Lecture 5 Floating Point Continued CS 230 - Spring 2020 1-1 - - PowerPoint PPT Presentation

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Lecture 5 Floating Point Continued CS 230 - Spring 2020 1-1 - - PowerPoint PPT Presentation

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CS 230 Introduction to Computers and Computer Systems Lecture 5 Floating Point Continued CS 230 - Spring 2020 1-1 Normalized Representation Why do we need to move the radix point to after the first 1 bit? Why is there a 1 in the

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CS 230 – Introduction to Computers and Computer Systems Lecture 5 – Floating Point Continued

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Normalized Representation

 Why do we need to move the radix point to after

the first 1 bit? Why is there a 1 in the formula?

 (-1)S * 1.F * 2E-B

 Let’s pretend it’s a zero: consider 1.012  In our example format it could be

 (-1)0 * 0.1010 * 24-3  (-1)0 * 0.0101 * 25-3

 Now we can’t compare bits for equality and we

waste the potential free 1 bit

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Normalized Representation Problem

 What is the smallest positive number we can

represent in our example format?

 (-1)S * 1.F * 2E-B

 S=0 E=000 F=0001  (-1)0 * 1.0001 * 20-3 = 1.0001 * 2-3 = 0.00100012

 How do we get rid of that 1?

 It’s blocking all the small numbers  We can’t just always use zero instead

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Solution: Subnormal

 Only use zero when the number is too small  Two forms of the floating point formula

Normal: (-1)S * (1 + F) * 2E-B Subnormal: (-1)S * (0 + F) * 21-B

 How do we know when to use which form?

  • When E is all zeros use subnormal

 Why 1-B?

  • We used up E=0 to mean subnormal, so now smallest

exponent is E=1 and we don’t want to leave a gap

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Special Cases

 Overflow still possible (E too large)

 represent as +/- infinity  also for division by zero (by IEEE standard)

 Invalid result – Not a Number – NaN

 special cases, like 0/0 or ∞*0 or sqrt(-1)  can “safely” propagate during computation

 Both can propagate during computation

 no exception (like integer division by zero)

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IEEE-754 Special Cases

Exponent Fraction Case 000000… 000000… 000000… non-zero subnormal 111111… 000000… infinity (+ or -) 111111… non-zero NaN anything else anything normal

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Floating Point Arithmetic

 Addition

 align radix points  use normal addition

 Multiplication

 add exponents  multiply significands

 1.F if normal, 0.F if subnormal

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