15 213 The course that gives CMU its Zip Floating Point Arithmetic Feb 17 2000 Topics class10 ppt IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IA32 floating point Floating Point Puzzles For each of the following C expressions either Argue that is true for all argument values Explain why not true x int float x int x float f double d Assume neither d nor f is NAN x int double x f float double f d float d f f 2 3 2 3 0 d 0 0 d 2 0 0 d f f d d d 0 0 d f d f class10 ppt 2 CS 213 S 00 IEEE Floating Point IEEE Standard 754 Estabilished in 1985 as uniform standard for floating point arithmetic Before that many idiosyncratic formats Supported by all major CPUs Driven by Numerical Concerns Nice standards for rounding overflow underflow Hard to make go fast Numerical analysts predominated over hardware types in defining standard class10 ppt 3 CS 213 S 00 Fractional Binary 2 Numbers 2 i i 1 bi bi 1 b2 b1 4 2 1 b0 b 1 b 2 b 3 1 2 1 4 1 8 b j Representation 2 j Bits to right of binary point represent fractional powers of 2 i Represents rational number bk 2 k k j class10 ppt 4 CS 213 S 00 Value 5 3 4 Fractional Binary Number Examples Representation 2 7 8 63 64 101 112 10 1112 0 1111112 Observation Divide by 2 by shifting right Numbers of form 0 111111 2 just below 1 0 Use notation 1 0 Limitation Can only exactly represent numbers of the form x 2k Other numbers have repeating bit representations Value 1 3 Representation 0 0101010101 01 2 1 5 0 001100110011 0011 2 1 10 0 0001100110011 0011 2 class10 ppt 5 CS 213 S 00 Floating Point Representation Numerical Form 1s M 2E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range 1 0 2 0 Exponent E weights value by power of two Encoding s exp frac MSB is sign bit exp field encodes E frac field encodes M Sizes Single precision 8 exp bits 23 frac bits 32 bits total Double precision 11 exp bits 52 frac bits 64 bits total class10 ppt 6 CS 213 S 00 Normalized Numeric Values Condition exp 000 0 and exp 111 1 Exponent coded as biased value E Exp Bias Exp unsigned value denoted by exp Bias Bias value Single precision 127 Exp 1 254 E 126 127 Double precision 1023 Exp 1 2046 E 1022 1023 in general Bias 2m 1 1 where m is the number of exponent bits Significand coded with implied leading 1 m 1 xxx x2 xxx x bits of frac Minimum when 000 0 M 1 0 Maximum when 111 1 M 2 0 Get extra leading bit for free class10 ppt 7 CS 213 S 00 Normalized Encoding Example Value Float F 15213 0 1521310 111011011011012 1 1101101101101 2 X 213 Significand M frac 110110110110100000000002 Exponent E Bias Exp 1 1101101101101 2 13 127 140 100011002 Floating Point Representation Class 02 Hex Binary 0000 140 15213 class10 ppt 4 6 6 D B 4 0 0 0100 0110 0110 1101 1011 0100 0000 100 0110 0 1110 1101 1011 01 8 CS 213 S 00 Condition Denormalized Values exp 000 0 Value Exponent value E Bias 1 Significand value m 0 xxx x2 xxx x bits of frac Cases exp 000 0 frac 000 0 Represents value 0 Note that have distinct values 0 and 0 exp 000 0 frac 000 0 Numbers very close to 0 0 Lose precision as get smaller Gradual underflow class10 ppt 9 CS 213 S 00 Description Zero Interesting expNumbers frac Numeric Value 00 00 00 00 0 0 Smallest Pos Denorm 00 00 00 01 Single 1 4 X 10 45 Double 4 9 X 10 324 2 23 52 X 2 126 1022 Largest Denormalized 00 00 11 11 Single 1 18 X 10 38 Double 2 2 X 10 308 1 0 X 2 126 1022 Smallest Pos Normalized 00 01 00 00 Just larger than largest denormalized 1 0 X 2 126 1022 One 1 0 01 11 00 00 Largest Normalized 11 10 11 11 Single 3 4 X 1038 Double 1 8 X 10308 class10 ppt 10 2 0 X 2 127 1023 CS 213 S 00 Condition Special Values exp 111 1 Cases exp 111 1 frac 000 0 Represents value infinity Operation that overflows Both positive and negative E g 1 0 0 0 1 0 0 0 1 0 0 0 exp 111 1 frac 000 0 Not a Number NaN Represents case when no numeric value can be determined E g sqrt 1 class10 ppt 11 CS 213 S 00 Summary of Floating Point Real Number Encodings Normalized Denorm 0 0 Denorm Normalized NaN NaN class10 ppt 12 CS 213 S 00 Tiny floating point example Assume an 8 bit floating point representation where the sign bit is in the most significant bit the next four bits are the exponent with a bias of 7 the last three bits are the frac Otherwise the rules are the same as IEEE floating point format normalized denormalized representation of 0 NaN infinity 7 6 s class10 ppt 0 3 2 exp frac 13 CS 213 S 00 Values related to the exponent class10 ppt Exp exp E 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 6 denorms 6 5 4 3 2 1 0 1 2 3 4 5 6 7 inf Nan 14 CS 213 S 00 exp Denormalized numbers Normalized numbers class10 ppt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Dynamic E value Range 0000 000 0000 001 0000 010 n a 6 6 0 1 512 2 512 closest to zero 0000 0000 0001 0001 110 111 000 001 6 6 6 6 6 512 7 512 8 512 9 512 largest denorm smallest norm 0110 0110 0111 0111 0111 110 111 000 001 010 1 1 0 0 0 28 32 30 32 1 36 32 40 32 7 7 n a 224 240 inf 1110 110 1110 111 1111 000 15 closest to 1 below closest to 1 above largest norm CS 213 S 00 Special Properties of FP Zero Same as Integer Zero Encoding All bits 0 Can Almost Use Unsigned Integer Comparison Must first compare sign bits Must consider 0 0 NaNs problematic Will be greater than any other values What should comparison yield Otherwise OK Denorm vs normalized Normalized vs infinity class10 ppt 16 CS 213 S 00 Floating Point Operations Conceptual View First compute exact result Make it fit into desired precision Possibly overflow if exponent too large Possibly round to fit into frac Rounding Modes illustrate with rounding Zero Round down Round up Nearest Even default …
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