lecture 5 floating point continued
play

Lecture 5 Floating Point Continued CS 230 - Spring 2020 1-1 - PowerPoint PPT Presentation

Sep 19, 2022 •257 likes •935 views

CS 230 Introduction to Computers and Computer Systems Lecture 5 Floating Point Continued CS 230 - Spring 2020 1-1 Normalized Representation Why do we need to move the radix point to after the first 1 bit? Why is there a 1 in the

  1. CS 230 – Introduction to Computers and Computer Systems Lecture 5 – Floating Point Continued CS 230 - Spring 2020 1-1

  2. Normalized Representation  Why do we need to move the radix point to after the first 1 bit? Why is there a 1 in the formula?  (-1) S * 1 .F * 2 E-B  Let’s pretend it’s a zero: consider 1.01 2  In our example format it could be  (-1) 0 * 0 .1010 * 2 4-3  (-1) 0 * 0 .0101 * 2 5-3  Now we can’t compare bits for equality and we waste the potential free 1 bit CS 230 - Spring 2020 1-2

  3. Normalized Representation Problem  What is the smallest positive number we can represent in our example format?  (-1) S * 1 .F * 2 E-B  S=0 E=000 F=0001  (-1) 0 * 1 .0001 * 2 0-3 = 1 .0001 * 2 -3 = 0.00 1 0001 2  How do we get rid of that 1?  It’s blocking all the small numbers  We can’t just always use zero instead CS 230 - Spring 2020 1-3

  4. Solution: Subnormal  Only use zero when the number is too small  Two forms of the floating point formula Normal: (-1) S * (1 + F) * 2 E-B Subnormal: (-1) S * (0 + F) * 2 1-B  How do we know when to use which form? - When E is all zeros use subnormal  Why 1-B? - We used up E=0 to mean subnormal, so now smallest exponent is E=1 and we don’t want to leave a gap CS 230 - Spring 2020 1-4

  5. Special Cases  Overflow still possible (E too large)  represent as +/- infinity  also for division by zero (by IEEE standard)  Invalid result – Not a Number – NaN  special cases, like 0/0 or ∞ *0 or sqrt(-1)  can “safely” propagate during computation  Both can propagate during computation  no exception (like integer division by zero) CS 230 - Spring 2020 1-5

  6. IEEE-754 Special Cases Exponent Fraction Case 000000… 000000… 0 000000… non-zero subnormal 111111… 000000… infinity (+ or -) 111111… non-zero NaN anything else anything normal CS 230 - Spring 2020 1-6

  7. CS 230 - Spring 2020 1-7

  8. CS 230 - Spring 2020 1-8

  9. CS 230 - Spring 2020 1-9

  10. CS 230 - Spring 2020 1-10

  11. CS 230 - Spring 2020 1-11

  12. CS 230 - Spring 2020 1-12

  13. CS 230 - Spring 2020 1-13

  14. CS 230 - Spring 2020 1-14

  15. CS 230 - Spring 2020 1-15

  16. CS 230 - Spring 2020 1-16

  17. CS 230 - Spring 2020 1-17

  18. CS 230 - Spring 2020 1-18

  19. CS 230 - Spring 2020 1-19

  20. CS 230 - Spring 2020 1-20

  21. CS 230 - Spring 2020 1-21

  22. CS 230 - Spring 2020 1-22

  23. CS 230 - Spring 2020 1-23

  24. CS 230 - Spring 2020 1-24

  25. CS 230 - Spring 2020 1-25

  26. CS 230 - Spring 2020 1-26

  27. CS 230 - Spring 2020 1-27

  28. CS 230 - Spring 2020 1-28

  29. CS 230 - Spring 2020 1-29

  30. CS 230 - Spring 2020 1-30

  31. CS 230 - Spring 2020 1-31

  32. CS 230 - Spring 2020 1-32

  33. CS 230 - Spring 2020 1-33

  34. CS 230 - Spring 2020 1-34

  35. Floating Point Arithmetic  Addition  align radix points  use normal addition  Multiplication  add exponents  multiply significands  1.F if normal, 0.F if subnormal CS 230 - Spring 2019 1-35

