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CS61C

Summary

And in Conclusion…\dots

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

Reprint of the 32-bit IEEE 754 single-precision layout: sign bit, exponent field, and fraction bits ordered from most significant to least significant as in the floating-point representation section.

Single-Precision (32-bit) Floating Point Representation (reprint of #fig-float 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 (00 for positive, 11 for negative).

  • The exponent is in biased notation. For single-precision floating point numbers, the bias is −127−127, which comes from −(2(8−1)−1)-(2^{(8−1)} −1). For double-precision floating point numbers, the bias is −1023−1023. 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.

  • For normalized floats:

Value=(−1)Sign×2Exp+Bias×1.Significand2\text{Value} = (−1)^{\text{Sign}} × 2^{\text{Exp}+\text{Bias}} × 1.\text{Significand}_2

Value=(−1)Sign×2Exp+Bias+1×0.Significand2\text{Value} = (−1)^{\text{Sign}} × 2^{\text{Exp}+\text{Bias}+1} × 0.\text{Significand}_2

When translating between binary and decimal floating point values, we must remember that there is a bias for the exponent.

Reprint of the table summarizing IEEE 754 single-precision exponent encodings for normalized, denormalized, infinity, and NaN cases from the special floating-point values section.

The IEEE 754 single-precision exponent field has values from 00 to $255 (reprint of #tab-float-exp-fields from this section).

Textbook Readings

P&H 3.5, 3.9

Additional References

Exercises

Check your knowledge!

Conceptual Review

Solution

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 +∞+\infty, −∞-\infty, NaN\text{NaN} (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

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

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 3232-bit storage there are 2323 bits for the significand, which means the LSB represents 2−232^{−23} 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 2−232^{−23}. If the exponent is 88 , the difference between two neighboring floats will be 2−152^{−15} because the mantissa is multiplied by 282 ^{8}. Limited precision makes binary floating-point numbersdiscontinuous; there are gaps between them.

Solution

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

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.