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

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

3. 3 Floating Point Notation Decimal 12.4568 ten (decimal notation) means 10*1 + 2 + 4/10 + 5/100 + 6/1000 + 8/10000 In scientific notation 12.4568 = 124568 * 10 -4 = 1245680 * 10 -5 = 12456.8 * 10 -3 = 1245.68 * 10 -2 = 124.568 * 10 -1 =12.4568 * 10 0 1.24568 * 10 1 1.24568*10 1 is an example of normalised scientific notation.

4. 4 Floating Point in Binary Binary 0.010011 two = (0/2) + (1/2 2 ) + (0/2 4 ) +(1/2 5 ) + (1/2 6 ) 0 + 1/4 + 0 + 1/32 + 1/64 = (0.25 + 0.03125 + 0.015625) ten = 0.296875 ten In scientific notation 10011*2 -6 = 1001.1*2 -5 = = 100.11*2 -4 = 1.0011*2 -2 normalised

5. 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. 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. 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 31 30230 22 sign bit S exponent 8 bits E significand: 23 bits F

8. 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= 0000 0000 is reserved for zero.

9. 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. 10 Bad Example: (1/2) > 2 ??? Representation of 1/2 is 0.1 two = 1.0*2 -1 (normalised) 0 1111 0000....0000 S E significand l Representation of 2 is »10 two = 1.0*2 1 (normalised) 0 0000 0001 0000....0000 S E significand

11. 11 Representation of Exponent Inappropriate to use twos complement for the exponent Ideally want 0000 0000 to represent most negative number, 1111 1111 most positive. Number range: 1111 1111 1110....... 0111 1111 0111 1110... 0000 use this for 2 0 positive negative 0111 1111 = 127 ten

12. 12 Biased Representation (IEEE FP Standard) The bias 127 represents 0 128 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. 13 Example 1 Represent 0.3125 ten = 5/16 5/16 = 1/4 + 1/16 = 0.0101 two = 1.01*2 -2 S = 0 E = ??? -2 = E-bias = E-127 E = 125 ten = 0111 1101 two Significand = 010.…000 0 0111 1101 010000...000

14. 14 Example 2 What does 0 0111 1101 010000...000 represent? S = 0 E = 0111 1101 = 125 ten Exponent = E-bias = 125-127 = -2 Significand = 1/4 (-1) S (1+sig.)2 E-bias = (1 + 1/4)*(1/4) = 5/16

15. 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, 11111 round to 4 bits).

16. 16 Addition Example 0.5 + 2.75 = 3.25 0.1 two + 10.11 two 1.0*2 -1 + 1.011*2 1 0.010*2 1 + 1.011*2 1 1.101*2 1 (already normalised) (1 + (1/2) + (1/8)) * 2 3.25

17. 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. 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.