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lecture4 - CMPSC/MATH 451 Numerical Computations Lecture 4...

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CMPSC/MATH 451 Numerical Computations Lecture 4 Aug 29, 2011 Prof. Kamesh Madduri
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REVIEW 2
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Floating-point numbers 3
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IEEE-754(1985) Representation Exponent is stored in “ biased format ”: a bias value is added to the actual exponent. Single-precision : bias is 127 ; the stored exponent in the 8-bit field lies in [0, 255]. Normalized representation: the leading ‘1’ is not stored. Single-precision example (Source: http:// en.wikipedia.org/wiki/IEEE_754-1985) 4
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Rounding rules 5
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e to the pi minus pi Source: XKCD, http://www.xkcd.com/217/ 6
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Machine precision 7
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Subnormals and Gradual Underflow Normalization causes gap around zero in floating-point system Denormalized numbers: Allow leading digit to be zero when exponent is at its minimum value (E=L) . Gap is then filled in by subnormal/ denormalized floating-point numbers A floating point system with denormalized numbers is said to exhibit gradual underflow 8
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Exceptional Values IEEE floating-point standard provides
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