sysdeps/powerpc/powerpc64/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, see <http://www.gnu.org/licenses/>.
  19  */
  20 /************************************************************************/
  21 /*  MODULE_NAME: mpa.c                                                  */
  22 /*                                                                      */
  23 /*  FUNCTIONS:                                                          */
  24 /*               mcr                                                    */
  25 /*               acr                                                    */
  26 /*               cr                                                     */
  27 /*               cpy                                                    */
  28 /*               cpymn                                                  */
  29 /*               norm                                                   */
  30 /*               denorm                                                 */
  31 /*               mp_dbl                                                 */
  32 /*               dbl_mp                                                 */
  33 /*               add_magnitudes                                         */
  34 /*               sub_magnitudes                                         */
  35 /*               add                                                    */
  36 /*               sub                                                    */
  37 /*               mul                                                    */
  38 /*               inv                                                    */
  39 /*               dvd                                                    */
  40 /*                                                                      */
  41 /* Arithmetic functions for multiple precision numbers.                 */
  42 /* Relative errors are bounded                                          */
  43 /************************************************************************/
  44
  45
  46 #include "endian.h"
  47 #include "mpa.h"
  48 #include "mpa2.h"
  49 #include <sys/param.h>  /* For MIN() */
  50 /* mcr() compares the sizes of the mantissas of two multiple precision  */
  51 /* numbers. Mantissas are compared regardless of the signs of the       */
  52 /* numbers, even if x->d[0] or y->d[0] are zero. Exponents are also     */
  53 /* disregarded.                                                         */
  54 static int mcr(const mp_no *x, const mp_no *y, int p) {
  55   long i;
  56   long p2 = p;
  57   for (i=1; i<=p2; i++) {
  58     if      (X[i] == Y[i])  continue;
  59     else if (X[i] >  Y[i])  return  1;
  60     else                    return -1; }
  61   return 0;
  62 }
  63
  64
  65
  66 /* acr() compares the absolute values of two multiple precision numbers */
  67 int __acr(const mp_no *x, const mp_no *y, int p) {
  68   long i;
  69
  70   if      (X[0] == ZERO) {
  71     if    (Y[0] == ZERO) i= 0;
  72     else                 i=-1;
  73   }
  74   else if (Y[0] == ZERO) i= 1;
  75   else {
  76     if      (EX >  EY)   i= 1;
  77     else if (EX <  EY)   i=-1;
  78     else                 i= mcr(x,y,p);
  79   }
  80
  81   return i;
  82 }
  83
  84
  85 /* cr90 compares the values of two multiple precision numbers           */
  86 int  __cr(const mp_no *x, const mp_no *y, int p) {
  87   int i;
  88
  89   if      (X[0] > Y[0])  i= 1;
  90   else if (X[0] < Y[0])  i=-1;
  91   else if (X[0] < ZERO ) i= __acr(y,x,p);
  92   else                   i= __acr(x,y,p);
  93
  94   return i;
  95 }
  96
  97
  98 /* Copy a multiple precision number. Set *y=*x. x=y is permissible.      */
  99 void __cpy(const mp_no *x, mp_no *y, int p) {
 100   long i;
 101
 102   EY = EX;
 103   for (i=0; i <= p; i++)    Y[i] = X[i];
 104
 105   return;
 106 }
 107
 108
 109 /* Copy a multiple precision number x of precision m into a */
 110 /* multiple precision number y of precision n. In case n>m, */
 111 /* the digits of y beyond the m'th are set to zero. In case */
 112 /* n<m, the digits of x beyond the n'th are ignored.        */
 113 /* x=y is permissible.                                      */
 114
 115 void __cpymn(const mp_no *x, int m, mp_no *y, int n) {
 116
 117   long i,k;
 118   long n2 = n;
 119   long m2 = m;
 120
 121   EY = EX;     k=MIN(m2,n2);
 122   for (i=0; i <= k; i++)    Y[i] = X[i];
