sysdeps/ieee754/dbl-64/dla.h

   1 /*
   2  * IBM Accurate Mathematical Library
   3  * Written by International Business Machines Corp.
   4  * Copyright (C) 2001, 2011 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, write to the Free Software
  18  * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
  19  */
  20
  21 /***********************************************************************/
  22 /*MODULE_NAME: dla.h                                                   */
  23 /*                                                                     */
  24 /* This file holds C language macros for 'Double Length Floating Point */
  25 /* Arithmetic'. The macros are based on the paper:                     */
  26 /* T.J.Dekker, "A floating-point Technique for extending the           */
  27 /* Available Precision", Number. Math. 18, 224-242 (1971).              */
  28 /* A Double-Length number is defined by a pair (r,s), of IEEE double    */
  29 /* precision floating point numbers that satisfy,                      */
  30 /*                                                                     */
  31 /*              abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)).              */
  32 /*                                                                     */
  33 /* The computer arithmetic assumed is IEEE double precision in         */
  34 /* round to nearest mode. All variables in the macros must be of type  */
  35 /* IEEE double.                                                        */
  36 /***********************************************************************/
  37
  38 /* We can use fma instructions if available.  */
  39 #if defined __x86_64__ || (defined __i386__ && defined __SSE2_MATH__)
  40 # ifdef __FMA4__
  41 #  define DLA_FMA(x,y,z) \
  42            ({ double __zz; \
  43               asm ("vfmsubsd %3, %2, %1, %0"                                  \
  44                    : "=x" (__zz) : "x" (x), "xm" (y), "x" (z));               \
  45               __zz; })
  46 # endif
  47 #endif
  48
  49
  50 /* CN = 1+2**27 = '41a0000002000000' IEEE double format */
  51 #define  CN   134217729.0
  52
  53
  54 /* Exact addition of two single-length floating point numbers, Dekker. */
  55 /* The macro produces a double-length number (z,zz) that satisfies     */
  56 /* z+zz = x+y exactly.                                                 */
  57
  58 #define  EADD(x,y,z,zz)  \
  59            z=(x)+(y);  zz=(ABS(x)>ABS(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x));
  60
  61
  62 /* Exact subtraction of two single-length floating point numbers, Dekker. */
  63 /* The macro produces a double-length number (z,zz) that satisfies        */
  64 /* z+zz = x-y exactly.                                                    */
  65
  66 #define  ESUB(x,y,z,zz)  \
  67            z=(x)-(y);  zz=(ABS(x)>ABS(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z)));
  68
  69
  70 /* Exact multiplication of two single-length floating point numbers,   */
  71 /* Veltkamp. The macro produces a double-length number (z,zz) that     */
  72 /* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary           */
  73 /* storage variables of type double.                                   */
  74
  75 #ifdef DLA_FMA
  76 # define  EMULV(x,y,z,zz,p,hx,tx,hy,ty)          \
  77            z=x*y; zz=DLA_FMA(x,y,z);
  78 #else
  79 # define  EMULV(x,y,z,zz,p,hx,tx,hy,ty)          \
  80            p=CN*(x);  hx=((x)-p)+p;  tx=(x)-hx; \
  81            p=CN*(y);  hy=((y)-p)+p;  ty=(y)-hy; \
  82            z=(x)*(y); zz=(((hx*hy-z)+hx*ty)+tx*hy)+tx*ty;
  83 #endif
  84
  85
  86 /* Exact multiplication of two single-length floating point numbers, Dekker. */
  87 /* The macro produces a nearly double-length number (z,zz) (see Dekker)      */
  88 /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary          */
  89 /* storage variables of type double.                                         */
  90
  91 #ifdef DLA_FMA
  92 # define  MUL12(x,y,z,zz,p,hx,tx,hy,ty,q)        \
  93            EMULV(x,y,z,zz,p,hx,tx,hy,ty)
  94 #else
  95 # define  MUL12(x,y,z,zz,p,hx,tx,hy,ty,q)        \
  96            p=CN*(x);  hx=((x)-p)+p;  tx=(x)-hx; \
  97            p=CN*(y);  hy=((y)-p)+p;  ty=(y)-hy; \
  98            p=hx*hy;  q=hx*ty+tx*hy; z=p+q;  zz=((p-z)+q)+tx*ty;
  99 #endif
 100
 101
 102 /* Double-length addition, Dekker. The macro produces a double-length   */
 103 /* number (z,zz) which satisfies approximately   z+zz = x+xx + y+yy.    */
 104 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy)       */
 105 /* are assumed to be double-length numbers. r,s are temporary           */
 106 /* storage variables of type double.                                    */
 107
 108 #define  ADD2(x,xx,y,yy,z,zz,r,s)                    \
 109            r=(x)+(y);  s=(ABS(x)>ABS(y)) ?           \
