Computers can represent only a finite number of digits. It is therefore impossible to precisely represent repeating binary-representation values such as 1/3 or 1/5 with the most common floating-point representation: binary floating point.
When precise computation is necessary, use alternative representations that can accurately represent the values. For example, if you are performing arithmetic on decimal values and need an exact decimal rounding, represent the values in binary-coded decimal instead of using floating-point values. Another option is decimal floating-point arithmetic as specified by ANSI/IEEE 754-2007. ISO/IEC WG14 has drafted a proposal to add support for decimal floating-point arithmetic to the C language [[ISO/IEC DTR 24732]].
When precise computation is necessary, carefully and methodically estimate the maximum cumulative error of the computations, regardless of whether decimal or binary is used, to ensure that the resulting error is within tolerances. Consider using numerical analysis to properly understand the problem. An introduction can be found in [[Goldberg 91]].
Noncompliant Code Example
This noncompliant code example takes the mean of 10 identical numbers and checks to see if the mean matches this number. It should, because the ten numbers are all 10.1. Yet, because of the imprecision of floating-point arithmetic, the computed mean does not match this number.
#include <stdio.h>
/* Returns the mean value of the array */
float mean(float array[], int size) {
float total = 0.0;
size_t i;
for (i = 0; i < size; i++) {
total += array[i];
printf("array[%d] = %f and total is %f\n", i, array[i], total);
}
if (size != 0)
return total / size;
else
return 0.0;
}
enum { array_size = 10 };
float array_value = 10.1;
int main(void) {
float array[array_size];
float avg;
size_t i;
for (i = 0; i < array_size; i++) {
array[i] = array_value;
}
avg = mean( -array, array_size);
printf("mean is %f\n", avg);
if (avg == array[0]) {
printf("array[0] is the mean\n");
} else {
printf("array[0] is not the mean\n");
}
return 0;
}
On a 64-bit Linux machine using GCC 4.1, this program yields the following output:
array[0] = 10.100000 and total is 10.100000 array[1] = 10.100000 and total is 20.200001 array[2] = 10.100000 and total is 30.300001 array[3] = 10.100000 and total is 40.400002 array[4] = 10.100000 and total is 50.500000 array[5] = 10.100000 and total is 60.599998 array[6] = 10.100000 and total is 70.699997 array[7] = 10.100000 and total is 80.799995 array[8] = 10.100000 and total is 90.899994 array[9] = 10.100000 and total is 100.999992 mean is 10.099999 array[0] is not the mean
Compliant Solution
This code may be fixed by replacing the floating-point numbers with integers for the internal additions. Floats are used only when printing results and when doing the division to compute the mean.
#include <stdio.h>
/* Returns the mean value of the array */
float mean(int array[], int size) {
int total = 0;
size_t i;
for (i = 0; i < size; i++) {
total += array[i];
printf("array[%d] = %f and total is %f\n", i, array[i] / 100.0, total / 100.0);
}
if (size != 0)
return ((float) total) / size;
else
return 0.0;
}
enum {array_size = 10};
int array_value = 1010;
int main(void) {
int array[array_size];
float avg;
size_t i;
for (i = 0; i < array_size; i++) {
array[i] = array_value;
}
avg = mean( -array, array_size);
printf("mean is %f\n", avg / 100.0);
if (avg == array[0]) {
printf("array[0] is the mean\n");
} else {
printf("array[0] is not the mean\n");
}
return 0;
}
On a 64-bit Linux machine using GCC 4.1, this program yields the following expected output:
array[0] = 10.100000 and total is 10.100000 array[1] = 10.100000 and total is 20.200000 array[2] = 10.100000 and total is 30.300000 array[3] = 10.100000 and total is 40.400000 array[4] = 10.100000 and total is 50.500000 array[5] = 10.100000 and total is 60.600000 array[6] = 10.100000 and total is 70.700000 array[7] = 10.100000 and total is 80.800000 array[8] = 10.100000 and total is 90.900000 array[9] = 10.100000 and total is 101.000000 mean is 10.100000 array[0] is the mean
Risk Assessment
Using a representation other than floating point may allow for more accurate results.
Recommendation |
Severity |
Likelihood |
Remediation Cost |
Priority |
Level |
|---|---|---|---|---|---|
FLP02-C |
low |
probable |
high |
P2 |
L3 |
Automated Detection
Compass/ROSE can detect violations of this recommendation. In particular, it checks to see if the arguments to an equality operator are of a floating point type.
Related Vulnerabilities
Search for vulnerabilities resulting from the violation of this rule on the CERT website.
Other Languages
This rule appears in the C++ Secure Coding Standard as FLP02-CPP. Avoid using floating point numbers when precise computation is needed.
References
[[Goldberg 91]]
[[IEEE 754 2006]]
[[ISO/IEC JTC1/SC22/WG11]]
[[ISO/IEC PDTR 24772]] "PLF Floating Point Arithmetic"
[[ISO/IEC DTR 24732]]
FLP01-C. Take care in rearranging floating point expressions 05. Floating Point (FLP) FLP03-C. Detect and handle floating point errors