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Most implementations of C use the IEEE 754 standard for floating-point representation. In this representation, floats are encoded using 1 sign bit, 8 exponent bits, and 23 mantissa bits. Doubles are encoded and used exactly the same way, except they use 1 sign bit, 11 exponent bits, and 52 mantissa bits. These bits encode the values of s, the sign; M, the significand; and E, the exponent. Floating-point numbers are then calculated as (-1−1)^s ^s * M * 2^E2 ^E.

Ordinarily, all of the mantissa bits are used to express significant figures, in addition to a leading 1, which is implied and therefore left out. Thus Consequently, floats ordinarily have 24 significant bits of precision, and doubles ordinarily have 53 significant bits of precision. Such numbers are called normalized numbers. All floating-point numbers are limited in the sense that they have fixed precision. See FLP00-C. Understand the limitations of floating-point numbers.

Mantissa bits are used However, to express extremely small numbers that are too small to encode normally because not enough of the lack of available exponent bits are available . Using mantissa bits are used to extend extends the possible range of exponents. These Because these bits no longer function as significant bits of precision so , the total precision of extremely small numbers is smaller less than usual. Such numbers are called denormalized, and they are more limited than normalized numbers. However, even using normalized numbers where precision is required can pose a risk. See FLP02-C. Avoid using floating-point numbers when precise computation is needed for more information.

Denormalized Using denormalized numbers can severely impair the precision of floating-point numbers and should not be used. FLP00-C. Understand the limitations of floating point numbers
FLP02-C. Avoid using floating point numbers when precise computation is needed

Noncompliant Code Example

This code attempts to reduce a floating-point number to a denormalized value and then restore the value. This operation is very imprecise.

#include <stdio.h>
float x = 1/3.0;
printf("Original      : %e\n", x);
x = x * 7e-45;
printf("Denormalized? : %e\n", x);
x = x / 7e-45;
printf("Restored      : %e\n", x);

This code produces the following output on implementations that use IEEE 754 floats:

Original      : 3.333333e-01
Denormalized? : 2.802597e-45
Restored      : 4.003710e-01

Compliant Solution

Don't produce code that could use Do not allow code to produce denormalized numbers. If floats are producing denormalized numbers, use doubles instead.

#include <stdio.h>
double x = 1/3.0;
printf("Original      : %e\n", x);
x = x * 7e-45;
printf("Denormalized? : %e\n", x);
x = x / 7e-45;
printf("Restored      : %e\n", x);

Original      : 3.333333e-01
Denormalized? : 2.333333e-45
Restored      : 3.333333e-01

If using doubles also produces denormalized numbers, using long doubles may or may not help. (On some implementations, long double has the same exponent range as double.) If using long doubles produces denormalized numbers, some other solution must be found.

Printing Denormalized Numbers

Denormalized numbers can also be troublesome because some functions have implementation-defined behavior when used with denormalized values. For example, using the %a or $%A conversion specifier in a format string can produce implementation-defined results when applied to denormalized numbers.

According to Subclause 7.21.6.1, paragraph 8, of the C Standard [ISO/IEC 9899:TC3 §7.19.6.12011], states:

A double argument representing a floating-point number is converted in the style ?0xh[-]0xh.hhhh p±dp±d, where there is one hexadecimal digit (which is nonzero if the argument is a normalized floating-point number and is otherwise unspecified) before the decimal-point character.

Relying on the %a and %A specifiers to produce values without a leading zero is error prone.

#include<stdio.h>
float x = 0x1p-125;
double y = 0x1p-1020;

printf("normalized float with %%e    : %e\n", x);
printf("normalized float with %%a    : %a\n", x);
x = 0x1p-140;
printf("denormalized float with %%e  : %e\n", x);
printf("denormalized float with %%a  : %a\n", x);
printf("normalized double with %%e   : %e\n", y);
printf("normalized double with %%a   : %a\n", y);
y = 0x1p-1050;
printf("denormalized double with %%e : %e\n", y);
printf("denormalized double with %%a : %a\n", y);

Implementation Details

On a 32-bit Linux machine using GCC 4.3.2, this code produces the following output:

normalized float with %e    : 2.350989e-38
normalized float with %a    : 0x1p-125
denormalized float with %e  : 7.174648e-43
denormalized float with %a  : 0x1p-140
normalized double with %e   : 8.900295e-308
normalized double with %a   : 0x1p-1020
denormalized double with %e : 8.289046e-317
denormalized double with %a : 0x0.0000001p-1022

Risk Assessment

Floating-point numbers are an approximation; using subnormal floating-point number are a worse approximation.

Automated Detection

TODO

Related Vulnerabilities

Search for vulnerabilities resulting from the violation of this rule on the CERT website.

References

Related Guidelines

Bibliography

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Image Added Image Added Image Added Wiki Markup\[[IEEE 754|AA. C References#IEEE 754 2006]\] \[[Bryant 03|AA. C References#Bryant 03]\] Computer Systems: A Programmer's Perspective. Section 2.4 Floating Point \[[ISO/IEC 9899:1999]\]