sysdeps/ieee754/dbl-64/dla.h

   1 /*
   2  * IBM Accurate Mathematical Library
   3  * Written by International Business Machines Corp.
   4  * Copyright (C) 2001-2019 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 <https://www.gnu.org/licenses/>.
  18  */
  19
  20 #include <math.h>
  21
  22 /***********************************************************************/
  23 /*MODULE_NAME: dla.h                                                   */
  24 /*                                                                     */
  25 /* This file holds C language macros for 'Double Length Floating Point */
  26 /* Arithmetic'. The macros are based on the paper:                     */
  27 /* T.J.Dekker, "A floating-point Technique for extending the           */
  28 /* Available Precision", Number. Math. 18, 224-242 (1971).              */
  29 /* A Double-Length number is defined by a pair (r,s), of IEEE double    */
  30 /* precision floating point numbers that satisfy,                      */
  31 /*                                                                     */
  32 /*              abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)).              */
  33 /*                                                                     */
  34 /* The computer arithmetic assumed is IEEE double precision in         */
  35 /* round to nearest mode. All variables in the macros must be of type  */
  36 /* IEEE double.                                                        */
  37 /***********************************************************************/
  38
  39 /* CN = 1+2**27 = '41a0000002000000' IEEE double format.  Use it to split a
  40    double for better accuracy.  */
  41 #define  CN   134217729.0
  42
  43
  44 /* Exact addition of two single-length floating point numbers, Dekker. */
  45 /* The macro produces a double-length number (z,zz) that satisfies     */
  46 /* z+zz = x+y exactly.                                                 */
  47
  48 #define  EADD(x,y,z,zz)  \
  49            z=(x)+(y);  zz=(fabs(x)>fabs(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x));
  50
  51
  52 /* Exact subtraction of two single-length floating point numbers, Dekker. */
  53 /* The macro produces a double-length number (z,zz) that satisfies        */
  54 /* z+zz = x-y exactly.                                                    */
  55
  56 #define  ESUB(x,y,z,zz)  \
  57            z=(x)-(y);  zz=(fabs(x)>fabs(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z)));
  58
  59
  60 #ifdef __FP_FAST_FMA
  61 # define DLA_FMS(x, y, z) __builtin_fma (x, y, -(z))
  62 #endif
  63
  64 /* Exact multiplication of two single-length floating point numbers,   */
  65 /* Veltkamp. The macro produces a double-length number (z,zz) that     */
  66 /* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary           */
  67 /* storage variables of type double.                                   */
  68
  69 #ifdef DLA_FMS
  70 # define  EMULV(x, y, z, zz, p, hx, tx, hy, ty)          \
  71   z = x * y; zz = DLA_FMS (x, y, z);
  72 #else
  73 # define  EMULV(x, y, z, zz, p, hx, tx, hy, ty)          \
  74   p = CN * (x);  hx = ((x) - p) + p;  tx = (x) - hx; \
  75   p = CN * (y);  hy = ((y) - p) + p;  ty = (y) - hy; \
  76   z = (x) * (y); zz = (((hx * hy - z) + hx * ty) + tx * hy) + tx * ty;
  77 #endif
  78
  79
  80 /* Exact multiplication of two single-length floating point numbers, Dekker. */
  81 /* The macro produces a nearly double-length number (z,zz) (see Dekker)      */
  82 /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary          */
  83 /* storage variables of type double.                                         */
  84
  85 #ifdef DLA_FMS
  86 # define  MUL12(x,y,z,zz,p,hx,tx,hy,ty,q)        \
  87            EMULV(x,y,z,zz,p,hx,tx,hy,ty)
  88 #else
  89 # define  MUL12(x,y,z,zz,p,hx,tx,hy,ty,q)        \
  90            p=CN*(x);  hx=((x)-p)+p;  tx=(x)-hx; \
  91            p=CN*(y);  hy=((y)-p)+p;  ty=(y)-hy; \
  92            p=hx*hy;  q=hx*ty+tx*hy; z=p+q;  zz=((p-z)+q)+tx*ty;
  93 #endif
  94
  95
  96 /* Double-length addition, Dekker. The macro produces a double-length   */
  97 /* number (z,zz) which satisfies approximately   z+zz = x+xx + y+yy.    */
  98 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy)       */
  99 /* are assumed to be double-length numbers. r,s are temporary           */
 100 /* storage variables of type double.                                    */
 101
 102 #define  ADD2(x, xx, y, yy, z, zz, r, s)                   \
 103   r = (x) + (y);  s = (fabs (x) > fabs (y)) ?                \
 104                       (((((x) - r) + (y)) + (yy)) + (xx)) : \
 105                       (((((y) - r) + (x)) + (xx)) + (yy));  \
 106   z = r + s;  zz = (r - z) + s;
 107
 108
 109 /* Double-length subtraction, Dekker. The macro produces a double-length  */
 110 /* number (z,zz) which satisfies approximately   z+zz = x+xx - (y+yy).    */
