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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.

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Ordinarily, all of the mantissa bits are used to express significant figures, in addition to a leading 1, which is implied and consequently omitted. As a result, floats have 24 significant bits of precision, and doubles have 53 significant bits of precision. Such numbers are called normalized numbers.

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

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 NUM06 NUM53-J. Use the strictfp modifier for floating-point calculation consistency across platforms for additional information.

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Testing whether values of type double are denormalized is analogous.

Print Representation of Denormalized Numbers

Denormalized numbers can also be troublesome because 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.

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Code Block

normalized float with %e    : 2.350989e-38
normalized float with %a    : 0x1.0p-125
denormalized float with %e  : 7.174648e-43
denormalized float with %a  : 0x1.0p-140
normalized double with %e   : 8.900295e-308
normalized double with %a   : 0x1.0p-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.

Code Block

bgColor	#FFCCCC

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

Code Block
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.

Code Block

bgColor	#ccccff

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

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

Code Block
Original : 0.3333333333333333 Denormalized 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; denormalized floating-point numbers are a less 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.

Rule	Severity	Likelihood	Remediation Cost	Priority	Level
NUM05-J	low	probable	high	P2	L3

Related Vulnerabilities

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.