Carnegie Mellon Floating Point 15 213 Introduction to Computer Systems 3rd Lecture Aug 31 2010 Instructors Randy Bryant Dave O Hallaron Carnegie Mello Today Floating Point Background Fractional binary numbers IEEE floating point standard Definition Example and properties Rounding addition multiplication Floating point in C Summary 2 Carnegie Mello Fractional binary numbers What is 1011 1012 3 Carnegie Mello Fractional Binary Numbers 2i 2i 1 4 2 1 bi bi b2 b11 2b0 b 1 b 2 b 3 b j 1 Representation 1 4 1 8 2 j Bits to right of binary point represent fractional powers of 2 Represents rational number 4 Carnegie Mello Fractional Binary Numbers Examples Value 5 3 4 Representation 101 112 2 7 8 010 1112 63 64 001 01112 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 1 0 Use notation 1 0 5 Carnegie Mello Representable Numbers Limitation Can only exactly represent numbers of the form x 2k Other rational numbers have repeating bit representations Value 1 3 1 5 1 10 Representation 0 0101010101 01 2 0 001100110011 0011 2 0 0001100110011 0011 2 6 Carnegie Mello Today Floating Point Background Fractional binary numbers IEEE floating point standard Definition Example and properties Rounding addition multiplication Floating point in C Summary 7 Carnegie Mello 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 8 Carnegie Mello 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 9 Carnegie Mello Precisions Single precision 32 bits s exp 1 frac 8 bits Double precision 64 bits s exp 1 23 bits frac 11 bits 52 bits Extended precision 80 bits Intel only s exp 1 15 bits frac 63 or 64 bits 10 Carnegie Mello Normalized Values Condition exp 000 0 and exp 111 1 Exponent coded as biased value E Exp Bias Exp unsigned value exp Bias 2k 1 1 where k 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 11 Carnegie Mellon Normalized Encoding Example Value Float F 15213 0 1521310 111011011011012 1 11011011011012 x 213 Significand M frac 110110110110100000000002 Exponent E Bias Exp 1 11011011011012 13 127 140 100011002 Result s 10001100 exp frac 0 11011011011010000000000 12 Carnegie Mello 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 zero value Note distinct values 0 and 0 why exp 000 0 frac 000 0 Numbers very close to 0 0 Lose precision as get smaller 13 Carnegie Mello 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 0 14 Carnegie Mello Visualization Floating Point Encodings NaN Normalized Denorm 0 Denorm 0 Normalized NaN 15 Carnegie Mello Today Floating Point Background Fractional binary numbers IEEE floating point standard Definition Example and properties Rounding addition multiplication Floating point in C Summary 16 Carnegie Mello Tiny Floating Point Example s exp frac 1 4 bits 3 bits 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 17 Carnegie Mello Dynamic Range Positive Only s exp 0 0 Denormalized0 numbers 0 0 0 0 0 0 Normalized 0 0 numbers 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 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 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 zero largest denorm smallest norm closest to 1 below closest to 1 above largest norm 18 Carnegie Mello Distribution of Values 6 bit IEEE like format e 3 exponent bits f 2 fraction bits Bias is 23 1 1 3 s exp frac 1 3 bits 2 bits Notice how the distribution gets denser toward zero 8 values 15 10 5 Denormalized 0 5 Normalized Infinity 10 15 19 Carnegie Mello 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 frac 1 3 bits 2 bits 0 Normalized 0 5 1 Infinity 20 Carnegie Mello Interesting Numbers Description exp Zero frac Smallest Pos Denorm single double Numeric Value 00 00 00 00 0 0 00 00 00 01 2 23 52 x 2 00 00 11 11 1 0 x 2 126 1022 Single 1 4 x 10 45 Double 4 9 x 10 324 Largest Denormalized 126 1022 Single 1 18 x 10 38 Double 2 2 x 10 308 Smallest Pos Normalized 00 01 00 00 Just larger than largest denormalized One Largest Normalized 2 127 1023 Single 3 4 x 1038 Double 1 8 x 10308 01 11 00 00 11 10 11 11 1 0 x 2 126 1022 1 0 2 0 x 21 Carnegie Mello Special Properties of Encoding FP Zero Same as Integer Zero 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 22 Carnegie Mello Today Floating Point …
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