  36. CS 230 - Spring 2020 1-36

  37. CS 230 - Spring 2020 1-37

  38. CS 230 - Spring 2020 1-38

  39. CS 230 - Spring 2020 1-39

  40. CS 230 - Spring 2020 1-40

  41. CS 230 - Spring 2020 1-41

  42. CS 230 - Spring 2020 1-42

  43. CS 230 - Spring 2020 1-43

  44. CS 230 - Spring 2020 1-44

  45. CS 230 - Spring 2020 1-45

  46. CS 230 - Spring 2020 1-46

  47. CS 230 - Spring 2020 1-47

  48. CS 230 - Spring 2020 1-48

  49. CS 230 - Spring 2020 1-49

  50. CS 230 - Spring 2020 1-50

  51. CS 230 - Spring 2020 1-51

  52. CS 230 - Spring 2020 1-52

  53. CS 230 - Spring 2020 1-53

  54. CS 230 - Spring 2020 1-54

  55. CS 230 - Spring 2020 1-55

  56. CS 230 - Spring 2020 1-56

  57. CS 230 - Spring 2020 1-57

  58. CS 230 - Spring 2020 1-58

  59. CS 230 - Spring 2020 1-59

  60. CS 230 - Spring 2020 1-60

  61. CS 230 - Spring 2020 1-61

  62. CS 230 - Spring 2020 1-62

  63. CS 230 - Spring 2020 1-63

  64. CS 230 - Spring 2020 1-64

  65. CS 230 - Spring 2020 1-65

  66. CS 230 - Spring 2020 1-66

  67. CS 230 - Spring 2020 1-67

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur

Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur

ECE 0142 Computer Organization Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur floating-point programming. Floating point greatly simplifies working with large (e.g., 2 70 ) and small (e.g., 2 -17 )

638 views • 30 slides
Floating Point Data CSCI 125 & 161 / ENGR 144 Floating point numbers are not exact

Floating Point Data CSCI 125 & 161 / ENGR 144 Floating point numbers are not exact

Floating Point Data CSCI 125 & 161 / ENGR 144 Floating point numbers are not exact Value 0.1 is very close to 1/10, but not Lecture 13 precisely equal to it 0.1 10 = 0.000110011001100110011... 2 Cannot rely on accuracy of

313 views • 3 slides
Lecture 15 Floating point Announcements 2 Scott B. Baden / CSE 160 / Wi '16 Todays lecture

Lecture 15 Floating point Announcements 2 Scott B. Baden / CSE 160 / Wi '16 Todays lecture

Lecture 15 Floating point Announcements 2 Scott B. Baden / CSE 160 / Wi '16 Todays lecture SSE wrapping up Floating point 3 Scott B. Baden / CSE 160 / Wi '16 Recapping from last time: how do we vectorize? Original code

613 views • 33 slides
Floating-point numbers Fractional binary numbers IEEE floating-point standard Floating-point

Floating-point numbers Fractional binary numbers IEEE floating-point standard Floating-point

Floating-point numbers Fractional binary numbers IEEE floating-point standard Floating-point operations and rounding Lessons for programmers Many more details we will skip (its a 58-page standard) See CSAPP 2.4 for more detail. 1

381 views • 15 slides
15-213 The course that gives CMU its Zip! Floating Point Sept 6, 2006 Topics Topics

15-213 The course that gives CMU its Zip! Floating Point Sept 6, 2006 Topics Topics

15-213 The course that gives CMU its Zip! Floating Point Sept 6, 2006 Topics Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties class03.ppt 15-213, F06 Floating Point

1.03k views • 34 slides
lecture 2 - fixed point - IEEE floating point standard Wed. January 13, 2016 For those

lecture 2 - fixed point - IEEE floating point standard Wed. January 13, 2016 For those

lecture 2 - fixed point - IEEE floating point standard Wed. January 13, 2016 For those interested in finding out what research is all about, I encourage you to participate in studies such as these. Fixed point Fixed point means we have a

557 views • 30 slides
Floating point Today ! IEEE Floating Point Standard ! Rounding ! Floating Point Operations !

