Carnegie Mellon Introduction to Computer Systems 15 213 18 243 spring 2009 4th Lecture Jan 22nd Instructors Gregory Kesden and Markus P schel Carnegie Mellon Last Time Integers Representation unsigned and signed Conversion casting Bit representation maintained but reinterpreted Expanding truncating Truncating mod Addition negation multiplication shifting Operations are mod 2w Ring properties hold Associative commutative distributive additive 0 and inverse Ordering properties do not hold u 0 does not mean u v v u v 0 does not mean u v 0 Carnegie Mellon Today Floating Point Background Fractional binary numbers IEEE floating point standard Definition Example and properties Rounding addition multiplication Floating point in C Summary Carnegie Mellon Fractional binary numbers What is 1011 101 Carnegie Mellon Fractional Binary Numbers 2i 2i 1 4 2 1 bi bi 1 b2 b1 b0 b 1 b 2 b 3 b j 1 2 1 4 1 8 2 j Representation Bits to right of binary point represent fractional powers of 2 i Represents rational number k k bk 2 j Carnegie Mellon Fractional Binary Numbers 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 are just below 1 0 1 2 1 4 1 8 1 2i Use notation 1 0 1 0 Carnegie Mellon Representable Numbers Limitation Can only exactly represent numbers of the form x 2k Other rational numbers have repeating bit representations Value Representation 1 3 1 5 1 10 0 0101010101 01 2 0 001100110011 0011 2 0 0001100110011 0011 2 Carnegie Mellon Today Floating Point Background Fractional binary numbers IEEE floating point standard Definition Example and properties Rounding addition multiplication Floating point in C Summary Carnegie Mellon 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 fast in hardware Numerical analysts predominated over hardware designers in defining standard Carnegie Mellon Floating Point Representation Numerical Form 1 s 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 MSB s is sign bit s exp field encodes E but is not equal to E frac field encodes M but is not equal to M s exp frac Carnegie Mellon Precisions Single precision 32 bits s exp 1 8 23 Double precision 64 bits s exp 1 11 frac frac 52 Extended precision 80 bits Intel only s exp 1 15 frac 63 or 64 Carnegie Mellon Normalized Values Condition exp 000 0 and exp 111 1 Exponent coded as biased value E Exp Bias Exp unsigned value exp Bias 2e 1 1 where e is number of exponent bits Single precision 127 Exp 1 254 E 126 127 Double precision 1023 Exp 1 2046 E 1022 1023 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 Carnegie Mellon Normalized Encoding Example Value Float F 15213 0 1521310 111011011011012 1 11011011011012 x 213 Significand 1 11011011011012 110110110110100000000002 M frac Exponent E Bias Exp 13 127 140 100011002 Result 0 10001100 11011011011010000000000 s exp frac Carnegie Mellon Denormalized Values Condition exp 000 0 Exponent value E Bias 1 instead of E 0 Bias Significand coded with implied leading 0 M 0 xxx x2 xxx x bits of frac Cases exp 000 0 frac 000 0 Represents value 0 Note distinct values 0 and 0 why exp 000 0 frac 000 0 Numbers very close to 0 0 Lose precision as get smaller Equispaced Carnegie Mellon Special Values Condition exp 111 1 Case 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 Case exp 111 1 frac 000 0 Not a Number NaN Represents case when no numeric value can be determined E g sqrt 1 Carnegie Mellon Visualization Floating Point Encodings Normalized NaN Denorm Denorm 0 0 Normalized NaN Carnegie Mellon Today Floating Point Background Fractional binary numbers IEEE floating point standard Definition Example and properties Rounding addition multiplication Floating point in C Summary Carnegie Mellon Tiny Floating Point Example s exp 1 4 frac 3 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 Carnegie Mellon Dynamic Range Positive Only s exp Denormalized numbers Normalized numbers 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 frac 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 1110 110 1110 111 1111 000 closest to 1 below closest to 1 above largest norm Carnegie Mellon Distribution of Values 6 bit IEEE like format e 3 exponent bits f 2 fraction bits Bias is 23 1 1 3 s exp 1 3 frac 2 Notice how the distribution gets denser toward zero 15 10 5 Denormalized 0 5 Normalized Infinity 10 15 Carnegie Mellon Distribution of Values close up view 6 bit IEEE like format e 3 exponent bits f 2 fraction bits Bias is 3 1 0 5 Denormalized s exp 1 3 0 Normalized frac 2 0 5 Infinity 1 Carnegie Mellon Interesting Numbers Description exp frac Zero 00 00 00 00 Smallest Pos Denorm 00 00 00 01 Single 1 4 x 10 45 Double 4 9 x 10 324 Largest Denormalized 00 00 11 11 Single 1 18 x 10 38 Double 2 2 x 10 308 Smallest Pos Normalized 00 01 Just larger than largest denormalized One 01 11 00 00 Largest Normalized 11 10 11 11 Single 3 4 x 1038 Double 1 8 x 10308 single double Numeric Value 0 0 2 23 52 x 2 126 1022 1 0 x 2 126 1022 00 00 1 0 x 2 126 1022 1 0 2 0 x 2 127 1023 Carnegie Mellon Special Properties of Encoding FP Zero Same as Integer Zero All bits 0 Can Almost Use Unsigned Integer …
View Full Document
Unlocking...