Summary

1And in Conclusion$\dots$¶

The IEEE 754 standard defines a binary representation for floating point values using three fields.

• The sign determines the sign of the number ($0$ for positive, $1$ for negative).

• The exponent is in biased notation. For instance, the bias is $−127$, which comes from $-(2^{(8−1)} −1)$ for single-precision floating point numbers. For double-precision floating point numbers, the bias is $−1023$. An exponent of 00000000 represents a denormalized number and an exponent of 11111111 represents either NaN, if there is a non-zero mantissa, or infinity, if there is a zero mantissa.

• The significand is used to store a fraction instead of an integer and refers to the bits to the right of the leading “1” when normalized. For example, if a mantissa is 1.010011, its significand is 010011.

Figure 3 shows the bit breakdown for the single-precision (32-bit) representation. The leftmost bit is the MSB, and the rightmost bit is the LSB.

• For normalized floats:

• For denormalized floats, including zero:

Table 1 shows that the IEEE 754 exponent field has values from 0 to 255. When translating between binary and decimal floating point values, we must remember that there is a bias for the exponent.

2Textbook Readings¶

P&H 3.5, 3.9

3Additional References¶

• IEEE 754 Simulator