] = ONE; }
 314     else
 315       Z[--k] = ZERO;
 316   }
 317
 318   if (Z[1] == ZERO) {
 319     for (i=1; i<=p2; i++)    Z[i] = Z[i+1]; }
 320   else   EZ += ONE;
 321 }
 322
 323
 324 /*  sub_magnitudes() subtracts the magnitudes of *x & *y assuming that      */
 325 /*  abs(*x) > abs(*y) > 0.                                                  */
 326 /* The sign of the difference *z is undefined. x&y may overlap but not x&z  */
 327 /* or y&z. One guard digit is used. The error is less than one ulp.         */
 328 /* *x & *y are left unchanged.                                              */
 329
 330 static void sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 331
 332   long i,j,k;
 333   long p2 = p;
 334
 335   EZ = EX;
 336
 337   if (EX == EY) {
 338     i=j=k=p2;
 339     Z[k] = Z[k+1] = ZERO; }
 340   else {
 341     j= EX - EY;
 342     if (j > p2)  {__cpy(x,z,p);  return; }
 343     else {
 344       i=p2;   j=p2+1-j;   k=p2;
 345       if (Y[j] > ZERO) {
 346         Z[k+1] = RADIX - Y[j--];
 347         Z[k]   = MONE; }
 348       else {
 349         Z[k+1] = ZERO;
 350         Z[k]   = ZERO;   j--;}
 351     }
 352   }
 353
 354   for (; j>0; i--,j--) {
 355     Z[k] += (X[i] - Y[j]);
 356     if (Z[k] < ZERO) {
 357       Z[k]  += RADIX;
 358       Z[--k] = MONE; }
 359     else
 360       Z[--k] = ZERO;
 361   }
 362
 363   for (; i>0; i--) {
 364     Z[k] += X[i];
 365     if (Z[k] < ZERO) {
 366       Z[k]  += RADIX;
 367       Z[--k] = MONE; }
 368     else
 369       Z[--k] = ZERO;
 370   }
 371
 372   for (i=1; Z[i] == ZERO; i++) ;
 373   EZ = EZ - i + 1;
 374   for (k=1; i <= p2+1; )
 375     Z[k++] = Z[i++];
 376   for (; k <= p2; )
 377     Z[k++] = ZERO;
 378
 379   return;
 380 }
 381
 382
 383 /* Add two multiple precision numbers. Set *z = *x + *y. x&y may overlap  */
 384 /* but not x&z or y&z. One guard digit is used. The error is less than    */
 385 /* one ulp. *x & *y are left unchanged.                                   */
 386
 387 void __add(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 388
 389   int n;
 390
 391   if      (X[0] == ZERO)     {__cpy(y,z,p);  return; }
 392   else if (Y[0] == ZERO)     {__cpy(x,z,p);  return; }
 393
 394   if (X[0] == Y[0])   {
 395     if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] = X[0]; }
 396     else                     {add_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
 397   }
 398   else                       {
 399     if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] = X[0]; }
 400     else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
 401     else                      Z[0] = ZERO;
 402   }
 403   return;
 404 }
 405
 406
 407 /* Subtract two multiple precision numbers. *z is set to *x - *y. x&y may */
 408 /* overlap but not x&z or y&z. One guard digit is used. The error is      */
 409 /* less than one ulp. *x & *y are left unchanged.                         */
 410
 411 void __sub(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 412
 413   int n;
 414
 415   if      (X[0] == ZERO)     {__cpy(y,z,p);  Z[0] = -Z[0];  return; }
 416   else if (Y[0] == ZERO)     {__cpy(x,z,p);                 return; }
 417
 418   if (X[0] != Y[0])    {
 419     if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
 420     else                     {add_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
 421   }
 422   else                       {
 423     if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
 424     else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
 425     else                      Z[0] = ZERO;
 426   }
 427   return;
 428 }
 429
 430
 431 /* Multiply two multiple precision numbers. *z is set to *x * *y. x&y      */
 432 /* may overlap but not x&z or y&z. In case p=1,2,3 the exact result is     */
 433 /* truncated to p digits. In case p>3 the error is bounded by 1.001 ulp.   */
 434 /* *x & *y are left unchanged.                                             */
