1 | /* |
---|---|

2 | * IBM Accurate Mathematical Library |

3 | * Written by International Business Machines Corp. |

4 | * Copyright (C) 2001-2018 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 <http://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); } |

184 |