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Floating Point Representation

Floating Point Representation. Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Presented by: دکتر ابوالفضل رنجبر نوعی http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates.

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Floating Point Representation

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  1. Floating Point Representation Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Presented by: دکتر ابوالفضل رنجبر نوعی http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu

  2. Floating Point Representationhttp://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

  3. Floating Decimal Point : Scientific Form http://numericalmethods.eng.usf.edu

  4. Example The form is or Example: For http://numericalmethods.eng.usf.edu

  5. Floating Point Format for Binary Numbers 1 is not stored as it is always given to be 1. http://numericalmethods.eng.usf.edu

  6. Example 9 bit-hypothetical word • the first bit is used for the sign of the number, • the second bit for the sign of the exponent, • the next four bits for the mantissa, and • the next three bits for the exponent We have the representation as mantissa exponent Sign of the number Sign of the exponent http://numericalmethods.eng.usf.edu

  7. Machine Epsilon Defined as the measure of accuracy and found by difference between 1 and the next number that can be represented http://numericalmethods.eng.usf.edu

  8. Example Ten bit word • Sign of number • Sign of exponent • Next four bits for exponent • Next four bits for mantissa Next number http://numericalmethods.eng.usf.edu

  9. Relative Error and Machine Epsilon The absolute relative true error in representing a number will be less then the machine epsilon Example 10 bit word (sign, sign of exponent, 4 for exponent, 4 for mantissa) Sign of the number exponent mantissa Sign of the exponent http://numericalmethods.eng.usf.edu

  10. IEEE 754 Standards for Single Precision Representationhttp://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

  11. IEEE-754 Floating Point Standard • Standardizes representation of floating point numbers on different computers in single and double precision. • Standardizes representation of floating point operations on different computers. http://numericalmethods.eng.usf.edu

  12. One Great Reference What every computer scientist (and even if you are not) should know about floating point arithmetic! http://www.validlab.com/goldberg/paper.pdf http://numericalmethods.eng.usf.edu

  13. IEEE-754 Format Single Precision 32 bits for single precision Biased Exponent (e’) Sign (s) Mantissa (m) http://numericalmethods.eng.usf.edu 13

  14. Example#1 Biased Exponent (e’) Sign (s) Mantissa (m) http://numericalmethods.eng.usf.edu 14

  15. Example#2 Represent -5.5834x1010 as a single precision floating point number. Biased Exponent (e’) Sign (s) Mantissa (m) http://numericalmethods.eng.usf.edu 15

  16. Exponent for 32 Bit IEEE-754 8 bits would represent Bias is 127; so subtract 127 from representation http://numericalmethods.eng.usf.edu 16

  17. Exponent for Special Cases Actual range of and are reserved for special numbers Actual range of http://numericalmethods.eng.usf.edu

  18. Special Exponents and Numbers all zeros all ones http://numericalmethods.eng.usf.edu

  19. IEEE-754 Format The smallest number by magnitude Machine epsilon The largest number by magnitude http://numericalmethods.eng.usf.edu 19

  20. Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/floatingpoint_representation.html http://numericalmethods.eng.usf.edu

  21. THE END http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

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