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Using ISO Fixed Point Arithmetic for Neural Modelling Theory and Practice David Lester Manchester University December 2013 Research supported by EU-FET/EPSRC (BrainScales, Human Brain Project, BIMPC, BIMPA) David Lester (Manchester

  1. Using ISO Fixed Point Arithmetic for Neural Modelling Theory and Practice David Lester ∗ Manchester University December 2013 Research supported by EU-FET/EPSRC (BrainScales, Human Brain Project, BIMPC, BIMPA) ∗ David Lester ∗ (Manchester University) ISO Fixed Point December 2013 1 / 19

  2. Overview Introduction to the ISO/IEC draft standard. David Lester ∗ (Manchester University) ISO Fixed Point December 2013 2 / 19

  3. Overview Introduction to the ISO/IEC draft standard. How to write ‘C’ programs using fixed point arithmetic. David Lester ∗ (Manchester University) ISO Fixed Point December 2013 2 / 19

  4. Overview Introduction to the ISO/IEC draft standard. How to write ‘C’ programs using fixed point arithmetic. Further Improvements. David Lester ∗ (Manchester University) ISO Fixed Point December 2013 2 / 19

  5. Overview Introduction to the ISO/IEC draft standard. How to write ‘C’ programs using fixed point arithmetic. Further Improvements. Future Plans. David Lester ∗ (Manchester University) ISO Fixed Point December 2013 2 / 19

  6. ISO Fixed Point Arithmetic By ISO Fixed Point Arithmetic we mean document: ISO/IEC TR 18037 dated April 4, 2006, and produced under the auspices of ISO/IEC/ANSI. David Lester ∗ (Manchester University) ISO Fixed Point December 2013 3 / 19

  7. ISO Fixed Point Arithmetic By ISO Fixed Point Arithmetic we mean document: ISO/IEC TR 18037 dated April 4, 2006, and produced under the auspices of ISO/IEC/ANSI. Formal Title: “Programming languages - C - Extensions to support embedded processors”, ISO/IEC JTC1 SC22 WG14 N1169 David Lester ∗ (Manchester University) ISO Fixed Point December 2013 3 / 19

  8. ISO Fixed Point Arithmetic By ISO Fixed Point Arithmetic we mean document: ISO/IEC TR 18037 dated April 4, 2006, and produced under the auspices of ISO/IEC/ANSI. Formal Title: “Programming languages - C - Extensions to support embedded processors”, ISO/IEC JTC1 SC22 WG14 N1169 Important : “This document is an ISO/IEC draft Technical Report. It is not an ISO/IEC International Technical Report.” David Lester ∗ (Manchester University) ISO Fixed Point December 2013 3 / 19

  9. Fixed Point Data Types There are twelve primary fixed-point types . Six fractional types: unsigned short fract signed short fract unsigned fract signed fract unsigned long fract signed long fract David Lester ∗ (Manchester University) ISO Fixed Point December 2013 4 / 19

  10. Fixed Point Data Types There are twelve primary fixed-point types . Six fractional types: unsigned short fract signed short fract unsigned fract signed fract unsigned long fract signed long fract And six accum types: unsigned short accum signed short accum unsigned accum signed accum unsigned long accum signed long accum David Lester ∗ (Manchester University) ISO Fixed Point December 2013 4 / 19

  11. Fixed Point Data Types There are twelve primary fixed-point types . Six fractional types: unsigned short fract signed short fract unsigned fract signed fract unsigned long fract signed long fract And six accum types: unsigned short accum signed short accum unsigned accum signed accum unsigned long accum signed long accum For each of the primary types there is a corresponding (but different) saturating fixed-point type , e.g. unsigned long sat accum David Lester ∗ (Manchester University) ISO Fixed Point December 2013 4 / 19

  12. Fixed Point Data Type Minimal Formats We express the format of a fixed point number as � optional sign bit �� integer part � . � fractional part � David Lester ∗ (Manchester University) ISO Fixed Point December 2013 5 / 19

  13. Fixed Point Data Type Minimal Formats We express the format of a fixed point number as � optional sign bit �� integer part � . � fractional part � The minimum sizes for the fract types: unsigned short fract .7 signed short fract s.7 unsigned fract .15 signed fract s.15 unsigned long fract .23 signed long fract s.23 David Lester ∗ (Manchester University) ISO Fixed Point December 2013 5 / 19

  14. Fixed Point Data Type Minimal Formats We express the format of a fixed point number as � optional sign bit �� integer part � . � fractional part � The minimum sizes for the fract types: unsigned short fract .7 signed short fract s.7 unsigned fract .15 signed fract s.15 unsigned long fract .23 signed long fract s.23 The minimum sizes for the accum types: unsigned short accum 4.7 signed short accum s4.7 unsigned accum 4.15 signed accum s4.15 unsigned long accum 4.23 signed long accum s4.23 David Lester ∗ (Manchester University) ISO Fixed Point December 2013 5 / 19

