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Floating Point Hardware and Algorithms

Floating Point Hardware and Algorithms. Ch.3. Review. 4-bit multiplier using 4-bit adder 32-bit multiplier using 32 bit adder + 32 multiplicand, 32-bit multiplier, 64-bit product, and a control unit  this is a sequential multiplier Improve performance. How?

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Floating Point Hardware and Algorithms

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  1. Floating Point Hardware and Algorithms Ch.3

  2. Review • 4-bit multiplier using 4-bit adder • 32-bit multiplier using 32 bit adder + 32 multiplicand, 32-bit multiplier, 64-bit product, and a control unit  this is a sequential multiplier • Improve performance. How? • What is performance? Define performance. Speed, flops, power consumption, etc. • Improve performance through hardware optimization; • Example multiplier • Improve performance through better algorithms • Example: multiplier • Improve performance through better data representation.

  3. Floating point numbers • Why do we need them? • To represent very large number and very small number. • To represent a larger range of number with a given size in bits. • Lets consider a 4 bit container • Lets consider a 8 bit container • Now lets look at some data sizes, • An astronautical unit (Au)is the distance between the earth and the sun in meters. • 150,000,000,000 meters • Size of Helium atom: 1 angstrom: 10-10 meters • See more details on units of measurement here

  4. FP Representation S Exponent Fraction Value = (-1)S X F X 2 E 32 bit single precision 64 bit double precision On to IEEE 754 format

  5. Nothing is simple (IEEE 754 not yet) • Consider the representation of real number. • Eg.: -56.731 or -0.4539 • Lets see how we can go from this human readable form to IEEE 754 • Goals: you want to represent as large a range as possible • You want to include features that will optimize processing • Exponent, fraction, sign of exponent, sign of fraction -56.731 or -0.4539  -0.56731 X 102 -0.4539 X 100 0 2 1 56731

  6. Binary Representation • Now lets look at binary representation • Eg. 0.00001010 X 23 -0.0000000 X 2-4 How will you add these numbers? How will you multiply these? • When you answer these questions, you find that it will help to have the numbers in (i) fractional form, (ii) normalized, and (iii) exponents mapped to a positive range by adding a bias. • Also it is possible to represent a larger range by implying the leading 1 in the normalized fraction and a single zero.

  7. IEE754 Format A floating point number will be represented by a normalized fraction with implied leading 1, biased exponent mapped onto a positive range (no sign needed) and only sign bit is for the entire number. In other words: Fp = (S, normalized fraction, biased E) is equivalent (-1)S X (1 + fraction) X 2 (E-bias) Lets look at some examples.

  8. Floating Point Operation: Addition • Now lets look at floating point arithmetic unit. We will examine only addition.

  9. FP Addition and Multiplication • Components: operands, ALU, control, comparison, shifters, normalize, round • Control • ALUs

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