sysdeps/powerpc/powerpc64/power4/fpu/mpa.c

   1
   2 /*
   3  * IBM Accurate Mathematical Library
   4  * written by International Business Machines Corp.
   5  * Copyright (C) 2001-2013 Free Software Foundation, Inc.
   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 /*               cpy                                                    */
  27 /*               norm                                                   */
  28 /*               denorm                                                 */
  29 /*               mp_dbl                                                 */
  30 /*               dbl_mp                                                 */
  31 /*               add_magnitudes                                         */
  32 /*               sub_magnitudes                                         */
  33 /*               add                                                    */
  34 /*               sub                                                    */
  35 /*               mul                                                    */
  36 /*               inv                                                    */
  37 /*               dvd                                                    */
  38 /*                                                                      */
  39 /* Arithmetic functions for multiple precision numbers.                 */
  40 /* Relative errors are bounded                                          */
  41 /************************************************************************/
  42
  43
  44 #include "endian.h"
  45 #include "mpa.h"
  46 #include "mpa2.h"
  47 #include <sys/param.h>
  48
  49 const mp_no mpone = {1, {1.0, 1.0}};
  50 const mp_no mptwo = {1, {1.0, 2.0}};
  51
  52 /* Compare mantissa of two multiple precision numbers regardless of the sign
  53    and exponent of the numbers.  */
  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 /* Compare the absolute values of two multiple precision numbers.  */
  65 int __acr(const mp_no *x, const mp_no *y, int p) {
  66   long i;
  67
  68   if      (X[0] == ZERO) {
  69     if    (Y[0] == ZERO) i= 0;
  70     else                 i=-1;
  71   }
  72   else if (Y[0] == ZERO) i= 1;
  73   else {
  74     if      (EX >  EY)   i= 1;
  75     else if (EX <  EY)   i=-1;
  76     else                 i= mcr(x,y,p);
  77   }
  78
  79   return i;
  80 }
  81
  82 /* Copy multiple precision number X into Y.  They could be the same
  83    number.  */
  84 void __cpy(const mp_no *x, mp_no *y, int p) {
  85   long i;
  86
  87   EY = EX;
  88   for (i=0; i <= p; i++)    Y[i] = X[i];
  89
  90   return;
  91 }
  92
  93 /* Convert a multiple precision number *X into a double precision
  94    number *Y, normalized case  (|x| >= 2**(-1022))).  */
  95 static void norm(const mp_no *x, double *y, int p)
  96 {
  97   #define R  RADIXI
  98   long i;
  99   double a,c,u,v,z[5];
 100   if (p<5) {
 101     if      (p==1) c = X[1];
 102     else if (p==2) c = X[1] + R* X[2];
 103     else if (p==3) c = X[1] + R*(X[2]  +   R* X[3]);
 104     else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
 105   }
 106   else {
 107     for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
 108         {a *= TWO;   z[1] *= TWO; }
 109
 110     for (i=2; i<5; i++) {
 111       z[i] = X[i]*a;
 112       u = (z[i] + CUTTER)-CUTTER;
 113       if  (u > z[i])  u -= RADIX;
 114       z[i] -= u;
 115       z[i-1] += u*RADIXI;
 116     }
 117
 118     u = (z[3] + TWO71) - TWO71;
 119     if (u > z[3])   u -= TWO19;
 120     v = z[3]-u;
 121
 122     if (v == TWO18) {
 123       if (z[4] == ZERO) {
 124         for (i=5; i <= p; i++) {
 125           if (X[i] == ZERO)   continue;
 126           else                {z[3] += ONE;   break; }
 127         }
 128       }
 129       else              z[3] += ONE;
 130     }
 131
 132     c = (z[1] + R *(z[2] + R * z[3]))/a;
 133   }
 134
 135   c *= X[0];
 136
 137   for (i=1; i<EX; i++)   c *= RADIX;