  15. Fixed Point Data Type gcc Formats The formats chosen by gcc for the fract types on ARM : unsigned short fract .8 signed short fract s.7 unsigned fract .16 signed fract s.15 unsigned long fract .32 signed long fract s.32 David Lester ∗ (Manchester University) ISO Fixed Point December 2013 6 / 19

  16. Fixed Point Data Type gcc Formats The formats chosen by gcc for the fract types on ARM : unsigned short fract .8 signed short fract s.7 unsigned fract .16 signed fract s.15 unsigned long fract .32 signed long fract s.32 The formats chosen by gcc for the accum types on ARM : unsigned short accum 8.8 signed short accum s8.7 unsigned accum 16.16 signed accum s16.15 unsigned long accum 32.32 signed long accum s32.31 David Lester ∗ (Manchester University) ISO Fixed Point December 2013 6 / 19

  17. Fixed Point Data Type gcc Formats The formats chosen by gcc for the fract types on ARM : unsigned short fract .8 signed short fract s.7 unsigned fract .16 signed fract s.15 unsigned long fract .32 signed long fract s.32 The formats chosen by gcc for the accum types on ARM : unsigned short accum 8.8 signed short accum s8.7 unsigned accum 16.16 signed accum s16.15 unsigned long accum 32.32 signed long accum s32.31 There are two further non-ISO/IEC fract types: unsigned long long fract .64 signed long long fract s.63 David Lester ∗ (Manchester University) ISO Fixed Point December 2013 6 / 19

  18. Arithmetic Operations (I) Consider the following program: { fract x = 0.5; fract y; y = x + x; printf ("%k\n", y); } David Lester ∗ (Manchester University) ISO Fixed Point December 2013 7 / 19

  19. Arithmetic Operations (I) Consider the following program: { fract x = 0.5; fract y; y = x + x; printf ("%k\n", y); } What value is printed? David Lester ∗ (Manchester University) ISO Fixed Point December 2013 7 / 19

  20. Arithmetic Operations (I) Consider the following program: { fract x = 0.5; fract y; y = x + x; printf ("%k\n", y); } What value is printed? Note constants and printing format control. David Lester ∗ (Manchester University) ISO Fixed Point December 2013 7 / 19

  21. Arithmetic Operations (I) Consider the following program: { fract x = 0.5; fract y; y = x + x; printf ("%k\n", y); } What value is printed? Note constants and printing format control. 0.999984 . David Lester ∗ (Manchester University) ISO Fixed Point December 2013 7 / 19

  22. Arithmetic Operations (II) Consider the following program: { fract x = -1.0; fract y; y = x * x; printf ("%k\n", y); } David Lester ∗ (Manchester University) ISO Fixed Point December 2013 8 / 19

  23. Arithmetic Operations (II) Consider the following program: { fract x = -1.0; fract y; y = x * x; printf ("%k\n", y); } What value is printed? David Lester ∗ (Manchester University) ISO Fixed Point December 2013 8 / 19

  24. Arithmetic Operations (II) Consider the following program: { fract x = -1.0; fract y; y = x * x; printf ("%k\n", y); } What value is printed? 0.999984 (or maybe 1.000000 ). David Lester ∗ (Manchester University) ISO Fixed Point December 2013 8 / 19

  25. Arithmetic Operations (III) Consider the following program: fract x = ...; int y = (int)x; David Lester ∗ (Manchester University) ISO Fixed Point December 2013 9 / 19

  26. Arithmetic Operations (III) Consider the following program: fract x = ...; int y = (int)x; What value is the value of y ? David Lester ∗ (Manchester University) ISO Fixed Point December 2013 9 / 19

  27. Arithmetic Operations (III) Consider the following program: fract x = ...; int y = (int)x; What value is the value of y ? Probably what was intended was: fract x = ...; int y = bitsr (x); fract z = rbits (y); David Lester ∗ (Manchester University) ISO Fixed Point December 2013 9 / 19

  28. Comparisons (I) Consider the following program: #include <stdfix.h> accum test1 (accum x, accum y) { return ((x < y)? x: y); } David Lester ∗ (Manchester University) ISO Fixed Point December 2013 10 / 19

  29. Comparisons (I) Consider the following program: #include <stdfix.h> accum test1 (accum x, accum y) { return ((x < y)? x: y); } What arm code will be generated? David Lester ∗ (Manchester University) ISO Fixed Point December 2013 10 / 19

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