1. 1 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) CPRE 583 Reconfigurable Computing Lecture 14: Fri 10/12/2011 (Floating Point) Instructor: Dr. Phillip Jones (phjones@iastate.edu) Reconfigurable Computing Laboratory Iowa State University Ames, Iowa, USA http://class.ee.iastate.edu/cpre583/

2. 2 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Project Teams: Form by Monday 10/10 MP2 due Friday 10/14 Project Proposal due Friday 10/14 (midnight) –High-level topic, and high plan for execution –I’ll give feedback –Project proposal Class presentation on Wed 10/19 5-10 power point slides I plan to have exams back to you this Friday Announcements/Reminders

3. 3 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Project Grading Breakdown 50% Final Project Demo 30% Final Project Report –20% of your project report grade will come from your 5-6 project updates. Friday’s midnight 20% Final Project Presentation

4. 4 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) FPL FPT FCCM FPGA DAC ICCAD Reconfig RTSS RTAS ISCA Projects Ideas: Relevant conferences Micro Super Computing HPCA IPDPS

5. 5 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Teams Formed and Topic: Mon 10/10 –Project idea in Power Point 3-5 slides Motivation (why is this interesting, useful) What will be the end result High-level picture of final product –Project team list: Name, Responsibility High-level Plan/Proposal: Fri 10/14 –Power Point 5-10 slides (presentation to class Wed 10/19) System block diagrams High-level algorithms (if any) Concerns –Implementation –Conceptual Related research papers (if any) Projects: Target Timeline

6. 6 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Work on projects: 10/19 - 12/9 –Weekly update reports More information on updates will be given Presentations: Finals week –Present / Demo what is done at this point –15-20 minutes (depends on number of projects) Final write up and Software/Hardware turned in: Day of final (TBD) Projects: Target Timeline

7. 7 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Initial Project Proposal Slides (5-10 slides) Project team list: Name, Responsibility (who is project leader) –Team size: 3-4 (5 case-by-case) Project idea Motivation (why is this interesting, useful) What will be the end result High-level picture of final product High-level Plan –Break project into mile stones Provide initial schedule: I would initially schedule aggressively to have project complete by Thanksgiving. Issues will pop up to cause the schedule to slip. –System block diagrams –High-level algorithms (if any) –Concerns Implementation Conceptual Research papers related to you project idea

8. 8 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Weekly Project Updates The current state of your project write up –Even in the early stages of the project you should be able to write a rough draft of the Introduction and Motivation section The current state of your Final Presentation –Your Initial Project proposal presentation (Due Wed 10/19). Should make for a starting point for you Final presentation What things are work & not working What roadblocks are you running into

9. 9 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Common Questions

10. 10 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Common Questions

11. 11 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point on FPGAs (Chapter 21.4 and 31) –Why is it viewed as difficult?? –Options for mitigating issues Overview

12. 12 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point Format (IEEE-754) S exp Mantissa 11152 Double Precision S exp Mantissa 1823 Single Precision Floating point value = (-1) S * 2 (exp-127) * (1.Mantissa) 0 x”80” 110 x”00000” 23 Mantissa = b -1 b -2 b -3 ….b -23 = ∑ b -i 2 -i i=1 Example:= -1^0 * 2^128-127 * 1.(1/2 + 1/4) = -1^0 * 2^1 * 1.75 = 3.5 Floating point value = (-1) S * 2 (exp-1023) * (1.Mantissa)

13. 13 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Fixed Point 0 x”80” “110” x”00000” = 3.5 10-bit (Format 3.7) Fixed Point for 3.5 = ? Whole Fractional Example formats (W.F): 5.5, 10.12, 3.7 b -1 b -2 …. b -F b W-1 … b 1 b 0 Example fixed point 5.5 format: 01010 01100 = 10. 1/4 + 1/8 = 10.375 Compare floating point and fixed point Floating point: 011 1000000

14. 14 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Fixed Point (Addition) Whole Fractional Operand 1 Whole Fractional Operand 2 Whole Fractional sum +

15. 15 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Fixed Point (Addition) 0011 111 0000 Operand 1 = 3.875 0001 101 0000 Operand 2 = 1.625 sum + 11-bit 4.7 format 0101 100 0000 = 5.5 You can use a standard ripple-carry adder!

16. 16 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 +

17. 17 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 1.Common exponent (i.e. align binary point) Make x”80” -> x”7F” or visa-verse?

18. 18 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 1.Common exponent (i.e. align binary point) Make x”7F”->x”80”, lose least significant bits of Operand 2 - Add the difference of x”80” – x“7F” = 1 to x”7F” - Shift mantissa of Operand 2 by difference to the right. remember “implicit” 1 of the original mantissa 0 x”80” 111 x”80000” 0 x”80” 110 x”80000” Operand 1 = 3.875 Operand 2 = 1.625 +

19. 19 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 2.Add mantissas 0 x”80” 111 x”80000” 0 x”80” 110 x”80000” Operand 1 = 3.875 Operand 2 = 1.625 +

20. 20 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 2.Add mantissas 0 x”80” 111 x”80000” 0 x”80” 110 x”80000” Operand 1 = 3.875 Operand 2 = 1.625 + 1 110 x”00000” Overflow!