 138   for (i=1; i>EX; i--)   c *= RADIXI;
 139
 140   *y = c;
 141   return;
 142 #undef R
 143 }
 144
 145 /* Convert a multiple precision number *X into a double precision
 146    number *Y, Denormal case  (|x| < 2**(-1022))).  */
 147 static void denorm(const mp_no *x, double *y, int p)
 148 {
 149   long i,k;
 150   long p2 = p;
 151   double c,u,z[5];
 152
 153 #define R RADIXI
 154   if (EX<-44 || (EX==-44 && X[1]<TWO5))
 155      { *y=ZERO; return; }
 156
 157   if      (p2==1) {
 158     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=ZERO;  z[3]=ZERO;  k=3;}
 159     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=ZERO;  k=2;}
 160     else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
 161   }
 162   else if (p2==2) {
 163     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  z[3]=ZERO;  k=3;}
 164     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  z[3]=X[2];  k=2;}
 165     else              {z[1]=     TWO10;  z[2]=ZERO;  z[3]=X[1];  k=1;}
 166   }
 167   else {
 168     if      (EX==-42) {z[1]=X[1]+TWO10;  z[2]=X[2];  k=3;}
 169     else if (EX==-43) {z[1]=     TWO10;  z[2]=X[1];  k=2;}
 170     else              {z[1]=     TWO10;  z[2]=ZERO;  k=1;}
 171     z[3] = X[k];
 172   }
 173
 174   u = (z[3] + TWO57) - TWO57;
 175   if  (u > z[3])   u -= TWO5;
 176
 177   if (u==z[3]) {
 178     for (i=k+1; i <= p2; i++) {
 179       if (X[i] == ZERO)   continue;
 180       else {z[3] += ONE;   break; }
 181     }
 182   }
 183
 184   c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);
 185
 186   *y = c*TWOM1032;
 187   return;
 188
 189 #undef R
 190 }
 191
 192 /* Convert multiple precision number *X into double precision number *Y.  The
 193    result is correctly rounded to the nearest/even.  */
 194 void __mp_dbl(const mp_no *x, double *y, int p) {
 195
 196   if (X[0] == ZERO)  {*y = ZERO;  return; }
 197
 198   if      (EX> -42)                 norm(x,y,p);
 199   else if (EX==-42 && X[1]>=TWO10)  norm(x,y,p);
 200   else                              denorm(x,y,p);
 201 }
 202
 203 /* Get the multiple precision equivalent of X into *Y.  If the precision is too
 204    small, the result is truncated.  */
 205 void __dbl_mp(double x, mp_no *y, int p) {
 206
 207   long i,n;
 208   long p2 = p;
 209   double u;
 210
 211   /* Sign.  */
 212   if      (x == ZERO)  {Y[0] = ZERO;  return; }
 213   else if (x >  ZERO)   Y[0] = ONE;
 214   else                 {Y[0] = MONE;  x=-x;   }
 215
 216   /* Exponent.  */
 217   for (EY=ONE; x >= RADIX; EY += ONE)   x *= RADIXI;
 218   for (      ; x <  ONE;   EY -= ONE)   x *= RADIX;
 219
 220   /* Digits.  */
 221   n=MIN(p2,4);
 222   for (i=1; i<=n; i++) {
 223     u = (x + TWO52) - TWO52;
 224     if (u>x)   u -= ONE;
 225     Y[i] = u;     x -= u;    x *= RADIX; }
 226   for (   ; i<=p2; i++)     Y[i] = ZERO;
 227   return;
 228 }
 229
 230 /* Add magnitudes of *X and *Y assuming that abs (*X) >= abs (*Y) > 0.  The
 231    sign of the sum *Z is not changed.  X and Y may overlap but not X and Z or
 232    Y and Z.  No guard digit is used.  The result equals the exact sum,
 233    truncated.  */
 234 static void add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 235
 236   long i,j,k;
 237   long p2 = p;
 238
 239   EZ = EX;
 240
 241   i=p2;    j=p2+ EY - EX;    k=p2+1;
 242
 243   if (j<1)
 244      {__cpy(x,z,p);  return; }
 245   else   Z[k] = ZERO;
 246
 247   for (; j>0; i--,j--) {
 248     Z[k] += X[i] + Y[j];
 249     if (Z[k] >= RADIX) {
 250       Z[k]  -= RADIX;
 251       Z[--k] = ONE; }
 252     else
 253       Z[--k] = ZERO;
 254   }
 255
 256   for (; i>0; i--) {
 257     Z[k] += X[i];
 258     if (Z[k] >= RADIX) {
 259       Z[k]  -= RADIX;
 260       Z[--k] = ONE; }
 261     else
 262       Z[--k] = ZERO;
 263   }
 264
 265   if (Z[1] == ZERO) {
 266     for (i=1; i<=p2; i++)    Z[i] = Z[i+1]; }
 267   else   EZ += ONE;
 268 }
 269
 270 /* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0.
