Using ISO Fixed Point Arithmetic for Neural Modelling Theory and - - PowerPoint PPT Presentation

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Using ISO Fixed Point Arithmetic for Neural Modelling Theory and - - PowerPoint PPT Presentation

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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

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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 University) ISO Fixed Point December 2013 1 / 19

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SLIDE 2

Overview

Introduction to the ISO/IEC draft standard.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 2 / 19

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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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

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SLIDE 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

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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

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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

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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

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Fixed Point Data Type Minimal Formats

We express the format of a fixed point number as

  • ptional sign bitinteger part.fractional part

David Lester∗ (Manchester University) ISO Fixed Point December 2013 5 / 19

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Fixed Point Data Type Minimal Formats

We express the format of a fixed point number as

  • ptional sign bitinteger 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

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SLIDE 14

Fixed Point Data Type Minimal Formats

We express the format of a fixed point number as

  • ptional sign bitinteger 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

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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

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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

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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

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SLIDE 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

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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

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SLIDE 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

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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

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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

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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

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SLIDE 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

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Arithmetic Operations (III)

Consider the following program: fract x = ...; int y = (int)x;

David Lester∗ (Manchester University) ISO Fixed Point December 2013 9 / 19

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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

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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

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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

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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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Comparisons (I)

Disassembly of section .text: 00000000 <test1>: 0: e92d4038 push {r3, r4, r5, lr} 4: e1a04000 mov r4, r0 8: e1a05001 mov r5, r1 c: e1a00001 mov r0, r1 10: e1a01004 mov r1, r4 14: ebfffffe bl 0 <__gnu_cmpsa2> 18: e3500001 cmp r0, #1 1c: c1a00004 movgt r0, r4 20: d1a00005 movle r0, r5 24: e8bd4038 pop {r3, r4, r5, lr} 28: e12fff1e bx lr

David Lester∗ (Manchester University) ISO Fixed Point December 2013 11 / 19

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Comparisons (II)

As before, but using bitsk for comparison: #include <stdfix.h> accum test2 (accum x, accum y) { return ((bitsk(x) < bitsk(y))? x: y); }

David Lester∗ (Manchester University) ISO Fixed Point December 2013 12 / 19

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Comparisons (II)

Now we get: 00000000 <test2>: 0: e92d4070 push {r4, r5, r6, lr} 4: e1a04001 mov r4, r1 8: e1a05000 mov r5, r0 c: ebfffffe bl 0 <bitsk> 10: e1a06000 mov r6, r0 14: e1a00004 mov r0, r4 18: ebfffffe bl 0 <bitsk> 1c: e1560000 cmp r6, r0 20: a1a00004 movge r0, r4 24: b1a00005 movlt r0, r5 28: e8bd4070 pop {r4, r5, r6, lr} 2c: e12fff1e bx lr

David Lester∗ (Manchester University) ISO Fixed Point December 2013 13 / 19

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Comparisons (III)

Now including my extra header file: #include <stdfix.h> #include <stdfix-full-iso.h> accum test3 (accum x, accum y) { return ((bitsk(x) < bitsk(y))? x: y); }

David Lester∗ (Manchester University) ISO Fixed Point December 2013 14 / 19

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Comparisons (III)

Now including my extra header file: #include <stdfix.h> #include <stdfix-full-iso.h> accum test3 (accum x, accum y) { return ((bitsk(x) < bitsk(y))? x: y); } Now we get: 00000000 <test3>: 0: e1510000 cmp r1, r0 4: b1a00001 movlt r0, r1 8: a1a00000 movge r0, r0 c: e12fff1e bx lr

David Lester∗ (Manchester University) ISO Fixed Point December 2013 14 / 19

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Getting a compiler with fixed point support

Not available with armcc.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 15 / 19

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Getting a compiler with fixed point support

Not available with armcc. Can be compiled into gcc.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 15 / 19

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Getting a compiler with fixed point support

Not available with armcc. Can be compiled into gcc. gcc-4.7.5 available from Code Sourcery (Mentor Graphics).

David Lester∗ (Manchester University) ISO Fixed Point December 2013 15 / 19

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Getting a compiler with fixed point support

Not available with armcc. Can be compiled into gcc. gcc-4.7.5 available from Code Sourcery (Mentor Graphics). Or roll your own:

David Lester∗ (Manchester University) ISO Fixed Point December 2013 15 / 19

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Getting a compiler with fixed point support

Not available with armcc. Can be compiled into gcc. gcc-4.7.5 available from Code Sourcery (Mentor Graphics). Or roll your own: Note: this will not work unless you are compiling for “bare-metal”. Don’t ask why.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 15 / 19

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Getting a compiler with fixed point support (II)

Target: arm-none-eabi Configured with: ../gcc-4.8.2/configure

  • -prefix=/home/dave/Project/x-tools
  • -target=arm-none-eabi --disable-shared --disable-nls
  • -disable-threads --disable-libssp --with-gcc
  • -disable-libstdcxx-pch
  • -disable-libmudflap
  • -disable-libgomp --disable-libquadmath --with-gnu-ld
  • -disable-libstdc++-v3 -v --enable-languages=c
  • -with-arch=armv5te --with-cpu=arm968e-s
  • -with-mode=arm --disable-bootstrap
  • -enable-interwork --disable-multilib --with-gnu-as
  • -enable-fixed-point
  • -disable-decimal-float
  • -without-long-double-128 --with-dwarf2
  • -with-pkgversion=’SpiNNaker ArmBall V0.2 (12th December 2013)’

Thread model: single gcc version 4.8.2 (GCC 4.8.2 for arm-ball)

David Lester∗ (Manchester University) ISO Fixed Point December 2013 16 / 19

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Next steps

Provide fixed point support for non-bare metal platforms.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 17 / 19

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Next steps

Provide fixed point support for non-bare metal platforms. Improve code quality emitted by gcc for fixed point arithmetic.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 17 / 19

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Next steps

Provide fixed point support for non-bare metal platforms. Improve code quality emitted by gcc for fixed point arithmetic. Provide transcendental support.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 17 / 19

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Next steps: Progress

Platforms Not yet attempted.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 18 / 19

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Next steps: Progress

Platforms Not yet attempted. Code Quality As has been seen, an improved library has already been started.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 18 / 19

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Next steps: Progress

Platforms Not yet attempted. Code Quality As has been seen, an improved library has already been started. Transcendentals Kahan’s reciprt has been implemented.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 18 / 19

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Why are we still interested in Fix Point?

Energy efficiency. Two to six times more energy efficient than floating point hardware.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 19 / 19

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Why are we still interested in Fix Point?

Energy efficiency. Two to six times more energy efficient than floating point hardware. If weights in synaptic matrices are held as 16-bit integer quantities, we can use ARM Neon SIMD units as 8 × 16 lanes.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 19 / 19

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Why are we still interested in Fix Point?

Energy efficiency. Two to six times more energy efficient than floating point hardware. If weights in synaptic matrices are held as 16-bit integer quantities, we can use ARM Neon SIMD units as 8 × 16 lanes. But for neural processing we will provide floating point hardware. Neon permits us vectorize with four lanes.

David Lester∗ (Manchester University) ISO Fixed Point December 2013 19 / 19