Signed Multiplication Multiply the positives Negate result if signs of operand are different

Transcription

1 Another Improvement Save on space: Put multiplier in product saves on speed: only single shift needed Figure: Improved hardware for multiplication

2 Signed Multiplication Multiply the positives Negate result if signs of operand are different

3 Multiplication and Division in MIPS mult, multu; div, divu Result in LO and HI Use mflo and mfhi for result

4 Floating Point Real numbers (π) ( ) (e)

5 Floating Point Real numbers (π) ( ) (e) Floating numbers: position of binary point is not fixed. Just like float in C. vs. fixed-point systems

6 Floating Point Real numbers (π) ( ) (e) Floating numbers: position of binary point is not fixed. Just like float in C. vs. fixed-point systems Scientific notation

7 Floating Point Real numbers (π) ( ) (e) Floating numbers: position of binary point is not fixed. Just like float in C. vs. fixed-point systems Scientific notation Normalized no leading 0

8 Floating Point Real numbers (π) ( ) (e) Floating numbers: position of binary point is not fixed. Just like float in C. vs. fixed-point systems Scientific notation Normalized no leading 0 Exponent no. of positions to move the point in the fraction

9 Binary Floating Numbers Binary point (analogous to decimal point) two 2 4

10 Binary Floating Numbers Binary point (analogous to decimal point) two 2 4 In general 1.xxxxxxx two 2 yyyy

11 Binary Floating Numbers Binary point (analogous to decimal point) two 2 4 In general 1.xxxxxxx two 2 yyyy Why 1 in fraction? (Will use exponent in decimal for simplicity)

12 Binary Floating Numbers In design: compromise between sizes of fraction and exponent between precision and range since fixed word size

13 Binary Floating Numbers In design: compromise between sizes of fraction and exponent between precision and range since fixed word size Represent in (floating) binary word as: S (sign bit): 1 bit (31st bit) ( 1) S F 2 E E (exponent): 8 bits (bits 23 to 30) F (significand, fraction): 23 bits (bits 0 to 22)

14 Binary Floating Numbers In design: compromise between sizes of fraction and exponent between precision and range since fixed word size Represent in (floating) binary word as: S (sign bit): 1 bit (31st bit) ( 1) S F 2 E E (exponent): 8 bits (bits 23 to 30) F (significand, fraction): 23 bits (bits 0 to 22) Not just MIPS formats: IEEE 754 floating-point standard

15 Overflow & Underflow Range: 2.0 ten to 2.0 ten 10 38

16 Overflow & Underflow Range: 2.0 ten to 2.0 ten Overflow: Too large to represent exponent too large to fit in 8 bits

17 Overflow & Underflow Range: 2.0 ten to 2.0 ten Overflow: Too large to represent exponent too large to fit in 8 bits Underflow: Too accurate to represent Negative exponent too large to fit

18 double format double-precision floating-point

19 double format double-precision floating-point vs. single-precision

20 double format double-precision floating-point vs. single-precision Uses two MIPS words

21 double format double-precision floating-point vs. single-precision Uses two MIPS words S: 31st bit of 1st register E: bits 30 to 20 of 1st register F: rest 20 bits of 1st register + 32 bits of 2nd

22 double format double-precision floating-point vs. single-precision Uses two MIPS words S: 31st bit of 1st register E: bits 30 to 20 of 1st register F: rest 20 bits of 1st register + 32 bits of 2nd Increased range: 2.0 ten to 2.0 ten

23 Another Optimization Normalized Make leading 1-bit implicit

24 Another Optimization Normalized Make leading 1-bit implicit 24 bits for significand 53 bits for double-precision

25 Another Optimization Normalized Make leading 1-bit implicit 24 bits for significand 53 bits for double-precision Also use biased notation for exponent instead of two s complement

26 Another Optimization Normalized Make leading 1-bit implicit 24 bits for significand 53 bits for double-precision Also use biased notation for exponent instead of two s complement Subtract 127 from actual value 1023 for double precision

27 Another Optimization Normalized Make leading 1-bit implicit 24 bits for significand 53 bits for double-precision Also use biased notation for exponent instead of two s complement Subtract 127 from actual value 1023 for double precision is for is for infinity

28 Another Optimization Normalized Make leading 1-bit implicit 24 bits for significand 53 bits for double-precision Also use biased notation for exponent instead of two s complement Subtract 127 from actual value 1023 for double precision is for is for infinity

29 IEEE 754 Representation Final representation: ( 1) S (1 + F) 2 ( E 127)

30 Advantages of Normalized Scientific Notation Simplifies exchange of floating point data Simplifies arithmetic Increases accuracy: unnecessary leading 0 s are replaced by real numbers on the right