IEEE Floating Point Standard

The IEEE Standard 754 floating-point arithmetic offers users greater control over computation than does any other kind of floating-point arithmetic. The IEEE Standard 754 simplifies the task of writing numerically sophisticated, portable programs not only by imposing rigorous requirements on conforming implementations. The Standard also allows such implementations to provide refinements and enhancements to the Standard itself.

The IEEE single format consists of three fields: a 23-bit fraction, f; an 8-bit biased exponent, e; and a 1-bit sign, s. These fields are stored contiguously in one 32-bit word. Bits 0:22 contain the 23-bit fraction, f, with bit 0 being the least significant bit of the fraction and bit 22 being the most significant; bits 23:30 contain the 8-bit biased exponent, e, with bit 23 being the least significant bit of the biased exponent and bit 30 being the most significant; and the highest-order bit 31 contains the sign bit.

The table below shows the correspondence between the values of the three constituent fields s, e and f and the value of the IEEE single precision number.

Notice that when e < 255, the value assigned to the single format bit pattern is formed by inserting the binary radix point immediately to the left of the fraction's most significant bit, and inserting an implicit bit immediately to the left of the binary point, thus representing in binary positional notation a mixed number (whole number plus fraction, wherein 0 <= fraction < 1).

The mixed number thus formed is called the single-format significand. The implicit bit is so named because its value is not explicitly given in the single- format bit pattern, but is implied by the value of the biased exponent field.

For the single format, the difference between a normal number and a subnormal number is that the leading bit of the significand (the bit to left of the binary point) of a normal number is 1, whereas the leading bit of the significand of a subnormal number is 0. Single-format subnormal numbers were called single-format denormalized numbers in IEEE Standard 754.

The 23-bit fraction combined with the implicit leading significand bit provides 24 bits of precision in single-format normal numbers.

Examples of important bit patterns in the single-storage format are shown in the table below. The maximum positive normal number is the largest finite number representable in IEEE single format. The minimum positive subnormal number is the smallest positive number representable in IEEE single format. The minimum positive normal number is often referred to as the underflow threshold. (The decimal values for the maximum and minimum normal and subnormal numbers are approximate; they are correct to the number of figures shown.)

Bit Patterns in Single-Storage Format and their IEEE Values

Common Name	Bit Pattern (Hex)	Decimal Value
+ 0	00000000	0.0
- 0	80000000	-0.0
1	3f800000	1.0
2	40000000	2.0
maximum normal number	7f7fffff	3.40282347e+38
minimum positive normal number	00800000	1.17549435e-38
maximum subnormal number	007fffff	1.17549421e-38
minimum positive subnormal number	00000001	1.40129846e-45
+	7f800000	Infinity
-	ff800000	-Infinity
Not-a-Number*	7fc00000	NaN

*A NaN (Not a Number) can be represented with any of the many bit patterns that satisfy the definition of a NaN. The hex value of the NaN shown in the table is just one of the many bit patterns that can be used to represent a NaN.