Cs 232 computer architecture ii
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CS 232: Computer Architecture II. Prof. Laxmikant (Sanjay) Kale Floating point arithmetic. Floating Point (a brief look). We need a way to represent numbers with fractions, e.g., 3.1416 very small numbers, e.g., .000000001 very large numbers, e.g., 3.15576  10 9 Representation:

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CS 232: Computer Architecture II

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Cs 232 computer architecture ii

CS 232: Computer Architecture II

Prof. Laxmikant (Sanjay) Kale

Floating point arithmetic


Floating point a brief look

Floating Point (a brief look)

  • We need a way to represent

    • numbers with fractions, e.g., 3.1416

    • very small numbers, e.g., .000000001

    • very large numbers, e.g., 3.15576  109

  • Representation:

    • sign, exponent, significand: (–1)signsignificand 2exponent

    • more bits for significand gives more accuracy

    • more bits for exponent increases range

  • IEEE 754 floating point standard:

    • single precision: 8 bit exponent, 23 bit significand

    • double precision: 11 bit exponent, 52 bit significand


Floating point representation

Floating point representation:

  • The idea is to normalize all numbers, so the significand has exactly one digit to the left of the decimal point.

    • 12345 = 1.2345 * 10^4

    • .0000012345 = 1.2345 * 10^-6

    • Do this in binary: 1.01110 x 2^(1011)

  • IEEE FP representation

    • (+/-) 1.0101010101010101010101 * 2 ^ ( 10101010)

    • This is single precision

    • Double precision: 64 bits in all.

  • Where does one need accuracy of that level?


Floating point numbers

Floating point numbers

  • Representation issues:

    • sign bit, exponent, significand

    • Question: how to represent each field

    • Question: which order to lay them out in a word?

    • Factor: should be easy to do comparisons (for sorting)

      • For arithmetic, we will have special hardware anyway

    • Choice:

      • Sign + magnitude representation

      • Sign bit, followed by exponent, then significand (why?)

      • exponent: represented with a “bias”: add 127 (1023 for double precision)

      • significand: assume implicit 1. (so 00001 means 1.00001)


Floating point representation1

Floating point representation

  • So:

    • (+/-) x (1 + significand) x 2 ^ (exponent - bias) is the value of a floating point number

    • Example: 0 00001000 01010000000000000000000

    • Example: convert -.41 to single precision form


Ieee 754 floating point standard

IEEE 754 floating-point standard

  • Leading “1” bit of significand is implicit

  • Exponent is “biased” to make sorting easier

    • all 0s is smallest exponent all 1s is largest

    • bias of 127 for single precision and 1023 for double precision

    • summary: (–1)signsignificand) 2exponent – bias

  • Example:

    • decimal: -.75 = -3/4 = -3/22

    • binary: -.11 = -1.1 x 2-1

    • floating point: exponent = 126 = 01111110

    • IEEE single precision: 10111111010000000000000000000000


Floating point addition

Floating point addition

  • The problem is: the exponents of numbers being added may be different

    • 2.0 * 10^1 + 3.0 * 10^(-1)

    • 2.0 * 10^1 + .03 * 10^ 1 : Now we can add them

    • 2.03 * 10 ^1

    • But we are not necessarily done!

    • E.g. 9.74 * 10^0 + 3.3 * 10^(-1)

    • 10.07 * 10^0 is not correct form!

    • Shift again to get the correct form: 1.037 * 10^1


You can get different results

You can get different results

  • A + B + C = A + (B+C) = (A+B) + C

    • Right?

  • Can you see a problem?

  • When do you lose bits?


Floating point multiplication

Floating point multiplication

  • Add exponents, but subtract bias

  • Then multiply significands

  • Then normalize


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