Unit3--FP-EE357-Nazarian-Fall09

# Unit3--FP-EE357-Nazarian-Fall09 - University University of...

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University of Southern California Viterbi School of Engineering EE357 asic Organization of Computer Systems Basic Organization of Computer Systems EEE 54 loatin Point Representation IEEE 754 Floating Point Representation Floating Point Arithmetic References: 1) Textbook ) ark Redekopp’s slide series Shahin Nazarian Fall 2009 2) Mark Redekopp s slide series

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Floating Point Programming languages support numbers with fractions (aka real in mathematics) Floating point is used to represent very small numbers (fractions) and very large numbers vogadro’s Number: + 247 * 0 23 Avogadro s Number: +6.0247 10 Planck’s Constant: +6.6254 * 10 -27 Floating Point representation is used in HLL’s like C by declaring variables as float or double ote: 2 r 4 it integers can’t represent Note: 32 or 64-bit integers can t represent this range Shahin Nazarian/EE357/Fall 2009 2
Fixed Point Unsigned and 2’s complement fall under a category of representations called “Fixed Point” The radix point is assumed to be in a fixed location for all numbers ntegers: 0011101 inary point to right of LSB) Integers: 10011101. (binary point to right of LSB) For 32-bits, unsigned range is 0 to ~4 billion Fractions: .10011101 (binary point to left of MSB) Range [0 to 1) Main point: By fixing the radix point, we limit the range of numbers that can be represented Floating point allows the radix point to be in a different location for each value Shahin Nazarian/EE357/Fall 2009 3

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Floating Point Representation Similar to scientific notation used with decimal numbers ± D.DDD * 10 ± exp Floating Point representation uses the following form ± b.bbbb * 2 ± exp 3 Fields: sign, exponent, fraction (also called mantissa or significand) Shahin Nazarian/EE357/Fall 2009 4 S Exp. fraction Overall Sign of #
Normalized FP Numbers Decimal Example +0.754*10 15 is not correct scientific notation Must have exactly one significant digit before decimal point: +7.54*10 14 In binary the only significant digit is ‘1’ Thus normalized FP format is: ± 1.bbbbbb * 2 ± exp P numbers will always be normalized before being stored in FP numbers will always be normalized before being stored in memory or a reg. The 1. is actually not stored but assumed since we will always store normalized numbers If HW calculates a result of 0.001101*2 5 it must normalize to 1.101000*2 2 before storing Shahin Nazarian/EE357/Fall 2009 5

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IEEE Floating Point Formats Single Precision (32-bit format) Double Precision (64-bit format) 1 Sign bit (0=p/1=n) 8 Exponent bits (Excess-127 1 Sign bit (0=p/1=n) 11 Exponent bits (Excess-1023 representation) 23 fraction (significand or representation) 52 fraction (significand or mantissa) bits Equiv. Decimal Range: 7 digits x 10 ± 38 mantissa) bits Equiv. Decimal Range: 16 digits x 10 ± 308 7 gts 6 gts 18 2 3 11 1 5 2 Shahin Nazarian/EE357/Fall 2009 6 S Fraction Exp. S Fraction Exp.
Exponent Representation Exponent includes its own sign (+/-) Rather than using 2’s comp. system, Single-Precision uses Excess-127 while 2’s comp.

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## This note was uploaded on 09/14/2010 for the course EE 357 at USC.

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Unit3--FP-EE357-Nazarian-Fall09 - University University of...

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