 123   for (   ; i <= n2; i++)    Y[i] = ZERO;
 124
 125   return;
 126 }
 127
 128 /* Convert a multiple precision number *x into a double precision */
 129 /* number *y, normalized case  (|x| >= 2**(-1022))) */
 130 static void norm(const mp_no *x, double *y, int p)
 131 {
 132   #define R  radixi.d
 133   long i;
 134 #if 0
 135   int k;
 136 #endif
 137   double a,c,u,v,z[5];
 138   if (p<5) {
 139     if      (p==1) c = X[1];
 140     else if (p==2) c = X[1] + R* X[2];
 141     else if (p==3) c = X[1] + R*(X[2]  +   R* X[3]);
 142     else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
 143   }
 144   else {
 145     for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
 146         {a *= TWO;   z[1] *= TWO; }
 147
 148     for (i=2; i<5; i++) {
 149       z[i] = X[i]*a;
 150       u = (z[i] + CUTTER)-CUTTER;
 151       if  (u > z[i])  u -= RADIX;
 152       z[i] -= u;
 153       z[i-1] += u*RADIXI;
 154     }
 155
 156     u = (z[3] + TWO71) - TWO71;
 157     if (u > z[3])   u -= TWO19;
 158     v = z[3]-u;
 159
 160     if (v == TWO18) {
 161       if (z[4] == ZERO) {
 162         for (i=5; i <= p; i++) {
 163           if (X[i] == ZERO)   continue;
 164           else                {z[3] += ONE;   break; }
 165         }
 166       }
 167       else              z[3] += ONE;
 168     }
 169
 170     c = (z[1] + R *(z[2] + R * z[3]))/a;
 171   }
 172
 173   c *= X[0];
 174
 175   for (i=1; i<EX; i++)   c *= RADIX;
 176   for (i=1; i>EX; i--)   c *= RADIXI;
 177
 178   *y = c;
 179   return;
 180 #undef R
 181 }
 182
 183 /* Convert a multiple precision number *x into a double precision */
 184 /* number *y, denormalized case  (|x| < 2**(-1022))) */
 185 static void denorm(const mp_no *x, double *y, int p)
 186 {
 187   long i,k;
 188   long p2 = p;
 189   double c,u,z[5];
 190 #if 0
 191   double a,v;
 192 #endif
 193
 194 #define R  radixi.d
 195   if (EX<-44 || (EX==-44 && X[1]<TWO5))
 196      { *y=ZERO; return; }
 197
 198   if      (p2==1) {
 199     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=ZERO;  z[3]=ZERO;  k=3;}
 200     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=ZERO;  k=2;}
 201     else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
 202   }
 203   else if (p2==2) {
 204     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  z[3]=ZERO;  k=3;}
 205     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=X[2];  k=2;}
 206     else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
 207   }
 208   else {
 209     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  k=3;}
 210     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  k=2;}
 211     else              {z[1]=     TWO10;  z[2]=ZERO;  k=1;}
 212     z[3] = X[k];
 213   }
 214
 215   u = (z[3] + TWO57) - TWO57;
 216   if  (u > z[3])   u -= TWO5;
 217
 218   if (u==z[3]) {
 219     for (i=k+1; i <= p2; i++) {
 220       if (X[i] == ZERO)   continue;
 221       else {z[3] += ONE;   break; }
 222     }
 223   }
 224
 225   c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);
 226
 227   *y = c*TWOM1032;
 228   return;
 229
 230 #undef R
 231 }
 232
 233 /* Convert a multiple precision number *x into a double precision number *y. */
 234 /* The result is correctly rounded to the nearest/even. *x is left unchanged */
 235
 236 void __mp_dbl(const mp_no *x, double *y, int p) {
 237 #if 0
 238   int i,k;
 239   double a,c,u,v,z[5];
 240 #endif
 241
 242   if (X[0] == ZERO)  {*y = ZERO;  return; }
 243
 244   if      (EX> -42)                 norm(x,y,p);
 245   else if (EX==-42 && X[1]>=TWO10)  norm(x,y,p);
 246   else                              denorm(x,y,p);
 247 }
 248
 249
 250 /* dbl_mp() converts a double precision number x into a multiple precision  */
 251 /* number *y. If the precision p is too small the result is truncated. x is */
 252 /* left unchanged.                                                          */
 253
 254 void __dbl_mp(double x, mp_no *y, int p) {
 255
 256   long i,n;
 257   long p2 = p;
 258   double u;
 259
 260   /* Sign */
 261   if      (x == ZERO)  {Y[0] = ZERO;  return; }
 262   else if (x >  ZERO)   Y[0] = ONE;
 263   else                 {Y[0] = MONE;  x=-x;   }
 264
 265   /* Exponent */
 266   for (EY=ONE; x >= RADIX; EY += ONE)   x *= RADIXI;
 267   for (      ; x <  ONE;   EY -= ONE)   x *= RADIX;
 268
 269   /* Digits */
 270   n=MIN(p2,4);
 271   for (i=1; i<=n; i++) {
 272     u = (x + TWO52) - TWO52;