 271    The sign of the difference *Z is not changed.  X and Y may overlap but not X
 272    and Z or Y and Z.  One guard digit is used.  The error is less than one
 273    ULP.  */
 274 static void sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 275
 276   long i,j,k;
 277   long p2 = p;
 278
 279   EZ = EX;
 280
 281   if (EX == EY) {
 282     i=j=k=p2;
 283     Z[k] = Z[k+1] = ZERO; }
 284   else {
 285     j= EX - EY;
 286     if (j > p2)  {__cpy(x,z,p);  return; }
 287     else {
 288       i=p2;   j=p2+1-j;   k=p2;
 289       if (Y[j] > ZERO) {
 290         Z[k+1] = RADIX - Y[j--];
 291         Z[k]   = MONE; }
 292       else {
 293         Z[k+1] = ZERO;
 294         Z[k]   = ZERO;   j--;}
 295     }
 296   }
 297
 298   for (; j>0; i--,j--) {
 299     Z[k] += (X[i] - Y[j]);
 300     if (Z[k] < ZERO) {
 301       Z[k]  += RADIX;
 302       Z[--k] = MONE; }
 303     else
 304       Z[--k] = ZERO;
 305   }
 306
 307   for (; i>0; i--) {
 308     Z[k] += X[i];
 309     if (Z[k] < ZERO) {
 310       Z[k]  += RADIX;
 311       Z[--k] = MONE; }
 312     else
 313       Z[--k] = ZERO;
 314   }
 315
 316   for (i=1; Z[i] == ZERO; i++) ;
 317   EZ = EZ - i + 1;
 318   for (k=1; i <= p2+1; )
 319     Z[k++] = Z[i++];
 320   for (; k <= p2; )
 321     Z[k++] = ZERO;
 322
 323   return;
 324 }
 325
 326 /* Add *X and *Y and store the result in *Z.  X and Y may overlap, but not X
 327    and Z or Y and Z.  One guard digit is used.  The error is less than one
 328    ULP.  */
 329 void __add(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 330
 331   int n;
 332
 333   if      (X[0] == ZERO)     {__cpy(y,z,p);  return; }
 334   else if (Y[0] == ZERO)     {__cpy(x,z,p);  return; }
 335
 336   if (X[0] == Y[0])   {
 337     if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] = X[0]; }
 338     else                     {add_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
 339   }
 340   else                       {
 341     if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] = X[0]; }
 342     else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = Y[0]; }
 343     else                      Z[0] = ZERO;
 344   }
 345   return;
 346 }
 347
 348 /* Subtract *Y from *X and return the result in *Z.  X and Y may overlap but
 349    not X and Z or Y and Z.  One guard digit is used.  The error is less than
 350    one ULP.  */
 351 void __sub(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 352
 353   int n;
 354
 355   if      (X[0] == ZERO)     {__cpy(y,z,p);  Z[0] = -Z[0];  return; }
 356   else if (Y[0] == ZERO)     {__cpy(x,z,p);                 return; }
 357
 358   if (X[0] != Y[0])    {
 359     if (__acr(x,y,p) > 0)      {add_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
 360     else                     {add_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
 361   }
 362   else                       {
 363     if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p);  Z[0] =  X[0]; }
 364     else if (n == -1)        {sub_magnitudes(y,x,z,p);  Z[0] = -Y[0]; }
 365     else                      Z[0] = ZERO;
 366   }
 367   return;
 368 }
 369
 370 /* Multiply *X and *Y and store result in *Z.  X and Y may overlap but not X
 371    and Z or Y and Z.  For P in [1, 2, 3], the exact result is truncated to P
 372    digits.  In case P > 3 the error is bounded by 1.001 ULP.  */