 110                        (((((x)-r)+(y))+(yy))+(xx)) : \
 111                        (((((y)-r)+(x))+(xx))+(yy));  \
 112            z=r+s;  zz=(r-z)+s;
 113
 114
 115 /* Double-length subtraction, Dekker. The macro produces a double-length  */
 116 /* number (z,zz) which satisfies approximately   z+zz = x+xx - (y+yy).    */
 117 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy)         */
 118 /* are assumed to be double-length numbers. r,s are temporary             */
 119 /* storage variables of type double.                                      */
 120
 121 #define  SUB2(x,xx,y,yy,z,zz,r,s)                    \
 122            r=(x)-(y);  s=(ABS(x)>ABS(y)) ?           \
 123                        (((((x)-r)-(y))-(yy))+(xx)) : \
 124                        ((((x)-((y)+r))+(xx))-(yy));  \
 125            z=r+s;  zz=(r-z)+s;
 126
 127
 128 /* Double-length multiplication, Dekker. The macro produces a double-length  */
 129 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)*(y+yy).       */
 130 /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy)               */
 131 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are         */
 132 /* temporary storage variables of type double.                               */
 133
 134 #define  MUL2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc)  \
 135            MUL12(x,y,c,cc,p,hx,tx,hy,ty,q)          \
 136            cc=((x)*(yy)+(xx)*(y))+cc;   z=c+cc;   zz=(c-z)+cc;
 137
 138
 139 /* Double-length division, Dekker. 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. p,hx,tx,hy,ty,q,c,cc,u,uu        */
 143 /* are temporary storage variables of type double.                           */
 144
 145 #define  DIV2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc,u,uu)  \
 146            c=(x)/(y);   MUL12(c,y,u,uu,p,hx,tx,hy,ty,q)  \
 147            cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y);   z=c+cc;   zz=(c-z)+cc;
 148
 149
 150 /* Double-length addition, slower but more accurate than ADD2.               */
 151 /* The macro produces a double-length                                        */
 152 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)+(y+yy).       */
 153 /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy)                 */
 154 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w                 */
 155 /* are temporary storage variables of type double.                           */
 156
 157 #define  ADD2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w)                        \
 158            r=(x)+(y);                                                  \
 159            if (ABS(x)>ABS(y)) { rr=((x)-r)+(y);  s=(rr+(yy))+(xx); }   \
 160            else               { rr=((y)-r)+(x);  s=(rr+(xx))+(yy); }   \
 161            if (rr!=0.0) {                                              \
 162              z=r+s;  zz=(r-z)+s; }                                     \
 163            else {                                                      \
 164              ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)+(yy)) : (((yy)-s)+(xx)); \
 165              u=r+s;                                                    \
 166              uu=(ABS(r)>ABS(s))   ? ((r-u)+s)   : ((s-u)+r)  ;         \
 167              w=uu+ss;  z=u+w;                                          \
 168              zz=(ABS(u)>ABS(w))   ? ((u-z)+w)   : ((w-z)+u)  ; }
 169
 170
 171 /* Double-length subtraction, slower but more accurate than SUB2.            */
 172 /* The macro produces a double-length                                        */
 173 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)-(y+yy).       */
 174 /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy)               */
 175 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w                 */
 176 /* are temporary storage variables of type double.                           */
 177
 178 #define  SUB2A(x,xx,y,yy,z,zz,r,rr,s,ss,u,uu,w)                        \
 179            r=(x)-(y);                                                  \
 180            if (ABS(x)>ABS(y)) { rr=((x)-r)-(y);  s=(rr-(yy))+(xx); }   \
 181            else               { rr=(x)-((y)+r);  s=(rr+(xx))-(yy); }   \
 182            if (rr!=0.0) {                                              \
 183              z=r+s;  zz=(r-z)+s; }                                     \
 184            else {                                                      \
 185              ss=(ABS(xx)>ABS(yy)) ? (((xx)-s)-(yy)) : ((xx)-((yy)+s)); \
 186              u=r+s;                                                    \
 187              uu=(ABS(r)>ABS(s))   ? ((r-u)+s)   : ((s-u)+r)  ;         \
 188              w=uu+ss;  z=u+w;                                          \
 189              zz=(ABS(u)>ABS(w))   ? ((u-z)+w)   : ((w-z)+u)  ; }