 273     if (u>x)   u -= ONE;
 274     Y[i] = u;     x -= u;    x *= RADIX; }
 275   for (   ; i<=p2; i++)     Y[i] = ZERO;
 276   return;
 277 }
 278
 279
 280 /*  add_magnitudes() adds the magnitudes of *x & *y assuming that           */
 281 /*  abs(*x) >= abs(*y) > 0.                                                 */
 282 /* The sign of the sum *z is undefined. x&y may overlap but not x&z or y&z. */
 283 /* No guard digit is used. The result equals the exact sum, truncated.      */
 284 /* *x & *y are left unchanged.                                              */
 285
 286 static void add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 287
 288   long i,j,k;
 289   long p2 = p;
 290
 291   EZ = EX;
 292
 293   i=p2;    j=p2+ EY - EX;    k=p2+1;
 294
 295   if (j<1)
 296      {__cpy(x,z,p);  return; }
 297   else   Z[k] = ZERO;
 298
 299   for (; j>0; i--,j--) {
 300     Z[k] += X[i] + Y[j];
 301     if (Z[k] >= RADIX) {
 302       Z[k]  -= RADIX;
 303       Z[--k] = ONE; }
 304     else
 305       Z[--k] = ZERO;
 306   }
 307
 308   for (; i>0; i--) {
 309     Z[k] += X[i];
 310     if (Z[k] >= RADIX) {
 311       Z[k]  -= RADIX;
 312       Z[--k] = ONE; }
 313     else
 314       Z[--k] = ZERO;
 315   }
 316
 317   if (Z[1] == ZERO) {
 318     for (i=1; i<=p2; i++)    Z[i] = Z[i+1]; }
 319   else   EZ += ONE;
 320 }
 321
 322
 323 /*  sub_magnitudes() subtracts the magnitudes of *x & *y assuming that      */
 324 /*  abs(*x) > abs(*y) > 0.                                                  */
 325 /* The sign of the difference *z is undefined. x&y may overlap but not x&z  */
 326 /* or y&z. One guard digit is used. The error is less than one ulp.         */
 327 /* *x & *y are left unchanged.                                              */
 328
 329 static void sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 330
 331   long i,j,k;
 332   long p2 = p;
 333
 334   EZ = EX;
 335
 336   if (EX == EY) {
 337     i=j=k=p2;
 338     Z[k] = Z[k+1] = ZERO; }
 339   else {
 340     j= EX - EY;
 341     if (j > p2)  {__cpy(x,z,p);  return; }
 342     else {
 343       i=p2;   j=p2+1-j;   k=p2;
 344       if (Y[j] > ZERO) {
 345         Z[k+1] = RADIX - Y[j--];
 346         Z[k]   = MONE; }
 347       else {
 348         Z[k+1] = ZERO;
 349         Z[k]   = ZERO;   j--;}
 350     }
 351   }
 352
 353   for (; j>0; i--,j--) {
 354     Z[k] += (X[i] - Y[j]);
 355     if (Z[k] < ZERO) {
 356       Z[k]  += RADIX;
 357       Z[--k] = MONE; }
 358     else
 359       Z[--k] = ZERO;
 360   }
 361
 362   for (; i>0; i--) {
 363     Z[k] += X[i];
 364     if (Z[k] < ZERO) {
 365       Z[k]  += RADIX;
 366       Z[--k] = MONE; }
 367     else
 368       Z[--k] = ZERO;
 369   }
 370
 371   for (i=1; Z[i] == ZERO; i++) ;
 372   EZ = EZ - i + 1;
 373   for (k=1; i <= p2+1; )
 374     Z[k++] = Z[i++];
 375   for (; k <= p2; )
 376     Z[k++] = ZERO;
 377
 378   return;
 379 }
 380
 381
 382 /* Add two multiple precision numbers. Set *z = *x + *y. x&y may overlap  */
 383 /* but not x&z or y&z. One guard digit is used. The error is less than    */
 384 /* one ulp. *x & *y are left unchanged.                                   */
 385
 386 void __add(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 387
 388   int n;
 389
 390   if      (X[0] == ZERO)     {__cpy(y,z,p);  return; }
 391   else if (Y[0] == ZERO)     {__cpy(x,z,p);  return; }
 392
 393   if (X[0] == Y[0])   {
 394     if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] = X[0]; }
 395     else                     {add_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
 396   }
 397   else                       {
 398     if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] = X[0]; }
 399     else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
 400     else                      Z[0] = ZERO;
 401   }
 402   return;
 403 }
 404
 405
 406 /* Subtract two multiple precision numbers. *z is set to *x - *y. x&y may */
 407 /* overlap but not x&z or y&z. One guard digit is used. The error is      */
 408 /* less than one ulp. *x & *y are left unchanged.                         */
 409
 410 void __sub(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 411
 412   int n;
 413
 414   if      (X[0] == ZERO)     {__cpy(y,z,p);  Z[0] = -Z[0];  return; }
 415   else if (Y[0] == ZERO)     {__cpy(x,z,p);                 return; }
 416
 417   if (X[0] != Y[0])    {
 418     if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
