1 /* 2 * IBM Accurate Mathematical Library 3 * Copyright (C) 2001-2022 Free Software Foundation, Inc. 4 * 5 * This program is free software; you can redistribute it and/or modify 6 * it under the terms of the GNU Lesser General Public License as published by 7 * the Free Software Foundation; either version 2.1 of the License, or 8 * (at your option) any later version. 9 * 10 * This program is distributed in the hope that it will be useful, 11 * but WITHOUT ANY WARRANTY; without even the implied warranty of 12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 13 * GNU Lesser General Public License for more details. 14 * 15 * You should have received a copy of the GNU Lesser General Public License 16 * along with this program; if not, see <https://www.gnu.org/licenses/>. 17 */ 18 19 #include <math.h> 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 /* CN = 1+2**27 = '41a0000002000000' IEEE double format. Use it to split a 39 double for better accuracy. */ 40 #define CN 134217729.0 41 42 43 /* Exact addition of two single-length floating point numbers, Dekker. */ 44 /* The macro produces a double-length number (z,zz) that satisfies */ 45 /* z+zz = x+y exactly. */ 46 47 #define EADD(x,y,z,zz) \ 48 z=(x)+(y); zz=(fabs(x)>fabs(y)) ? (((x)-(z))+(y)) : (((y)-(z))+(x)); 49 50 51 /* Exact subtraction of two single-length floating point numbers, Dekker. */ 52 /* The macro produces a double-length number (z,zz) that satisfies */ 53 /* z+zz = x-y exactly. */ 54 55 #define ESUB(x,y,z,zz) \ 56 z=(x)-(y); zz=(fabs(x)>fabs(y)) ? (((x)-(z))-(y)) : ((x)-((y)+(z))); 57 58 59 #ifdef __FP_FAST_FMA 60 # define DLA_FMS(x, y, z) __builtin_fma (x, y, -(z)) 61 #endif 62 63 /* Exact multiplication of two single-length floating point numbers, */ 64 /* Veltkamp. The macro produces a double-length number (z,zz) that */ 65 /* satisfies z+zz = x*y exactly. p,hx,tx,hy,ty are temporary */ 66 /* storage variables of type double. */ 67 68 #ifdef DLA_FMS 69 # define EMULV(x, y, z, zz) \ 70 z = x * y; zz = DLA_FMS (x, y, z); 71 #else 72 # define EMULV(x, y, z, zz) \ 73 ({ __typeof__ (x) __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 }) 78 #endif 79 80 81 /* Exact multiplication of two single-length floating point numbers, Dekker. */ 82 /* The macro produces a nearly double-length number (z,zz) (see Dekker) */ 83 /* that satisfies z+zz = x*y exactly. p,hx,tx,hy,ty,q are temporary */ 84 /* storage variables of type double. */ 85 86 #ifdef DLA_FMS 87 # define MUL12(x, y, z, zz) \ 88 EMULV(x, y, z, zz) 89 #else 90 # define MUL12(x, y, z, zz) \ 91 ({ __typeof__ (x) __p, hx, tx, hy, ty, __q; \ 92 __p=CN*(x); hx=((x)-__p)+__p; tx=(x)-hx; \ 93 __p=CN*(y); hy=((y)-__p)+__p; ty=(y)-hy; \ 94 __p=hx*hy; __q=hx*ty+tx*hy; z=__p+__q; zz=((__p-z)+__q)+tx*ty; \ 95 }) 96 #endif 97 98 99 /* Double-length addition, Dekker. The macro produces a double-length */ 100 /* number (z,zz) which satisfies approximately z+zz = x+xx + y+yy. */ 101 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */ 102 /* are assumed to be double-length numbers. r,s are temporary */ 103 /* storage variables of type double. */ 104 105 #define ADD2(x, xx, y, yy, z, zz, r, s) \ 106 r = (x) + (y); s = (fabs (x) > fabs (y)) ? \ 107 (((((x) - r) + (y)) + (yy)) + (xx)) : \ 108 (((((y) - r) + (x)) + (xx)) + (yy)); \ 109 z = r + s; zz = (r - z) + s; 110 111 112 /* Double-length subtraction, Dekker. The macro produces a double-length */ 113 /* number (z,zz) which satisfies approximately z+zz = x+xx - (y+yy). */ 114 /* An error bound: (abs(x+xx)+abs(y+yy))*4.94e-32. (x,xx), (y,yy) */ 115 /* are assumed to be double-length numbers. r,s are temporary */ 116 /* storage variables of type double. */ 117 118 #define SUB2(x, xx, y, yy, z, zz, r, s) \ 119 r = (x) - (y); s = (fabs (x) > fabs (y)) ? \ 120 (((((x) - r) - (y)) - (yy)) + (xx)) : \ 121 ((((x) - ((y) + r)) + (xx)) - (yy)); \ 122 z = r + s; zz = (r - z) + s; 123 124 125 /* Double-length multiplication, Dekker. The macro produces a double-length */ 126 /* number (z,zz) which satisfies approximately z+zz = (x+xx)*(y+yy). */ 127 /* An error bound: abs((x+xx)*(y+yy))*1.24e-31. (x,xx), (y,yy) */ 128 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc are */ 129 /* temporary storage variables of type double. */ 130 131 #define MUL2(x, xx, y, yy, z, zz, c, cc) \ 132 MUL12 (x, y, c, cc); \ 133 cc = ((x) * (yy) + (xx) * (y)) + cc; z = c + cc; zz = (c - z) + cc; 134 135 136 /* Double-length division, Dekker. The macro produces a double-length */ 137 /* number (z,zz) which satisfies approximately z+zz = (x+xx)/(y+yy). */ 138 /* An error bound: abs((x+xx)/(y+yy))*1.50e-31. (x,xx), (y,yy) */ 139 /* are assumed to be double-length numbers. p,hx,tx,hy,ty,q,c,cc,u,uu */ 140 /* are temporary storage variables of type double. */ 141 142 #define DIV2(x, xx, y, yy, z, zz, c, cc, u, uu) \ 143 c=(x)/(y); MUL12(c,y,u,uu); \ 144 cc=(((((x)-u)-uu)+(xx))-c*(yy))/(y); z=c+cc; zz=(c-z)+cc; 145 146 147 /* Double-length addition, slower but more accurate than ADD2. */ 148 /* The macro produces a double-length */ 149 /* number (z,zz) which satisfies approximately z+zz = (x+xx)+(y+yy). */ 150 /* An error bound: abs(x+xx + y+yy)*1.50e-31. (x,xx), (y,yy) */ 151 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */ 152 /* are temporary storage variables of type double. */ 153 154 #define ADD2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \ 155 r = (x) + (y); \ 156 if (fabs (x) > fabs (y)) { rr = ((x) - r) + (y); s = (rr + (yy)) + (xx); } \ 157 else { rr = ((y) - r) + (x); s = (rr + (xx)) + (yy); } \ 158 if (rr != 0.0) { \ 159 z = r + s; zz = (r - z) + s; } \ 160 else { \ 161 ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) + (yy)) : (((yy) - s) + (xx));\ 162 u = r + s; \ 163 uu = (fabs (r) > fabs (s)) ? ((r - u) + s) : ((s - u) + r); \ 164 w = uu + ss; z = u + w; \ 165 zz = (fabs (u) > fabs (w)) ? ((u - z) + w) : ((w - z) + u); } 166 167 168 /* Double-length subtraction, slower but more accurate than SUB2. */ 169 /* The macro produces a double-length */ 170 /* number (z,zz) which satisfies approximately z+zz = (x+xx)-(y+yy). */ 171 /* An error bound: abs(x+xx - (y+yy))*1.50e-31. (x,xx), (y,yy) */ 172 /* are assumed to be double-length numbers. r,rr,s,ss,u,uu,w */ 173 /* are temporary storage variables of type double. */ 174 175 #define SUB2A(x, xx, y, yy, z, zz, r, rr, s, ss, u, uu, w) \ 176 r = (x) - (y); \ 177 if (fabs (x) > fabs (y)) { rr = ((x) - r) - (y); s = (rr - (yy)) + (xx); } \ 178 else { rr = (x) - ((y) + r); s = (rr + (xx)) - (yy); } \ 179 if (rr != 0.0) { \ 180 z = r + s; zz = (r - z) + s; } \ 181 else { \ 182 ss = (fabs (xx) > fabs (yy)) ? (((xx) - s) - (yy)) : ((xx) - ((yy) + s)); \ 183 u = r + s; \ 184 uu = (fabs (r) > fabs (s)) ? ((r - u) + s) : ((s - u) + r); \ 185 w = uu + ss; z = u + w; \ 186 zz = (fabs (u) > fabs (w)) ? ((u - z) + w) : ((w - z) + u); } 187