15 213 Floating Point Arithmetic August 31 2009 Topics class03 ppt IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For each of the following C expressions either Argue that it 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 2 15 213 Intro to Computer Systems Fall 2009 IEEE Floating Point IEEE Standard 754 Established 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 3 15 213 Intro to Computer Systems Fall 2009 Fractional Binary Numbers 2i 2i 1 bi bi 1 b2 b1 4 2 1 b0 b 1 b 2 b 3 1 2 1 4 1 8 b j 2 j Representation Bits to right of binary point represent fractional powers of 2 Represents rational number i bk 2 k k j 4 15 213 Intro to Computer Systems Fall 2009 Frac Binary Number Examples Value 5 3 4 2 7 8 63 64 Representation 101 112 10 1112 0 1111112 Observations Divide by 2 by shifting right Multiply by 2 by shifting left Numbers of form 0 111111 2 just below 1 0 1 2 1 4 1 8 1 2i 1 0 Use notation 1 0 5 15 213 Intro to Computer Systems Fall 2009 Representable Numbers Limitation Can only exactly represent numbers of the form x 2k Other numbers have repeating bit representations Value 1 3 1 5 1 10 6 Representation 0 0101010101 01 2 0 001100110011 0011 2 0 0001100110011 0011 2 15 213 Intro to Computer Systems Fall 2009 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 7 exp frac MSB is sign bit exp field encodes E frac field encodes M 15 213 Intro to Computer Systems Fall 2009 Floating Point Precisions 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 Extended precision 15 exp bits 63 frac bits Only found in Intel compatible machines Stored in 80 bits 1 bit wasted 8 15 213 Intro to Computer Systems Fall 2009 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 2e 1 1 where e is 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 9 15 213 Intro to Computer Systems Fall 2009 Normalized Encoding Example Value Float F 15213 0 1521310 111011011011012 1 11011011011012 X 213 Significand M frac Exponent E Bias Exp 1 11011011011012 110110110110100000000002 13 127 140 100011002 Floating Point Representation Hex Binary 0000 140 10 15213 4 6 6 D B 4 0 0 0100 0110 0110 1101 1011 0100 0000 100 0110 0 1110 1101 1011 01 15 213 Intro to Computer Systems Fall 2009 Denormalized Values Condition 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 11 Gradual underflow 15 213 Intro to Computer Systems Fall 2009 Special Values Condition 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 12 15 213 Intro to Computer Systems Fall 2009 Summary of Floating Point Real Number Encodings NaN 13 Normalized Denorm Denorm 0 0 15 213 Intro to Computer Systems Fall 2009 Normalized NaN Tiny Floating Point Example 8 bit Floating Point Representation 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 Same General Form as IEEE Format normalized denormalized representation of 0 NaN infinity 7 6 s 14 0 3 2 exp frac 15 213 Intro to Computer Systems Fall 2009 Values Related to the Exponent 15 Exp exp E 2E 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 6 5 4 3 2 1 0 1 2 3 4 5 6 7 n a 1 64 1 64 1 32 1 16 1 8 1 4 1 2 1 2 4 8 16 32 64 128 denorms inf NaN 15 213 Intro to Computer Systems Fall 2009 Dynamic Range E Value 0000 000 0000 001 0000 010 6 6 6 0 1 8 1 64 1 512 2 8 1 64 2 512 closest to zero 0000 0000 0001 0001 110 111 000 001 6 6 6 6 6 8 1 64 7 8 1 64 8 8 1 64 9 8 1 64 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 14 8 1 2 15 8 1 2 8 8 1 9 8 1 10 8 1 14 16 15 16 1 9 8 10 8 7 7 n a 14 8 128 224 15 8 128 240 inf s exp 0 0 Denormalized 0 numbers 0 0 0 0 0 0 Normalized 0 numbers 0 0 0 0 0 16 frac 1110 110 1110 111 1111 000 15 213 Intro to Computer Systems Fall 2009 closest to 1 below closest to 1 above largest norm Distribution of Values 6 bit IEEE like format e 3 exponent bits f 2 fraction bits Bias is 3 Notice how the distribution gets denser toward zero 15 10 5 Denormalized 17 0 5 Normalized 15 213 Intro to Computer Systems Fall 2009 10 Infinity 15 Distribution of Values close up view 6 bit IEEE like format e 3 exponent bits f …
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