 111 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy)         */
 112 /* are assumed to be double-length numbers. r,s are temporary             */
 113 /* storage variables of type double.                                      */
 114
 115 #define  SUB2(x, xx, y, yy, z, zz, r, s)                   \
 116   r = (x) - (y);  s = (fabs (x) > fabs (y)) ?                \
 117                       (((((x) - r) - (y)) - (yy)) + (xx)) : \
 118                       ((((x) - ((y) + r)) + (xx)) - (yy));  \
 119   z = r + s;  zz = (r - z) + s;
 120
 121
 122 /* Double-length multiplication, Dekker. The macro produces a double-length  */
 123 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)*(y+yy).       */
 124 /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy)               */
 125 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are         */
 126 /* temporary storage variables of type double.                               */
 127
 128 #define  MUL2(x, xx, y, yy, z, zz, p, hx, tx, hy, ty, q, c, cc)  \
 129   MUL12 (x, y, c, cc, p, hx, tx, hy, ty, q)                      \
 130   cc = ((x) * (yy) + (xx) * (y)) + cc;   z = c + cc;   zz = (c - z) + cc;
 131
 132
 133 /* Double-length division, Dekker. The macro produces a double-length        */
 134 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)/(y+yy).       */
 135 /* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy)               */
 136 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu        */
 137 /* are temporary storage variables of type double.                           */
 138
 139 #define  DIV2(x,xx,y,yy,z,zz,p,hx,tx,hy,ty,q,c,cc,u,uu)  \
 140            c=(x)/(y);   MUL12(c,y,u,uu,p,hx,tx,hy,ty,q)  \
 141            cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y);   z=c+cc;   zz=(c-z)+cc;
 142
 143
 144 /* Double-length addition, slower but more accurate than ADD2.               */
 145 /* The macro produces a double-length                                        */
 146 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)+(y+yy).       */
 147 /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy)                 */
 148 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w                 */
 149 /* are temporary storage variables of type double.                           */
 150
 151 #define  ADD2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w)                 \
 152   r = (x) + (y);                                                            \
 153   if (fabs (x) > fabs (y)) { rr = ((x) - r) + (y);  s = (rr + (yy)) + (xx); } \
 154   else               { rr = ((y) - r) + (x);  s = (rr + (xx)) + (yy); }     \
 155   if (rr != 0.0) {                                                          \
 156       z = r + s;  zz = (r - z) + s; }                                       \
 157   else {                                                                    \
 158       ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) + (yy)) : (((yy) - s) + (xx));\
 159       u = r + s;                                                            \
 160       uu = (fabs (r) > fabs (s))   ? ((r - u) + s)   : ((s - u) + r);         \
 161       w = uu + ss;  z = u + w;                                              \
 162       zz = (fabs (u) > fabs (w))   ? ((u - z) + w)   : ((w - z) + u); }
 163
 164
 165 /* Double-length subtraction, slower but more accurate than SUB2.            */
 166 /* The macro produces a double-length                                        */
 167 /* number (z,zz) which satisfies approximately   z+zz = (x+xx)-(y+yy).       */
 168 /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy)               */
 169 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w                 */
 170 /* are temporary storage variables of type double.                           */
 171
 172 #define  SUB2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w)                   \
 173   r = (x) - (y);                                                              \
 174   if (fabs (x) > fabs (y)) { rr = ((x) - r) - (y);  s = (rr - (yy)) + (xx); }   \
 175   else               { rr = (x) - ((y) + r);  s = (rr + (xx)) - (yy); }       \
 176   if (rr != 0.0) {                                                            \
 177       z = r + s;  zz = (r - z) + s; }                                         \
 178   else {                                                                      \
 179       ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) - (yy)) : ((xx) - ((yy) + s)); \
 180       u = r + s;                                                              \
 181       uu = (fabs (r) > fabs (s))   ? ((r - u) + s)   : ((s - u) + r);           \
 182       w = uu + ss;  z = u + w;                                                \
 183       zz = (fabs (u) > fabs (w))   ? ((u - z) + w)   : ((w - z) + u); }