# CS 232: Computer Architecture II - PowerPoint PPT Presentation

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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

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  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:

• 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

• 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 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

• 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

• 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

• A + B + C = A + (B+C) = (A+B) + C

• Right?

• Can you see a problem?

• When do you lose bits?

### Floating point multiplication

• Add exponents, but subtract bias

• Then multiply significands

• Then normalize