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/*
* IBM Accurate Mathematical Library
* Written by International Business Machines Corp.
* Copyright (C) 2001-2020 Free Software Foundation, Inc.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2.1 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, see <https://www.gnu.org/licenses/>.
*/
#include <math.h>
/***********************************************************************/
/*MODULE_NAME: dla.h */
/* */
/* This file holds C language macros for 'Double Length Floating Point */
/* Arithmetic'. The macros are based on the paper: */
/* T.J.Dekker, "A floating-point Technique for extending the */
/* Available Precision", Number. Math. 18, 224-242 (1971). */
/* A Double-Length number is defined by a pair (r,s), of IEEE double */
/* precision floating point numbers that satisfy, */
/* */
/* abs(s) <= abs(r+s)*2**(-53)/(1+2**(-53)). */
/* */
/* The computer arithmetic assumed is IEEE double precision in */
/* round to nearest mode. All variables in the macros must be of type */
/* IEEE double. */
/***********************************************************************/
/* CN = 1+2**27 = '41a0000002000000' IEEE double format. Use it to split a
double for better accuracy. */
#define CN 134217729.0
/* Exact addition of two single-length floating point numbers, Dekker. */
/* The macro produces a double-length number (z,zz) that satisfies */
/* z+zz = x+y exactly. */
#define EADD(x,y,z,zz) \
z=(x)+(y); zz=(fabs(x)>fabs(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x));
/* Exact subtraction of two single-length floating point numbers, Dekker. */
/* The macro produces a double-length number (z,zz) that satisfies */
/* z+zz = x-y exactly. */
#define ESUB(x,y,z,zz) \
z=(x)-(y); zz=(fabs(x)>fabs(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z)));
#ifdef __FP_FAST_FMA
# define DLA_FMS(x, y, z) __builtin_fma (x, y, -(z))
#endif
/* Exact multiplication of two single-length floating point numbers, */
/* Veltkamp. The macro produces a double-length number (z,zz) that */
/* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary */
/* storage variables of type double. */
#ifdef DLA_FMS
# define EMULV(x, y, z, zz) \
z = x * y; zz = DLA_FMS (x, y, z);
#else
# define EMULV(x, y, z, zz) \
({ __typeof__ (x) __p, hx, tx, hy, ty; \
__p = CN * (x); hx = ((x) - __p) + __p; tx = (x) - hx; \
__p = CN * (y); hy = ((y) - __p) + __p; ty = (y) - hy; \
z = (x) * (y); zz = (((hx * hy - z) + hx * ty) + tx * hy) + tx * ty; \
})
#endif
/* Exact multiplication of two single-length floating point numbers, Dekker. */
/* The macro produces a nearly double-length number (z,zz) (see Dekker) */
/* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary */
/* storage variables of type double. */
#ifdef DLA_FMS
# define MUL12(x, y, z, zz) \
EMULV(x, y, z, zz)
#else
# define MUL12(x, y, z, zz) \
({ __typeof__ (x) __p, hx, tx, hy, ty, __q; \
__p=CN*(x); hx=((x)-__p)+__p; tx=(x)-hx; \
__p=CN*(y); hy=((y)-__p)+__p; ty=(y)-hy; \
__p=hx*hy; __q=hx*ty+tx*hy; z=__p+__q; zz=((__p-z)+__q)+tx*ty; \
})
#endif
/* Double-length addition, Dekker. The macro produces a double-length */
/* number (z,zz) which satisfies approximately z+zz = x+xx + y+yy. */
/* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */
/* are assumed to be double-length numbers. r,s are temporary */
/* storage variables of type double. */
#define ADD2(x, xx, y, yy, z, zz, r, s) \
r = (x) + (y); s = (fabs (x) > fabs (y)) ? \
(((((x) - r) + (y)) + (yy)) + (xx)) : \
(((((y) - r) + (x)) + (xx)) + (yy)); \
z = r + s; zz = (r - z) + s;
/* Double-length subtraction, Dekker. The macro produces a double-length */
/* number (z,zz) which satisfies approximately z+zz = x+xx - (y+yy). */
/* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */
/* are assumed to be double-length numbers. r,s are temporary */
/* storage variables of type double. */
#define SUB2(x, xx, y, yy, z, zz, r, s) \
r = (x) - (y); s = (fabs (x) > fabs (y)) ? \
(((((x) - r) - (y)) - (yy)) + (xx)) : \
((((x) - ((y) + r)) + (xx)) - (yy)); \
z = r + s; zz = (r - z) + s;
/* Double-length multiplication, Dekker. The macro produces a double-length */
/* number (z,zz) which satisfies approximately z+zz = (x+xx)*(y+yy). */
/* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy) */
/* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are */
/* temporary storage variables of type double. */
#define MUL2(x, xx, y, yy, z, zz, c, cc) \
MUL12 (x, y, c, cc); \
cc = ((x) * (yy) + (xx) * (y)) + cc; z = c + cc; zz = (c - z) + cc;
/* Double-length division, Dekker. The macro produces a double-length */
/* number (z,zz) which satisfies approximately z+zz = (x+xx)/(y+yy). */
/* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy) */
/* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu */
/* are temporary storage variables of type double. */
#define DIV2(x, xx, y, yy, z, zz, c, cc, u, uu) \
c=(x)/(y); MUL12(c,y,u,uu); \
cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y); z=c+cc; zz=(c-z)+cc;
/* Double-length addition, slower but more accurate than ADD2. */
/* The macro produces a double-length */
/* number (z,zz) which satisfies approximately z+zz = (x+xx)+(y+yy). */
/* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy) */
/* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */
/* are temporary storage variables of type double. */
#define ADD2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \
r = (x) + (y); \
if (fabs (x) > fabs (y)) { rr = ((x) - r) + (y); s = (rr + (yy)) + (xx); } \
else { rr = ((y) - r) + (x); s = (rr + (xx)) + (yy); } \
if (rr != 0.0) { \
z = r + s; zz = (r - z) + s; } \
else { \
ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) + (yy)) : (((yy) - s) + (xx));\
u = r + s; \
uu = (fabs (r) > fabs (s)) ? ((r - u) + s) : ((s - u) + r); \
w = uu + ss; z = u + w; \
zz = (fabs (u) > fabs (w)) ? ((u - z) + w) : ((w - z) + u); }
/* Double-length subtraction, slower but more accurate than SUB2. */
/* The macro produces a double-length */
/* number (z,zz) which satisfies approximately z+zz = (x+xx)-(y+yy). */
/* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy) */
/* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */
/* are temporary storage variables of type double. */
#define SUB2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \
r = (x) - (y); \
if (fabs (x) > fabs (y)) { rr = ((x) - r) - (y); s = (rr - (yy)) + (xx); } \
else { rr = (x) - ((y) + r); s = (rr + (xx)) - (yy); } \
if (rr != 0.0) { \
z = r + s; zz = (r - z) + s; } \
else { \
ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) - (yy)) : ((xx) - ((yy) + s)); \
u = r + s; \
uu = (fabs (r) > fabs (s)) ? ((r - u) + s) : ((s - u) + r); \
w = uu + ss; z = u + w; \
zz = (fabs (u) > fabs (w)) ? ((u - z) + w) : ((w - z) + u); }
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