Notes

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IEEE-754 Double Breakdown

Formula: (-1)^S * 1.M * 2^(E-1023)

• Sign: (-1)^S

• 1 is understood: 1.

• Mantissa (w/o leading 1): .M

• Base: 2

• Exponent: E-1023

Edge Cases

Properties

• Same basic idea as scientific notation

• Precise representation at the bit level

• Precise behavior of arithmetic operations

• Efficiency (Space & Time)

• Special values: not-a-number, +/-infinity, etc.

• Correct rounding

• Sortable without FP hardware

• Edge case for tiny numbers: E==0 is special, so 2-126 is smallest exponent; we use underflow case for anything smaller, like 2-127; note E==0 is computed as E==1

• the 32-bit form the exponent is biased by 127

NaN Properties

• Suppose A is a floating point number set to NaN

• A != B is true, when B is another floating point number, NaN, infinity, or anything

• Interestingly A != A is true as well

• Also, if A or B is NaN, the following are always false:
  • A < B, A > B, A == B

Converting variable types

• In CS 1331 we noted:

  float f;
  int i;
  f = i;
  i = (int)f;

• What does this imply?
  1. Converting floats to ints we may lose information
  2. Converting ints to floats we may lose information
  3. Converting either way we may lose information

Comparing Two FP Numbers

• The layout of the representation allows certain operations (like > ) to be performed with no conversion

• If either is NaN, the comparison is defined as “unordered” (all comparisons to it except != are false)

• If either is -0.0, replace with +0.0

• If the signs (high bit) are different, the positive number is bigger

• Compare the rest of the bits as integers

• If the signs are both negative reverse the comparison

• Example:

  A 0 10111000 11001010 11001010 1100101

  B 0 10111000 11101010 11001010 1100101

  |B| > |A|

  • Treat as 31-bit unsigned whole numbers
    • without the sign bits (bit 31)
  • And compare the two magnitudes bit by bit, left to right - until you find different values
  • Both are positive, so B>A

Questions & Answers

What’s the formula for finding what value a regular floating point number represents in terms of its sign, exponent, and mantissa?

How is the exponent represented? What are the range of possible values for the exponent?

• Challenge conceptual question: why do we not use a 2's complement number for the exponent part?

Edge cases

• Given the sign, exponent bits, and mantissa, how can you tell when a floating point number is the following:
  • Infinity/negative infinity
  • Subnormal (a.k.a. denormalized)
  • 0.0 or -0.0
  • Power of 2
  • NaN
  • A regular number (none of the above)

• What does "normalization" mean in the context of IEEE-754?

• What is a subnormal number? How is the formula for computing the value of a subnormal number different from the usual formula?

What are the different values that an IEEE-754 floating point number could represent if:

• It had a mantissa of 00000000000000000000000?

• It had an exponent of 00000000?

• It had an exponent of 11111111?