1 Floating Point Representation and Arithmetic (see Patterson Chapter 4)

2 Outline Review of floating point scientific notation Floating point binary IEEE Floating Point Standard Addition in Floating Point Remarks about multiplication

3 Floating Point Notation Decimal ten (decimal notation) means 10* /10 + 5/ / /10000 In scientific notation = * = * = * = * = * = * * *10 1 is an example of normalised scientific notation.

4 Floating Point in Binary Binary two = (0/2) + (1/2 2 ) + (0/2 4 ) +(1/2 5 ) + (1/2 6 ) 0 + 1/ /32 + 1/64 = ( ) ten = ten In scientific notation 10011*2 -6 = *2 -5 = = *2 -4 = *2 -2 normalised

5 Normalised Notation In normalised binary scientific notation unless the number is 0 always have 1.sssssss...sss * 2 E sss...sss is the significand E is the exponent The significand s 1 s 2...s n represents

6 Representation Note that it is impossible to exactly represent all decimal numbers in this way (eg 0.3) Problem of representation of floating point numbers in fixed word length need to represent sign significand exponent in one word (32 bits).

7 Representation Represents floating point number: (-1) S * (1.0+F) * 2 E S is 1 bit (if S=1 then negative) F is 23 bits E is 8 bits sign bit S exponent 8 bits E significand: 23 bits F

8 Squeezing out More from the Bits Since every non-zero binary f.p. number (normalised) is of the form: 1.sss...sss *2 E We do not have to represent explicitly the 1 in the word, and can therefore interpret the bit- pattern as: (-1) S (1 + significand) * 2 E thus reclaiming an extra bit! E= is reserved for zero.

9 Requirements As far as possible the ALU should be able to reuse integer machinery in implementation of f.p. Eg, comparison with zero easy because of sign bit fp numbers can be easily classified as negative, zero or positive without additional hardware. Comparison of two fp numbers x<y not so straightforward - how are negative exponents to be formed?

10 Bad Example: (1/2) > 2 ??? Representation of 1/2 is 0.1 two = 1.0*2 -1 (normalised) S E significand l Representation of 2 is »10 two = 1.0*2 1 (normalised) S E significand

11 Representation of Exponent Inappropriate to use twos complement for the exponent Ideally want to represent most negative number, most positive. Number range: use this for 2 0 positive negative = 127 ten

12 Biased Representation (IEEE FP Standard) The bias 127 represents to 255 represent positive exponents 1 to 127 represent negative exponents (remember 0 is reserved for the entire number being zero). The actual exponent is therefore: E - bias (-1) S * (1 + significand) * 2 E-bias

13 Example 1 Represent ten = 5/16 5/16 = 1/4 + 1/16 = two = 1.01*2 -2 S = 0 E = ??? -2 = E-bias = E-127 E = 125 ten = two Significand = 010.…

14 Example 2 What does represent? S = 0 E = = 125 ten Exponent = E-bias = = -2 Significand = 1/4 (-1) S (1+sig.)2 E-bias = (1 + 1/4)*(1/4) = 5/16

15 Addition of FP Numbers Given two numbers: normalise them both adjust the floating point of the smaller number to match the larger one Add them together renormalise check for underflow/overflow of exponent if so then break; round significand to required number of bits might need renormalisation (eg, round to 4 bits).

16 Addition Example = two two 1.0* * * * *2 1 (already normalised) (1 + (1/2) + (1/8)) *

17 Remarks The IEEE FP standard represents floats in 32 bits, higher precision represented across two words (doubles). Multiplication is relatively easy, since the exponents add, and the significands can be done with integer multiplication. There can be huge pitfalls in reliably transferring floating point code to different hardware!

18 Summary FP scientific notation normalised representation in binary Bias to represent -ve to +ve range in exponent Addition Notice how a 32-bit binary string can represent many different entities in memory. Memory architectures NEXT.