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