1And in Conclusion¶
The IEEE 754 standard defines a binary representation for floating point values using three fields.
Single-Precision (32-bit) Floating Point Representation (reprint of Figure 3 from this section). The leftmost bit is the most significant bit; the rightmost bit is the least significant bit.
The sign determines the sign of the number ( for positive, for negative).
The exponent is in biased notation. For single-precision floating point numbers, the bias is 127, which comes from . For double-precision floating point numbers, the bias is 1023.
An exponent of
00000000represents either zero, if there is a zero mantissa, or a denormalized number, if there is a non-zero mantissa.An exponent of
11111111represents 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 is1.010011, its significand is010011.For normalized floats:
For denormalized floats, including zero:
When translating between binary and decimal floating point values, we must remember that there is a bias for the exponent.
| Biased Exponent | Significand field | Description |
|---|---|---|
0 (0000000) | all zeros | zero |
0 (0000000) | nonzero | Denormalized numbers, aka denorms |
| 1 – 254 | anything | Normalized floating point (mantissa has implicit leading 1) |
255 (1111111) | all zeros | infinity |
255 (1111111) | nonzero | NaNs |
The IEEE 754 single-precision exponent field has values from 0 to $255 (reprint of Table 1 from this section).
2Textbook Readings¶
P&H 3.5, 3.9
3Additional References¶
4Exercises¶
Check your knowledge!
4.1Conceptual Review¶
Solution to Exercise 1 #
True. Floating point:
Provides support for a wide range of values. (Both very small and very large)
Helps programmers deal with errors in real arithmetic because floating point can represent , , (Not a Number)
Keeps high precision. Recall that precision is a count of the number of bits in a computer word used to represent a value. IEEE 754 allocates a majority of bits for the significand, allowing for the use of a combination of negative powers of two to represent fractions.
Solution to Exercise 2 #
False. Floating Point can represent infinities as well as NaNs, so the total amount of representable numbers is lower than Two’s Complement, where every bit combination maps to a unique integer value.
Solution to Exercise 3 #
True. The uneven spacing is due to the exponent representation of floating point numbers. There are a fixed number of bits in the significand. In IEEE 32-bit storage there are bits for the significand, which means the LSB represents times 2 to the exponent. For example, if the exponent is zero (after allowing for the offset) the difference between two neighboring floats will be . If the exponent is , the difference between two neighboring floats will be because the mantissa is multiplied by . Limited precision makes binary floating-point numbers discontinuous; there are gaps between them.
Solution to Exercise 4 #
False. Because of rounding errors, you can find Big and Small numbers such that: (Small + Big) + Big != Small + (Big + Big)
FP approximates results because it only has 23 bits for the significand.
Solution to Exercise 5 #
A non-zero digit is required prior to the radix in scientific notation, and since the only non-zero digit in base-2 is 1, the normalized value will always start with a 1.