1. 14/02/2016 http://numericalmethods.eng.usf.edu 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

2. Floating Point Representation http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu 14/02/20162http://numericalmethods.eng.usf.e du

3. 3 Floating Decimal Point : Scientific Form 14/02/2016

4. http://numericalmethods.eng.usf.edu4 Example The form is or Example: For 14/02/2016

5. http://numericalmethods.eng.usf.edu5 Floating Point Format for Binary Numbers 1 is not stored as it is always given to be 1. 14/02/2016

6. http://numericalmethods.eng.usf.edu6 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 001011101 Sign of the number mantissa Sign of the exponent exponent 14/02/2016

7. http://numericalmethods.eng.usf.edu7 Machine Epsilon Defined as the measure of accuracy and found by difference between 1 and the next number that can be represented 14/02/2016

8. http://numericalmethods.eng.usf.edu8 Example Ten bit word  Sign of number  Sign of exponent  Next four bits for exponent  Next four bits for mantissa 0000000000 0000000001 Next number 14/02/2016

9. http://numericalmethods.eng.usf.edu9 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) 0101101100 Sign of the number mantissa Sign of the exponent exponent 14/02/2016

10. IEEE 754 Standards for Single Precision Representation http://numericalmethods.eng.usf.edu 14/02/201610http://numericalmethods.eng.usf.e du

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. 14/02/201611http://numericalmethods.eng.usf.e du

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 14/02/201612http://numericalmethods.eng.usf.e du

13. 13 IEEE-754 Format Single Precision 32 bits for single precision 00000000000000000000000000000000 Sign (s) Biased Exponent (e’) Mantissa (m) 14/02/201613http://numericalmethods.eng.usf.e du

14. 14 Example#1 11010001010100000000000000000000 Sign (s) Biased Exponent (e’) Mantissa (m) 14/02/201614http://numericalmethods.eng.usf.e du

15. 15 Example#2 ???????????????????????????????? Sign (s) Biased Exponent (e’) Mantissa (m) Represent -5.5834x10 10 as a single precision floating point number. 14/02/201615http://numericalmethods.eng.usf.e du

16. 16 Exponent for 32 Bit IEEE-754 8 bits would represent Bias is 127; so subtract 127 from representation 14/02/201616http://numericalmethods.eng.usf.e du

17. Exponent for Special Cases Actual range of andare reserved for special numbers Actual range of 14/02/201617http://numericalmethods.eng.usf.e du

18. Special Exponents and Numbers all zeros all ones smRepresents 0all zeros 0 1 -0 0all onesall zeros 1all onesall zeros 0 or 1all onesnon-zeroNaN 14/02/201618http://numericalmethods.eng.usf.e du

19. 19 IEEE-754 Format The largest number by magnitude The smallest number by magnitude Machine epsilon 14/02/201619http://numericalmethods.eng.usf.e du

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_re presentation.html 14/02/201620http://numericalmethods.eng.usf.e du

21. THE END http://numericalmethods.eng.usf.edu 14/02/201621http://numericalmethods.eng.usf.e du