Floating point Today ! IEEE Floating Point Standard ! Rounding ! Floating Point Operations !

Floating point Today ! IEEE Floating Point Standard ! Rounding ! Floating Point Operations ! Mathematical properties Next time ! The machine model Chris Riesbeck, Fall 2011 Monday, October 3, 2011 Checkpoint Monday, October 3, 2011 IEEE

1.07k views • 28 slides
9/20/2018 Today: Floating Point Background: Fractional binary numbers IEEE floating point

9/20/2018 Today: Floating Point Background: Fractional binary numbers IEEE floating point

9/20/2018 Today: Floating Point Background: Fractional binary numbers IEEE floating point standard: Definition Floating Point Example and properties Rounding, addition, multiplication CSci 2021: Machine Architecture and

339 views • 7 slides
2/10/2020 Today: Floating Point Background: Fractional binary numbers IEEE floating point

2/10/2020 Today: Floating Point Background: Fractional binary numbers IEEE floating point

2/10/2020 Today: Floating Point Background: Fractional binary numbers IEEE floating point standard: Definition Floating Point Example and properties Rounding, addition, multiplication CSci 2021: Machine Architecture and

411 views • 7 slides
ECE232: Hardware Organization and Design Lecture 9: Floating Point Adapted from Computer

ECE232: Hardware Organization and Design Lecture 9: Floating Point Adapted from Computer

ECE232: Hardware Organization and Design Lecture 9: Floating Point Adapted from Computer Organization and Design , Patterson & Hennessy, UCB Floating Point Representation for non-integral numbers Including very small and very large

256 views • 24 slides
CS 356 Unit 3 IEEE 754 Floating Point Representation 3.2 Floating Point Used to represent

CS 356 Unit 3 IEEE 754 Floating Point Representation 3.2 Floating Point Used to represent

3.1 CS 356 Unit 3 IEEE 754 Floating Point Representation 3.2 Floating Point Used to represent very small numbers (fractions) and very large numbers Avogadros Number: +6.0247 * 10 23 Plancks Constant: +6.6254 * 10 -27

789 views • 33 slides
Machine numbers: how floating point numbers are stored? Floating-point number representation

Machine numbers: how floating point numbers are stored? Floating-point number representation

Machine numbers: how floating point numbers are stored? Floating-point number representation What do we need to store when representing floating point numbers in a computer? ! = 1. & 2 ! Initially, different floating-point

379 views • 18 slides
Floating point How arithmetic operations mathematics involving floating point numbers

Floating point How arithmetic operations mathematics involving floating point numbers

Numerical and Scientific Computing with Applications David F . Gleich CS 314, Purdue September 7, 2016 In this class: Floating point How arithmetic operations mathematics involving floating point numbers work. Next class IEEE

812 views • 11 slides
Unit 3 IEEE 754 Floating Point Representation 3.2 Floating Point Used to represent very

Unit 3 IEEE 754 Floating Point Representation 3.2 Floating Point Used to represent very

3.1 Unit 3 IEEE 754 Floating Point Representation 3.2 Floating Point Used to represent very small numbers (fractions) and very large numbers Avogadros Number: +6.022 10 23 Boltzmanns Constant: +1.38 10 -23 32 or

1.13k views • 38 slides
Formal verification of floating-point algorithms John Harrison Intel Corporation Floating

Formal verification of floating-point algorithms John Harrison Intel Corporation Floating

Formal verification of floating-point algorithms 1 Formal verification of floating-point algorithms John Harrison Intel Corporation Floating point algorithm verification HOL Light Floating point numbers and formats HOL floating

636 views • 28 slides
lecture 12 MIPS assembly language 5 - coprocessor 1 (floating point unit FPU) -

lecture 12 MIPS assembly language 5 - coprocessor 1 (floating point unit FPU) -

lecture 12 MIPS assembly language 5 - coprocessor 1 (floating point unit FPU) - coprocessor 0 (kernel use, exception handing) February 17, 2016 lecture 12 MIPS assembly language 5 - coprocessor 1 (floating point unit FPU) -