 435
 436 void __mul(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 437
 438   long i, i1, i2, j, k, k2;
 439   long p2 = p;
 440   double u, zk, zk2;
 441
 442                       /* Is z=0? */
 443   if (X[0]*Y[0]==ZERO)
 444      { Z[0]=ZERO;  return; }
 445
 446                        /* Multiply, add and carry */
 447   k2 = (p2<3) ? p2+p2 : p2+3;
 448   zk = Z[k2]=ZERO;
 449   for (k=k2; k>1; ) {
 450     if (k > p2)  {i1=k-p2; i2=p2+1; }
 451     else        {i1=1;   i2=k;   }
 452 #if 1
 453     /* rearange this inner loop to allow the fmadd instructions to be
 454        independent and execute in parallel on processors that have
 455        dual symetrical FP pipelines.  */
 456     if (i1 < (i2-1))
 457     {
 458         /* make sure we have at least 2 iterations */
 459         if (((i2 - i1) & 1L) == 1L)
 460         {
 461                 /* Handle the odd iterations case.  */
 462                 zk2 = x->d[i2-1]*y->d[i1];
 463         }
 464         else
 465                 zk2 = zero.d;
 466         /* Do two multiply/adds per loop iteration, using independent
 467            accumulators; zk and zk2.  */
 468         for (i=i1,j=i2-1; i<i2-1; i+=2,j-=2)
 469         {
 470                 zk += x->d[i]*y->d[j];
 471                 zk2 += x->d[i+1]*y->d[j-1];
 472         }
 473         zk += zk2; /* final sum.  */
 474     }
 475     else
 476     {
 477         /* Special case when iterations is 1.  */
 478         zk += x->d[i1]*y->d[i1];
 479     }
 480 #else
 481     /* The orginal code.  */
 482     for (i=i1,j=i2-1; i<i2; i++,j--)  zk += X[i]*Y[j];
 483 #endif
 484
 485     u = (zk + CUTTER)-CUTTER;
 486     if  (u > zk)  u -= RADIX;
 487     Z[k]  = zk - u;
 488     zk = u*RADIXI;
 489     --k;
 490   }
 491   Z[k] = zk;
 492
 493                  /* Is there a carry beyond the most significant digit? */
 494   if (Z[1] == ZERO) {
 495     for (i=1; i<=p2; i++)  Z[i]=Z[i+1];
 496     EZ = EX + EY - 1; }
 497   else
 498     EZ = EX + EY;
 499
 500   Z[0] = X[0] * Y[0];
 501   return;
 502 }
 503
 504
 505 /* Invert a multiple precision number. Set *y = 1 / *x.                     */
 506 /* Relative error bound = 1.001*r**(1-p) for p=2, 1.063*r**(1-p) for p=3,   */
 507 /* 2.001*r**(1-p) for p>3.                                                  */
 508 /* *x=0 is not permissible. *x is left unchanged.                           */
 509
 510 void __inv(const mp_no *x, mp_no *y, int p) {
 511   long i;
 512 #if 0
 513   int l;
 514 #endif
 515   double t;
 516   mp_no z,w;
 517   static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
 518                             4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};
 519   const mp_no mptwo = {1,{1.0,2.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
 520                          0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
 521                          0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
 522                          0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
 523
 524   __cpy(x,&z,p);  z.e=0;  __mp_dbl(&z,&t,p);
 525   t=ONE/t;   __dbl_mp(t,y,p);    EY -= EX;
 526
 527   for (i=0; i<np1[p]; i++) {
 528     __cpy(y,&w,p);
 529     __mul(x,&w,y,p);
 530     __sub(&mptwo,y,&z,p);
 531     __mul(&w,&z,y,p);
 532   }
 533   return;
 534 }
 535
 536
 537 /* Divide one multiple precision number by another.Set *z = *x / *y. *x & *y */
 538 /* are left unchanged. x&y may overlap but not x&z or y&z.                   */
 539 /* Relative error bound = 2.001*r**(1-p) for p=2, 2.063*r**(1-p) for p=3     */
 540 /* and 3.001*r**(1-p) for p>3. *y=0 is not permissible.                      */
 541
 542 void __dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 543
 544   mp_no w;
 545
 546   if (X[0] == ZERO)    Z[0] = ZERO;
 547   else                {__inv(y,&w,p);   __mul(x,&w,z,p);}
 548   return;
 549 }