cs33-floatingpoint

cs33-floatingpoint - loating Point loating Point Floating...

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Unformatted text preview: loating Point loating Point Floating Point Floating Point Chapter 2 of B&O Some notes adopted from Bryant and O’Hallaron Floating Point Puzzles Floating Point Puzzles – For each of the following C expressions, either: • Argue that it is true for all argument values xplain why not true • Explain why not true • x == (int)(float) x • x == (int)(double) x int x = …; • f == (float)(double) f • d == (float) d ) float f = …; double d = …; • f == -(-f); • 2/3 == 2/3.0 • d < 0.0 ((d*2) < 0.0) Assume neither d nor f is NaN • d > f -f > -d • d * d >= 0.0 • (d+f)-d == f Fractional Binary Numbers Fractional Binary Numbers 2 i –1 2 i 1 2 4 • • • b i b i –1 b 2 b 1 b b –1 b –2 b –3 b – j • • • • • • . • • 1/2 /4 • • 1/4 1/8 2 – j • Representation – Bits to right of “binary point” represent fractional i powers of 2 – Represents rational number: b k 2 k k j Frac. Binary Number Examples Frac. Binary Number Examples •Value Representation 5 3 / 4 101.11 2 2 7 / 8 10.111 2 63 / 64 0.111111 2 •Observations –Divide by 2 by shifting right –Multiply by 2 by shifting left –Numbers of form 0.111111… 2 just below 1.0 /2 1/4 1/8 1/2 •1/2 + 1/4 + 1/8 + … + 1/2 i + … 1.0 Representable Numbers Representable Numbers •Limitation –Can only exactly represent numbers of the form x /2 k –Other numbers have repeating bit representations alue epresentation •Value Representation 1/3 0.0101010101[01]… 2 /5 001100110011[0011] 1/5 0.001100110011[0011]… 2 1/10 0.0001100110011[0011]… 2 IEEE 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 Floating Point Representation Floating Point Representation •Numerical Form – 1 s M 2 E •Sign bit s determines whether number is negative or positive •Significand M normally a fractional value in range [1.0,2.0). xponent eights value by power of two •Exponent E weights value by power of two •Encoding xp rac –MSB is sign bit xp eld encodes s exp frac –exp field encodes E –frac field encodes M Floating Point Precisions 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 “Normalized” Numeric Values “Normalized” Numeric Values • Condition – e x p 000…0 and exp 111…1 ponent coded as biased value • Exponent coded as biased value – E = Exp – Bias • Exp : unsigned value denoted by exp • Bias : Bias value – Single precision: 127 (Exp: 1…254, E:...
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cs33-floatingpoint - loating Point loating Point Floating...

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