21. 21 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 2.Add mantissas -You can’t just overflow mantissa into exponent field -You are actually overflowing the implicit “1” of Operand 1, so you sort of have an implicit “2” (i.e. “10”). 0 x”80” 111 x”80000” 0 x”80” 110 x”80000” Operand 1 = 3.875 Operand 2 = 1.625 + 1 110 x”00000” Overflow!

22. 22 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 2.Add mantissas -Deal with overflow of Mantissa by normalizing. -Shift mantissa right by 1 (shift a “0” in because of implicit “2”) -Increment exponent by 1 0 x”80” 111 x”80000” 0 x”80” 110 x”80000” Operand 1 = 3.875 Operand 2 = 1.625 + 0 x”81” 011 x”00000”

23. 23 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition) 0 x”80” 111 x”80000” 0 x”7F” 101 x”00000” Operand 1 = 3.875 Operand 2 = 1.625 + 2.Add mantissas -Deal with overflow of Mantissa by normalizing. -Shift mantissa right by 1 (shift a “0” in because of implicit “2”) -Increment exponent by 1 0 x”80” 111 x”80000” 0 x”80” 110 x”80000” Operand 1 = 3.875 Operand 2 = 1.625 + 0 x”81” 011 x”00000” = 5.5

24. 24 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition): Other concerns Special ValueSignExponentMantissa Zero0/100 Infinity0MAX0 -Infinity1MAX0 NaN0/1MAXNon-zero Denormal0/10nonzero S exp Mantissa 1823 Single Precision

25. 25 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Fixed Point (Addition): Hardware

26. 26 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point (Addition): High-level Hardware E0 E1 M0 M1 SWAP Difference Mux Right Shift Add/Sub Priority Encoder Round Left Shift value Left Shift M Greater Than Shift value Denormal? Sub/const E Standard Adder from previous slidet

27. 27 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Floating Point Both Xilinx and Altera supply floating point soft-cores (which I believe are IEEE-754 compliant). So don’t get too afraid if you need floating point in your class projects Also there should be floating point open cores that are freely available.

28. 28 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Fixed Point vs. Floating Point Floating Point advantages: –Application designer does not have to think “much” about the math Floating point format supports a wide range of numbers (+/- 3x10 38 to +/-1x10 -38 ), single precision –If IEEE-754 compliant, then easier to accelerate existing floating point base applications Floating Point disadvantages –Ease of use at great hardware expense 32-bit fix point add (~32 DFF + 32 LUTs) 32-bit single precision floating point add (~250 DFF + 250 LUTs). About 10x more resources, thus 1/10 possible best case parallelism. Floating point typically needs massive pipeline to achieve high clock rates (i.e. high throughput) –No hard-resouces such as carry-chain to take advantage of

29. 29 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Fixed Point vs. Floating Point Range example: (using decimal for clarity) –Assume we can only use 3 digit For fixed point, all 3 digits used for whole part (3.0 format) For floating point, 2 digits used for mantissa, 1 digit for exponent What is the largest number you can represent for each? Precision example: (using decimal for clarity) –For the same format above, represent 125

30. 30 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Mitigating Floating Point Disadvantages Only support a subset of the IEEE-754 standard –Could use software to off-load special cases Modify floating point format to support a smaller data type (e.g. 18-bit instead of 32-bit) –Link to Cornell class: http://instruct1.cit.cornell.edu/courses/ece576/FloatingPoint/index.html Add hardware support in the FPGA for floating point –Hardcore multipliers: Added by companies early 2000’s –Altera: Hard shared paths for floating point (Stratix-V 2011) How to get 1-TFLOP throughput on FPGAs article –http://www.eetimes.com/design/programmable-logic/4207687/How-to-achieve-1- trillion-floating-point-operations-per-second-in-an-FPGAhttp://www.eetimes.com/design/programmable-logic/4207687/How-to-achieve-1- trillion-floating-point-operations-per-second-in-an-FPGA –http://www.altera.com/education/webcasts/all/wc-2011-floating-point-dsp- fpga.htmlhttp://www.altera.com/education/webcasts/all/wc-2011-floating-point-dsp- fpga.html

31. 31 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Mitigating Fixed Point Disadvantages (21.4) Block Floating Point (mitigating range issue) –All data in a block share an exponent –Periodic rescaling –Makes use of Fix-point hardware Useful in application where data is processed in stages, and a limited value range can be placed on each stage (e.g. FFT)

32. 32 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Next Lecture Review Mid-term EDK (XPS) tool overview

33. 33 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Questions/Comments/Concerns Write down –Main point of lecture –One thing that’s still not quite clear –If everything is clear, then give an example of how to apply something from lecture OR

34. 34 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Lecture Notes Altera App Notes on computing FLOPs for Stratix-III or Stratix-IV Altera old app Notes on floating point add/mult Link to floating point single precision calculator Block (fixed) floating point (build slide explanation example) Number comparing CPU/FPGA/GPU floating point throughput Pre-make showing some examples of Fix Point advantage for: -Representing the precision of a number -And precision for add a convoluted type of number 1M.0001

35. 35 - CPRE 583 (Reconfigurable Computing): Floating Point Iowa State University (Ames) Lecture Notes Points: Original 286, 386: Not floating point HW Next: Floating point coprocessor (on a separate chip) Next: Floating point on same chip Why carry ripple used over my advanced “high” performing generate-propagate adders (.1 for 4- LUTs vs.4ns for 1 LUT