CS 105 Tour of the Black Holes of Computing Floating Point Topics floats ppt Overview of Floating Point 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 2 CS 105 Fractional Binary Numbers 2 i 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 3 CS 105 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 4 CS 105 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 5 Representation 0 0101010101 01 2 0 001100110011 0011 2 0 0001100110011 0011 2 CS 105 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 6 exp frac MSB is sign bit exp field encodes E frac field encodes M CS 105 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 7 CS 105 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 8 CS 105 Normalized Encoding Ex 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 Class 02 Hex Binary 0000 140 9 15213 4 6 6 D B 4 0 0 0100 0110 0110 1101 1011 0100 0000 100 0110 0 1110 1101 1011 01 CS 105 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 1 40 1 60 1 50 2 50 1 50 Zero 1 1 1 2 1 Round down 1 1 1 2 2 Round up 2 2 2 3 1 Nearest Even default 1 2 2 2 2 Note 1 Round down rounded result is close to but no greater than true result 2 Round up rounded result is close to but no less than true result 10 CS 105 Floating Point in C C Guarantees Two Levels float single precision double double precision Conversions Casting between int float and double changes numeric values Double or float to int Truncates fractional part Like rounding toward zero Not defined when out of range Generally saturates to TMin or TMax int to double Exact conversion as long as int has 53 bit word size int to float Will round according to rounding mode 11 CS 105 Ariane 5 Exploded 37 seconds after liftof Cargo worth 500 million Why Computed horizontal velocity as floating point number Converted to 16 bit integer Worked OK for Ariane 4 Overflowed for Ariane 5 Used same software 12 CS 105 Summary IEEE Floating Point Has Clear Mathematical Properties Represents numbers of form M X 2E Can reason about operations independent of implementation As if computed with perfect precision and then rounded Not the same as real arithmetic Violates associativity distributivity Makes life difficult for compilers serious numerical applications programmers 13 CS 105
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