1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
|
#include <stdlib.h>
#include <math.h>
#include <stdio.h>
#include <float.h>
#include <stdbool.h>
#define EPS 2*DBL_EPSILON
typedef struct {double v; double e;} double_error;
double_error int_to_valuetype(int i){
double_error de;
de.v = (double) i;
if(abs(i) < 9999999) de.e = 0.0;
else de.e = EPS*abs(i);
return de;
}
double_error int_to_exptype(double d){
double_error de;
de.v = d;
de.e = EPS*fabs(d) + DBL_MIN;
return de;
}
double_error sum(double_error de1, double_error de2){
double_error de;
de.v = de1.v + de2.v;
de.e = de1.e + de2.e + ( fabs(de1.v) + fabs(de2.v) + de1.e + de2.e )*EPS + DBL_MIN;
return de;
}
double_error difference(double_error de1, double_error de2){
double_error de2m;
de2m.v = -de2.v;
de2m.e = de2.e;
return sum(de1,de2m);
}
bool is_greater_certainly(double_error de1, double_error de2){
return de1.v-de1.e > de2.v+de2.e + EPS*(fabs(de1.v)+fabs(de1.e)+fabs(de2.v)+fabs(de2.e)) + DBL_MIN;
}
bool is_greater_possibly(double_error de1, double_error de2){
return de1.v+de1.e + EPS*(fabs(de1.v)+fabs(de1.e)+fabs(de2.v)+fabs(de2.e)) + DBL_MIN > de2.v-de2.e;
}
double_error maximum(double_error de1, double_error de2){
if(is_greater_certainly(de1,de2)) return de1;
else if(is_greater_certainly(de2,de1)) return de2;
else{
double_error de;
if( de1.v >= de2.v ) de.v = de1.v;
else de.v = de2.v;
if( de1.e >= de2.e ) de.e = de1.e;
else de.e = de2.e;
return de;
}
}
double_error product(double_error de1, double_error de2){
double_error de;
de.v = de1.v * de2.v;
double p12 = fabs(de1.v)*de2.e*(1+EPS) + DBL_MIN;
double p21 = fabs(de2.v)*de1.e*(1+EPS) + DBL_MIN;
double p22 = de1.e*de2.e*(1+EPS) + DBL_MIN;
double s = (p12 + p21 + p22)*(1+2*EPS);
de.e = s + ( fabs(de.v) + s )*EPS + DBL_MIN;
return de;
}
double_error ratio(double_error de1, double_error de2){
double_error de2i;
de2i.v = 1/de2.v;
if(de2.e > fabs(de2.v)/2) de2i.e = INFINITY;
else de2i.e = 2*de2.e/(de2.v*de2.v) + fabs(de2i.v)*EPS + DBL_MIN; //2* is extra margin in case de2.e is near de2.v/2
return product(de1,de2i);
}
double_error absolute(double_error de){
double_error dea;
dea.v = fabs(de.v);
dea.e = de.e;
return dea;
}
double_error power(double_error de, double_error p) {
double_error dep;
dep.v = pow(de.v,p.v);
double dmin = (de.v - de.e)*(1-EPS);
double dmax = (de.v + de.e)*(1+EPS);
double pmin = (p.v - p.e)*(1-EPS);
double pmax = (p.v + p.e)*(1+EPS);
double rmin,rmax;
if(dmin >= 1){
rmin = pow(dmin,pmin);
rmax = pow(dmax,pmax);
} else if(dmax >= 1) {
rmin = pow(dmin,pmax);
rmax = pow(dmax,pmax);
} else {
rmin = pow(dmin,pmax);
rmax = pow(dmax,pmin);
}
if(dmin >= 0) dep.e = rmax-rmin + 2*EPS*rmax;
else dep.e = (1+2*EPS)*rmax;
return dep;
}
double valuetype_to_double(double_error de){ return de.v; }
double exptype_to_double(double_error de){ return de.v; }
int valuetype_to_string(char* s, double_error de){
sprintf(s,"%f… +/- %6.1e",de.v,de.e);
return 0;
}
int valuetype_to_latex(char* s, double_error de){
if(de.e == 0.0){
if(de.v == 0.0) sprintf(s,"$0$");
else if(de.v == 1.0) sprintf(s,"$1$");
}
else sprintf(s,"$%4.3f\\ldots\\pm\\texttt{%1.0e}$",de.v,de.e);
return 0;
}
int root_to_string(char* s, double_error de, double_error p){
double_error de1p = power(de,ratio(int_to_valuetype(1),p));
valuetype_to_string(s,de1p);
return 0;
}
int root_to_latex(char* s, double_error de, double_error p){
double_error de1p = power(de,ratio(int_to_valuetype(1),p));
valuetype_to_latex(s,de1p);
return 0;
}
|