344 views • 5 slides
CSC413/2516 Lecture 7: Generalization & Recurrent Neural Networks Jimmy Ba Jimmy Ba

CSC413/2516 Lecture 7: Generalization & Recurrent Neural Networks Jimmy Ba Jimmy Ba

CSC413/2516 Lecture 7: Generalization & Recurrent Neural Networks Jimmy Ba Jimmy Ba CSC413/2516 Lecture 7: Generalization & Recurrent Neural Networks 1 / 57 Overview Weve focused so far on how to optimize neural nets how to get

971 views • 60 slides
Half Flip 6D Lattice R. B. Palmer, Rick Fernow (BNL) Thursday 2/14/13 Introduction

Half Flip 6D Lattice R. B. Palmer, Rick Fernow (BNL) Thursday 2/14/13 Introduction

Half Flip 6D Lattice R. B. Palmer, Rick Fernow (BNL) Thursday 2/14/13 Introduction lattice types Parameters of Half-Flip lattices ICOOL simulation using matrices Conclusion 1 Cooling Scheme Advanced 4D Final Baseline 4D

461 views • 10 slides
Indium bonding @Selex G.Alimonti, INFN Milano Same technology used for about half the Atlas Pixel

Indium bonding @Selex G.Alimonti, INFN Milano Same technology used for about half the Atlas Pixel

Indium bonding @Selex G.Alimonti, INFN Milano Same technology used for about half the Atlas Pixel Detector modules but: Larger (~19x20 mm 2 instead of 7x11 mm 2 ) and thinner (100 m) chips (Atlas FE-I3 thinned down to about 180 m)

655 views • 14 slides
Assembly Services MPW Services Center for IC / MEMS Prototyping http://cmp.imag.fr Grenoble -

Assembly Services MPW Services Center for IC / MEMS Prototyping http://cmp.imag.fr Grenoble -

Development of Flip Chip Assembly Services MPW Services Center for IC / MEMS Prototyping http://cmp.imag.fr Grenoble - France CMP annual users meeting, 4 Feb. 2016, PARIS Summary ry Motivation and objectives Development of Flip Chip

594 views • 10 slides
VECTORISATION Adrian Jackson adrianj@epcc.ed.ac.uk @adrianjhpc Vectorisation Same operation

VECTORISATION Adrian Jackson adrianj@epcc.ed.ac.uk @adrianjhpc Vectorisation Same operation

VECTORISATION Adrian Jackson adrianj@epcc.ed.ac.uk @adrianjhpc Vectorisation Same operation on multiple data items Wide registers SIMD needed to approach FLOP peak performance, but your code must be capable of vectorisation

462 views • 35 slides
AM205: lecture 2 Nao and Martin will hold a Python tutorial on Monday September 10 from

AM205: lecture 2 Nao and Martin will hold a Python tutorial on Monday September 10 from

AM205: lecture 2 Nao and Martin will hold a Python tutorial on Monday September 10 from 3pm5pm in 60 Oxford Street, Room 330 Assignment 0 solutions now posted Assignment 1 will be posted this weekend Chris will hold office hours

694 views • 44 slides
CENG 342 Digital Systems Simplified Floating-point Adder Larry Pyeatt SDSM&T Binary

CENG 342 Digital Systems Simplified Floating-point Adder Larry Pyeatt SDSM&T Binary

CENG 342 Digital Systems Simplified Floating-point Adder Larry Pyeatt SDSM&T Binary Floating Point Representation Floating point number consists of three components: sign bit, exponent, and mantissa. Example: 12.75: sign is + , exponent

253 views • 12 slides
Io IoT Device ce Ar Arch chitect ecture e and Pr Programming 01219335 Data Acquisition and

Io IoT Device ce Ar Arch chitect ecture e and Pr Programming 01219335 Data Acquisition and

Io IoT Device ce Ar Arch chitect ecture e and Pr Programming 01219335 Data Acquisition and Integration Chaiporn J Chaipo n Jaik aikae aeo De Department of f Computer Engineering Kasetsart Unive versity Revised 2020-08-20 Ou

701 views • 32 slides

More recommend