sysdeps/ieee754/dbl-64/dla.h

   1 /*
   2  * IBM Accurate Mathematical Library
   3  * Written by International Business Machines Corp.
   4  * Copyright (C) 2001-2012 Free Software Foundation, Inc.
   5  *
   6  * This program is free software; you can redistribute it and/or modify
   7  * it under the terms of the GNU Lesser General Public License as published by
   8  * the Free Software Foundation; either version 2.1 of the License, or
   9  * (at your option) any later version.
  10  *
  11  * This program is distributed in the hope that it will be useful,
  12  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  13  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14  * GNU Lesser General Public License for more details.
  15  *
  16  * You should have received a copy of the GNU Lesser General Public License
  17  * along with this program; if not, see <http://www.gnu.org/licenses/>.
  18  */
  19
  20 /***********************************************************************/
  21 /*MODULE_NAME: dla.h                                                   */
  22 /*                                                                     */
  23 /* This file holds C language macros for 'Double Length Floating Point */
  24 /* Arithmetic'. The macros are based on the paper:                     */
  25 /* T.J.Dekker, "A floating-point Technique for extending the           */
  26 /* Available Precision", Number. Math. 18, 224-242 (1971).              */
  27 /* A Double-Length number is defined by a pair (r,s), of IEEE double    */
  28 /* precision floating point numbers that satisfy,                      */
  29 /*                                                                     */
  30 /*              abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)).              */
  31 /*                                                                     */
  32 /* The computer arithmetic assumed is IEEE double precision in         */
  33 /* round to nearest mode. All variables in the macros must be of type  */
  34 /* IEEE double.                                                        */
  35 /***********************************************************************/
  36
  37 /* CN = 1+2**27 = '41a0000002000000' IEEE double format.  Use it to split a
  38    double for better accuracy.  */
  39 #define  CN   134217729.0
  40
  41
  42 /* Exact addition of two single-length floating point numbers, Dekker. */
  43 /* The macro produces a double-length number (z,zz) that satisfies     */
  44 /* z+zz = x+y exactly.                                                 */
  45
  46 #define  EADD(x,y,z,zz)  \
  47            z=(x)+(y);  zz=(ABS(x)>ABS(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x));
  48
  49
  50 /* Exact subtraction of two single-length floating point numbers, Dekker. */
  51 /* The macro produces a double-length number (z,zz) that satisfies        */
  52 /* z+zz = x-y exactly.                                                    */
  53
  54 #define  ESUB(x,y,z,zz)  \
  55            z=(x)-(y);  zz=(ABS(x)>ABS(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z)));
  56
  57
  58 /* Exact multiplication of two single-length floating point numbers,   */
  59 /* Veltkamp. The macro produces a double-length number (z,zz) that     */
  60 /* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary           */
  61 /* storage variables of type double.                                   */
  62
  63 #ifdef DLA_FMS
  64 # define  EMULV(x,y,z,zz,p,hx,tx,hy,ty)          \
  65            z=x*y; zz=DLA_FMS(x,y,z);
  66 #else
  67 # define  EMULV(x,y,z,zz,p,hx,tx,hy,ty)          \
  68            p=CN*(x);  hx=((x)-p)+p;  tx=(x)-hx; \
  69            p=CN*(y);  hy=((y)-p)+p;  ty=(y)-hy; \
  70            z=(x)*(y); zz=(((hx*hy-z)+hx*ty)+tx*hy)+tx*ty;
  71 #endif
  72
  73
  74 /* Exact multiplication of two single-length floating point numbers, Dekker. */
  75 /* The macro produces a nearly double-length number (z,zz) (see Dekker)      */
  76 /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary          */
  77 /* storage variables of type double.                                         */
  78
  79 #ifdef DLA_FMS
  80 # define  MUL12(x,y,z,zz,p,hx,tx,hy,ty,q)        \
  81            EMULV(x,y,z,zz,p,hx,tx,hy,ty)
  82 #else
  83 # define  MUL12(x,y,z,zz,p,hx,tx,hy,ty,q)        \
  84            p=CN*(x);  hx=((x)-p)+p;  tx=(x)-hx; \
  85            p=CN*(y);  hy=((y)-p)+p;  ty=(y)-hy; \
  86            p=hx*hy;  q=hx*ty+tx*hy; z=p+q;  zz=((p-z)+q)+tx*ty;
  87 #endif
  88
  89
  90 /* Double-length addition, Dekker. The macro produces a double-length   */
  91 /* number (z,zz) which satisfies approximately   z+zz = x+xx + y+yy.    */
  92 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy)       */
  93 /* are assumed to be double-length numbers. r,s are temporary           */
  94 /* storage variables of type double.                                    */
  95
  96 #define  ADD2(x,xx,y,yy,z,zz,r,s)                    \
  97            r=(x)+(y);  s=(ABS(x)>ABS(y)) ?           \