 419     else                     {add_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
 420   }
 421   else                       {
 422     if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
 423     else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
 424     else                      Z[0] = ZERO;
 425   }
 426   return;
 427 }
 428
 429
 430 /* Multiply two multiple precision numbers. *z is set to *x * *y. x&y      */
 431 /* may overlap but not x&z or y&z. In case p=1,2,3 the exact result is     */
 432 /* truncated to p digits. In case p>3 the error is bounded by 1.001 ulp.   */
 433 /* *x & *y are left unchanged.                                             */
 434
 435 void __mul(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 436
 437   long i, i1, i2, j, k, k2;
 438   long p2 = p;
 439   double u, zk, zk2;
 440
 441                       /* Is z=0? */
 442   if (X[0]*Y[0]==ZERO)
 443      { Z[0]=ZERO;  return; }
 444
 445                        /* Multiply, add and carry */
 446   k2 = (p2<3) ? p2+p2 : p2+3;
 447   zk = Z[k2]=ZERO;
 448   for (k=k2; k>1; ) {
 449     if (k > p2)  {i1=k-p2; i2=p2+1; }
 450     else        {i1=1;   i2=k;   }
 451 #if 1
 452     /* rearange this inner loop to allow the fmadd instructions to be
 453        independent and execute in parallel on processors that have
 454        dual symetrical FP pipelines.  */
 455     if (i1 < (i2-1))
 456     {
 457         /* make sure we have at least 2 iterations */
 458         if (((i2 - i1) & 1L) == 1L)
 459         {
 460                 /* Handle the odd iterations case.  */
 461                 zk2 = x->d[i2-1]*y->d[i1];
 462         }
 463         else
 464                 zk2 = zero.d;
 465         /* Do two multiply/adds per loop iteration, using independent
 466            accumulators; zk and zk2.  */
 467         for (i=i1,j=i2-1; i<i2-1; i+=2,j-=2)
 468         {
 469                 zk += x->d[i]*y->d[j];
 470                 zk2 += x->d[i+1]*y->d[j-1];
 471         }
 472         zk += zk2; /* final sum.  */
 473     }
 474     else
 475     {
 476         /* Special case when iterations is 1.  */
 477         zk += x->d[i1]*y->d[i1];
 478     }
 479 #else
 480     /* The orginal code.  */
 481     for (i=i1,j=i2-1; i<i2; i++,j--)  zk += X[i]*Y[j];
 482 #endif
 483
 484     u = (zk + CUTTER)-CUTTER;
 485     if  (u > zk)  u -= RADIX;
 486     Z[k]  = zk - u;
 487     zk = u*RADIXI;
 488     --k;
 489   }
 490   Z[k] = zk;
 491
 492                  /* Is there a carry beyond the most significant digit? */
 493   if (Z[1] == ZERO) {
 494     for (i=1; i<=p2; i++)  Z[i]=Z[i+1];
 495     EZ = EX + EY - 1; }
 496   else
 497     EZ = EX + EY;
 498
 499   Z[0] = X[0] * Y[0];
 500   return;
 501 }
 502
 503
 504 /* Invert a multiple precision number. Set *y = 1 / *x.                     */
 505 /* Relative error bound = 1.001*r**(1-p) for p=2, 1.063*r**(1-p) for p=3,   */
 506 /* 2.001*r**(1-p) for p>3.                                                  */
 507 /* *x=0 is not permissible. *x is left unchanged.                           */
 508
 509 void __inv(const mp_no *x, mp_no *y, int p) {
 510   long i;
 511 #if 0
 512   int l;
 513 #endif
 514   double t;
 515   mp_no z,w;
 516   static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
 517                             4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};
 518   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,
 519                          0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
 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
 523   __cpy(x,&z,p);  z.e=0;  __mp_dbl(&z,&t,p);
 524   t=ONE/t;   __dbl_mp(t,y,p);    EY -= EX;
 525
 526   for (i=0; i<np1[p]; i++) {
 527     __cpy(y,&w,p);
 528     __mul(x,&w,y,p);
 529     __sub(&mptwo,y,&z,p);
 530     __mul(&w,&z,y,p);
 531   }
 532   return;
 533 }
 534
 535
 536 /* Divide one multiple precision number by another.Set *z = *x / *y. *x & *y */
 537 /* are left unchanged. x&y may overlap but not x&z or y&z.                   */
 538 /* Relative error bound = 2.001*r**(1-p) for p=2, 2.063*r**(1-p) for p=3     */
 539 /* and 3.001*r**(1-p) for p>3. *y=0 is not permissible.                      */
 540
 541 void __dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 542
 543   mp_no w;
 544
 545   if (X[0] == ZERO)    Z[0] = ZERO;
 546   else                {__inv(y,&w,p);   __mul(x,&w,z,p);}
 547   return;
 548 }