 373 void __mul(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 374
 375   long i, i1, i2, j, k, k2;
 376   long p2 = p;
 377   double u, zk, zk2;
 378
 379                       /* Is z=0? */
 380   if (X[0]*Y[0]==ZERO)
 381      { Z[0]=ZERO;  return; }
 382
 383                        /* Multiply, add and carry */
 384   k2 = (p2<3) ? p2+p2 : p2+3;
 385   zk = Z[k2]=ZERO;
 386   for (k=k2; k>1; ) {
 387     if (k > p2)  {i1=k-p2; i2=p2+1; }
 388     else        {i1=1;   i2=k;   }
 389 #if 1
 390     /* Rearrange this inner loop to allow the fmadd instructions to be
 391        independent and execute in parallel on processors that have
 392        dual symmetrical FP pipelines.  */
 393     if (i1 < (i2-1))
 394     {
 395         /* Make sure we have at least 2 iterations.  */
 396         if (((i2 - i1) & 1L) == 1L)
 397         {
 398                 /* Handle the odd iterations case.  */
 399                 zk2 = x->d[i2-1]*y->d[i1];
 400         }
 401         else
 402                 zk2 = 0.0;
 403         /* Do two multiply/adds per loop iteration, using independent
 404            accumulators; zk and zk2.  */
 405         for (i=i1,j=i2-1; i<i2-1; i+=2,j-=2)
 406         {
 407                 zk += x->d[i]*y->d[j];
 408                 zk2 += x->d[i+1]*y->d[j-1];
 409         }
 410         zk += zk2; /* Final sum.  */
 411     }
 412     else
 413     {
 414         /* Special case when iterations is 1.  */
 415         zk += x->d[i1]*y->d[i1];
 416     }
 417 #else
 418     /* The original code.  */
 419     for (i=i1,j=i2-1; i<i2; i++,j--)  zk += X[i]*Y[j];
 420 #endif
 421
 422     u = (zk + CUTTER)-CUTTER;
 423     if  (u > zk)  u -= RADIX;
 424     Z[k]  = zk - u;
 425     zk = u*RADIXI;
 426     --k;
 427   }
 428   Z[k] = zk;
 429
 430   /* Is there a carry beyond the most significant digit?  */
 431   if (Z[1] == ZERO) {
 432     for (i=1; i<=p2; i++)  Z[i]=Z[i+1];
 433     EZ = EX + EY - 1; }
 434   else
 435     EZ = EX + EY;
 436
 437   Z[0] = X[0] * Y[0];
 438   return;
 439 }
 440
 441 /* Invert *X and store in *Y.  Relative error bound:
 442    - For P = 2: 1.001 * R ^ (1 - P)
 443    - For P = 3: 1.063 * R ^ (1 - P)
 444    - For P > 3: 2.001 * R ^ (1 - P)
 445
 446    *X = 0 is not permissible.  */
 447 void __inv(const mp_no *x, mp_no *y, int p) {
 448   long i;
 449   double t;
 450   mp_no z,w;
 451   static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
 452                             4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};
 453
 454   __cpy(x,&z,p);  z.e=0;  __mp_dbl(&z,&t,p);
 455   t=ONE/t;   __dbl_mp(t,y,p);    EY -= EX;
 456
 457   for (i=0; i<np1[p]; i++) {
 458     __cpy(y,&w,p);
 459     __mul(x,&w,y,p);
 460     __sub(&mptwo,y,&z,p);
 461     __mul(&w,&z,y,p);
 462   }
 463   return;
 464 }
 465
 466 /* Divide *X by *Y and store result in *Z.  X and Y may overlap but not X and Z
 467    or Y and Z.  Relative error bound:
 468    - For P = 2: 2.001 * R ^ (1 - P)
 469    - For P = 3: 2.063 * R ^ (1 - P)
 470    - For P > 3: 3.001 * R ^ (1 - P)
 471
 472    *X = 0 is not permissible.  */
 473 void __dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {
 474
 475   mp_no w;
 476
 477   if (X[0] == ZERO)    Z[0] = ZERO;
 478   else                {__inv(y,&w,p);   __mul(x,&w,z,p);}
 479   return;
 480 }