  98                        (((((x)-r)+(y))+(yy))+(xx)) : \
  99                        (((((y)-r)+(x))+(xx))+(yy));  \
 100            z=r+s;  zz=(r-z)+s;
 101
 102
 103 /* Double-length subtraction, Dekker. The macro produces a double-length  */
 104 /* number (z,zz) which satisfies approximately   z+zz = x+xx - (y+yy).    */
 105 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy)         */
 106 /* are assumed to be double-length numbers. r,s are temporary             */
 107 /* storage variables of type double.                                      */
 108
 109 #define  SUB2(x,xx,y,yy,z,zz,r,s)                    \
 110            r=(x)-(y);  s=(ABS(x)>ABS(y)) ?           \
 111                        (((((x)-r)-(y))-(yy))+(xx)) : \
 112                        ((((x)-((y)+r))+(xx))-(yy));  \
 113            z=r+s;  zz=(r-z)+s;
 114
 115
 116 /* Double-length multiplication, Dekker. The macro produces a double-length  */
 117 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)*(y+yy).       */
 118 /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy)               */
 119 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are         */
 120 /* temporary storage variables of type double.                               */
 121
 122 #define  MUL2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc)  \
 123            MUL12(x,y,c,cc,p,hx,tx,hy,ty,q)          \
 124            cc=((x)*(yy)+(xx)*(y))+cc;   z=c+cc;   zz=(c-z)+cc;
 125
 126
 127 /* Double-length division, Dekker. The macro produces a double-length        */
 128 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)/(y+yy).       */
 129 /* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy)               */
 130 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu        */
 131 /* are temporary storage variables of type double.                           */
 132
 133 #define  DIV2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc,u,uu)  \
 134            c=(x)/(y);   MUL12(c,y,u,uu,p,hx,tx,hy,ty,q)  \
 135            cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y);   z=c+cc;   zz=(c-z)+cc;
 136
 137
 138 /* Double-length addition, slower but more accurate than ADD2.               */
 139 /* The macro produces a double-length                                        */
 140 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)+(y+yy).       */
 141 /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy)                 */
 142 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w                 */
 143 /* are temporary storage variables of type double.                           */
 144
 145 #define  ADD2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w)                        \
 146            r=(x)+(y);                                                  \
 147            if (ABS(x)>ABS(y)) { rr=((x)-r)+(y);  s=(rr+(yy))+(xx); }   \
 148            else               { rr=((y)-r)+(x);  s=(rr+(xx))+(yy); }   \
 149            if (rr!=0.0) {                                              \
 150              z=r+s;  zz=(r-z)+s; }                                     \
 151            else {                                                      \
 152              ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)+(yy)) : (((yy)-s)+(xx)); \
 153              u=r+s;                                                    \
 154              uu=(ABS(r)>ABS(s))   ? ((r-u)+s)   : ((s-u)+r)  ;         \
 155              w=uu+ss;  z=u+w;                                          \
 156              zz=(ABS(u)>ABS(w))   ? ((u-z)+w)   : ((w-z)+u)  ; }
 157
 158
 159 /* Double-length subtraction, slower but more accurate than SUB2.            */
 160 /* The macro produces a double-length                                        */
 161 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)-(y+yy).       */
 162 /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy)               */
 163 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w                 */
 164 /* are temporary storage variables of type double.                           */
 165
 166 #define  SUB2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w)                        \
 167            r=(x)-(y);                                                  \
 168            if (ABS(x)>ABS(y)) { rr=((x)-r)-(y);  s=(rr-(yy))+(xx); }   \
 169            else               { rr=(x)-((y)+r);  s=(rr+(xx))-(yy); }   \
 170            if (rr!=0.0) {                                              \
 171              z=r+s;  zz=(r-z)+s; }                                     \
 172            else {                                                      \
 173              ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)-(yy)) : ((xx)-((yy)+s)); \
 174              u=r+s;                                                    \
 175              uu=(ABS(r)>ABS(s))   ? ((r-u)+s)   : ((s-u)+r)  ;         \
 176              w=uu+ss;  z=u+w;                                          \
 177              zz=(ABS(u)>ABS(w))   ? ((u-z)+w)   : ((w-z)+u)  ; }