Data Types - Beginning Java 8 Fundamentals - page 88

Actual Significand: 1.fffffffffffffffffffffff (24 digits)
Stored Significand: fffffffffffffffffffffff (23 digits)

If you always represent the significand of a binary floating-point number in normalized form, there is a gap
around zero on the number line. The minimum number in magnitude that can be represented in IEEE
single-precision format can be computed as follows:

• Sign : It can be 0 or 1, denoting a pocsitive or negative number. For this example, let's
assume the sign bit as 0 to indicate a positive number.

• Exponent : The minimum exponent value is -126. Recall that the exponent values -127 and 128
are reserved to represent special floating-point numbers. The minimum biased exponent will
be -126 + 127 = 1. The binary representation of the biased exponent 1 in 8-bit is 00000001.

• Significand : The minimum value of significand in the normalized form will consist of the
leading 1 bit and all 23 fraction bits set to 0 as 1.00000000000000000000000.
If you combine the binary representation of a normalized floating-point number with the minimum possible values
for the exponent and significand, the actual number stored in the computer will look like the one shown in Table 3-7 .

Table 3-7. Minimum Possible Normalized Number

Sign Exponent Significand

0 00000001 00000000000000000000000

1-bit 8-bit 23-bit

The value of the minimum floating-point number in decimal is 1.0 x 2 -126 . Therefore, 1.0 x 2 -126 is the first
representable normalized number after zero, leaving a gap around zero on the number line.
If you store only normalized floating-point numbers using IEEE single-precision format, all numbers less than
1.0 x 2 -126 in magnitude must be rounded to zero. This will cause a serious problem when dealing with tiny numbers
in a program. In order to store numbers smaller than 1.0 x 2 -126 the numbers must be denormalized.

Special Floating-Point Numbers
This section describes special floating-point numbers and their representations in the IEEE single-precision format.

Signed Zeros
The IEEE floating-point format allows for two zeros, +0.0 (or 0.0) and -0.0. Zero is represented by the minimum exponent
value -127 for the single-precision format. The significand is 0.0 for zero. Since the sign bit can be 0 or 1, there are two
zeros: +0.0 and -0.0. The binary representations of zeros in the single-precision format are shown in Table 3-8 .

Table 3-8. Binary Representations of Positive and Negative Zeros in the Single-Precision Format

Number Sign Exponent Significand

0.0 0 00000000 00000000000000000000000

-0.0 1 00000000 00000000000000000000000