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Most implementations of C use Java uses 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)^s * M * 2 ^E.

Image Added

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. Thusconsequently omitted. As a result, 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 this sense that they have fixed precision. See recommendation FLP00-C. Understand the limitations of floating point numbers.

When the value to be represented is Mantissa bits are used to express extremely small numbers that are too small to encode normally because of the lack of available exponent bits. Using mantissa bits extends the possible range of exponents. Because these , it is encoded in denormalized form, indicated by an exponent value of Float.MIN_EXPONENT - 1 or Double.MIN_EXPONENT - 1. Denormalized floating-point numbers have an assumed 0 in the ones' place and have one or more leading zeros in the represented portion of their mantissa. These leading zero bits no longer function as significant bits of precision; consequently, the total precision of extremely small denormalized floating-point numbers is less than usual. Such numbers are called denormalized, and they are more limited than normalized numbers. However, that of normalized floating-point numbers. Note that even using normalized numbers where precision is required can pose a risk. See recommendation FLP02rule NUM04-C. Avoid using floating J. Do not use floating-point numbers when if precise computation is neededrequired for more information.

Using denormalized numbers can severely impair the precision of floating-point numbers and should calculations; as a result, denormalized numbers must not be used.

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 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.

Detecting Denormalized Numbers

The following code tests whether a float value is denormalized in FP-strict mode or for platforms that lack extended range support. Testing for denormalized numbers in the presence of extended range support is platform-dependent; see rule NUM53-J. Use the strictfp modifier for floating-point calculation consistency across platforms for additional information.

strictfp public static boolean isDenormalized(float val) {
  if (val == 0) {
    return false;
  }
  if ((val > -Float.MIN_NORMAL) && (val < Float.MIN_NORMAL)) {
    return true;
  }
  return false;
}

Testing whether values of type double are denormalized is analogous.

Print Representation of

...

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 ISO/IEC 9899:TC3 §7.19.6.1

A double argument representing a floating-point number is converted in the style ?0xh.hhhh p±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.

their printed representation is unusual. Floats and normalized doubles, when formatted with the %a specifier, begin with a leading nonzero digit. Denormalized doubles can begin with a leading zero to the left of the decimal point in the mantissa.

Here is a small program, along with its output, that demonstrates the print representation of denormalized numbers.

strictfp class FloatingPointFormats {
  public static void main(String[] args) {
    

#include<stdio.h>
float x = 0x1p-125125f;
    double y = 0x1p-1020;
printf    System.out.format("normalized float with %%e    : %e\n", x);
printf    System.out.format("normalized float with %%a    : %a\n", x);
    x = 0x1p-140140f;
printf    System.out.format("denormalized float with %%e  : %e\n", x);
printf    System.out.format("denormalized float with %%a  : %a\n", x);
printf    System.out.format("normalized double with %%e   : %e\n", y);
printf    System.out.format("normalized double with %%a   : %a\n", y);
    y = 0x1p-1050;
printf    System.out.format("denormalized double with %%e : %e\n", y);
printf    System.out.format("denormalized double with %%a : %a\n", y);

Implementation Details

...

  }
}

normalized float with %e    : 2.350989e-38
normalized float with %a    : 0x1p0x1.0p-125
denormalized float with %e  : 7.174648e-43
denormalized float with %a  : 0x1p0x1.0p-140
normalized double with %e   : 8.900295e-308
normalized double with %a   : 0x1p0x1.0p-1020
denormalized double with %e : 8.289046e-317
denormalized double with %a : 0x0.0000001p-1022

Noncompliant Code Example

This noncompliant code example attempts to reduce a floating-point number to a denormalized value and then restore the value.

float x = 1/3.0f;
System.out.println("Original    : " + x);
x = x * 7e-45f;
System.out.println("Denormalized: " + x);
x = x / 7e-45f;
System.out.println("Restored    : " + x);

Because this operation is imprecise, this code produces the following output when run in FP-strict mode:

Original    : 0.33333334
Denormalized: 2.8E-45
Restored    : 0.4

Compliant Solution

Do not use code that could use denormalized numbers. When calculations using float produce denormalized numbers, use of double can provide sufficient precision.

double x = 1/3.0;
System.out.println("Original  : " + x);
x = x * 7e-45;
System.out.println("Normalized: " + x);
x = x / 7e-45;
System.out.println("Restored  : " + x);

This code produces the following output in FP-strict mode:

Original  : 0.3333333333333333
Normalized: 2.333333333333333E-45
Restored  : 0.3333333333333333

Exceptions

NUM05-J-EX0: Denormalized numbers are acceptable when suitable numerical analysis demonstrates that the computed values meet all accuracy and behavioral requirements appropriate to the application.

Risk Assessment

Floating-point numbers are an approximation; using subnormal denormalized floating-point number numbers are a worse approximationless precise approximation. Use of denormalized numbers can cause unexpected loss of precision, possibly leading to incorrect or unexpected results. Although the severity for violations of this rule is low, applications that require accurate results should make every attempt to comply.

Automated Detection

TODO

Related Vulnerabilities

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

Related Guidelines

ISO/IEC 9899:1999

Bibliography

\[[IEEE 754|AA. Bibliography#IEEE 754 2006]\]
\[[Bryant 2003|AA. Bibliography#Bryant 03]\] Computer Systems: A Programmer's Perspective. Section 2.4 Floating Point

CVE-2010-4476 [CVE 2008 ] reports a vulnerability in the Double.parseDouble() method in Java 1.6 update 23 and earlier, Java 1.5 update 27 and earlier, and 1.4.2_29 and earlier. This vulnerability causes a denial of service when this method is passed a crafted string argument. The value 2.2250738585072012e-308 is close to the minimum normalized, positive, double-precision floating-point number; when encoded as a string it triggers an infinite loop of estimations during conversion to a normalized or denormalized double.

Related Guidelines

Bibliography

[Bryant 2003]	Computer Systems: A Programmer's Perspective, Section 2.4, Floating Point
[CVE 2008 ]	CVE-2010-4476
[IEEE 754]
[Seacord 2015]	Image Added NUM05-J. Do not use denormalized numbers LiveLesson

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Image Added Image Added Image AddedFLP04-C. Check floating point inputs for exceptional values      05. Floating Point (FLP)      FLP30-C. Do not use floating point variables as loop counters