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About floating point arithmetic

In general, to represent numbers with fractional parts, computers use a “floating point” binary representation. Floating point arithmetic is also used by AmiBroker for AFL calculations. For some more information about floating point representation in general see the following article, here we will only discuss some practical aspects.

http://en.wikipedia.org/wiki/Floating_point

Floating point calculations are performed in hardware by FPU (Floating Point Unit), which today is a part of your computer’s processor (CPU). The calculations are performed according to IEEE754 standard (see below) that all CPU manufacturers follow.

Internally in computers all numbers are represented in binary system.
This fact has some important consequences in practice. One of it is that some fractions that have finite representation in decimal system are not finite in binary.

For example, 0.1 is endless (infinite) fraction in binary system, as 1/3 or 2/3 fractions are in decimal system.

The mantissa of 0.1 fraction in binary system is cyclical:
1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 (1001)* …….

* represents endless cycle.

Since computers operate on limited word length, 32 bit binary representation of 0.1 is:
01111011 10011001100110011001100 (not rounded) (in decimal it is 0.09999999403953552)
or
01111011 10011001100110011001101 (rounded) (in decimal it is 0.10000000149011612)

Later is used by FPU (floating point processor) in your CPU because it has smaller relative error. If you add it nine times you will end up with 0.9000000134110449 which is of course higher than 0.9.

It will be easier for you to understand when explaining on decimal numbers. For example 2/3 represented in decimal is:
0.666666667

Now add THREE times this number (0.666666667 + 0.666666667 + 0.666666667) and what you will get?
2.000000001 and that is greater than 2.

Therefore, it is rule in programming, NEVER use fractions for loop counters. For that reason you should not use fractions in Optimize(), or if you need to use them use them in WISE way by adding HALF of “step” value to the “max” value.

step 0.1;
Optimize("x"0.50.10.9 step/2step);

Another thing to keep in mind is that 32-bit floating point number has only 7 significant digits (those digits that carry meaning contributing to its precision). So in 123.4567 all digits are significant and accurate, but in 123.456789, last two digits (‘8’ and ‘9’) are not significant and subject to floating point rounding – see links below).

See also:
About significant figures:
http://en.wikipedia.org/wiki/Significant_figures

IEEE754 conversion calculators:
http://babbage.cs.qc.edu/IEEE-754/Decimal.html
http://babbage.cs.qc.edu/IEEE-754/32bit.html

IEEE754 standard description:
http://en.wikipedia.org/wiki/IEEE_754-1985

Essay about comparing floatin point numbers:
http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm

Microsoft Knowledge Base: “Precision and Accuracy in Floating-Point Calculations”
http://support.microsoft.com/kb/125056

Using 32-bit floating point (as opposed to 64-bit double precision) has two main advantages:
a) consumes HALF of memory required for doubles (this *is* important, more important that you think, because if you have for example an array of 500000 elements, in floats it is 2MB and it fits into CPU cache, while in doubles it would be 4MB and may not fit into CPU cache).
b) 32-bit floating point numbers can be computed much faster. It simply takes less processor cycles to compute 32-bit float than 64-bit float. 64-bit version of AmiBroker is going even further and is using SSE2 instructions. This means that vectors of 4 single-precision numbers are processed in parallel using single SSE2 instruction on SINGLE processor. This gives more speed on single core than achievable using multiple cores.

Note also AmiBroker does use double precision (64bit/SSE2) or extended precision (80bit/x87) where it is necessary (for certain internal calculations).

Note also that due to the architectural differences between compilers for 32-bit and 64-bit programs,
small numerical differences may exist between 32-bit and 64-bit version of AmiBroker due to the fact that 32-bit version uses x87 FPU code while 64-bit version uses SSE2 code and the underlying floating point hardware is different and operate slightly differently.
x87 FPU code internally offers more (80bit) precision than SSE2 (64-bit).
For in-depth discussion see:
https://randomascii.wordpress.com/2012/03/21/intermediate-floating-point-precision/

AFL execution speed

NOTE: Benchmarks and timings given below are little outdated because article was written back in 2008. Today’s CPUs are faster and memory bandwidth is also higher.

AmiBroker Formula Language (AFL) thanks to its array processing model is able to run at the same speed as code written in assembler (i.e. machine code). The following article explains how.

AFL runs with native assembly speed when using array operations.
A simple array multiplication like this:

Close  H// array multiplication (each array element is multiplied)

gets compiled by AmiBroker to just 8 assembly instructions:

loop:
mov edx,dword ptr [esp+58h]
inc esi ; increase counters
add eax,4
cmp esi,edi
fld dword ptr [edx+esi*4-4] ; get element of close array
fmul dword ptr [eax+ecx-4] ; multiply by element of high array
fstp dword ptr [eax-4] ; store result
jl loop ; continue until all elements are processed

As you can see there are three 4 byte memory accesses per loop iteration (2 reads each 4 bytes long and 1 write 4 byte long)

With such tight loop, single processor core running an AFL formula is able to saturate memory bandwidth in majority of most common operations/functions if total array sizes used in given formula exceedes DATA cache size.

On my (2 year old) 2GHz Athlon x2 64 single iteration of this loop takes 6 nanoseconds (see benchmark code below). 6 nanoseconds to process one bar of data, or 166 million bars per second. So, during 6 nanoseconds we have 8 byte reads and 4 byte store. Thats (8/(6e-9)) bytes per second = 1333 MB per second read and 667 MB per second write simultaneously i.e. 2GB/sec combined !

Now if you look at memory benchmarks you will see that 2GB/s is the limit of system memory speed on Athlon x64 (DDR2 dual channel)
And that’s considering the fact that Athlon has superior-to-intel on-die integrated memory controller (hypertransfer)

// benchmark code
// for accurrate results run it on LARGE arrays -
// intraday database, 1-minute interval, 50K bars or more)
GetPerformanceCounter(1); 
for(
01000k++ )  H
"Time per single iteration [s]="+1e-3*GetPerformanceCounter()/(1000*BarCount); 

Only really complex operations that use *lots* of FPU (floating point) cycles such as trigonometric (sin/cos/tan) functions are slow enough for the memory to keep up.

QuickAFL facts

QuickAFL(tm) is a feature that allows faster AFL calculation under certain conditions. Initially (since 2003) it was available for indicators only, as of version 5.14+ it is available in Automatic Analysis too.

Initially the idea was to allow faster chart redraws through calculating AFL formula only for that part which is visible on the chart. In a similar manner, automatic analysis window can use subset of available quotations to calculate AFL, if selected “range” parameter is less than “All quotations”.

So, in short QuickAFL works so it calculates only part of the array that is currently visible (indicator) or within selected range (Automatic Analysis).

Your formulas, under QuickAFL, may or may NOT use all data bars available, but only visible (or “in-range”) bars (plus some extra to ensure calculation of used indicators), so when you are using Close[ 0 ] it represents not first bar in entire data set but first bar in array currently used (which is just a bit longer than visible, or ‘in-range’ area).

The QuickAFL is designed to be transparent, i.e. do not require any changes to the formulas you are using. To achieve this goal, AmiBroker in the first execution of given formula “learns” how many bars are really needed to calculate it correctly.

To find out the number of bars required to calculate formula AmiBroker internally uses two variables ‘backward ref’ and ‘forward ref’.

‘backward ref’ describes how many PREVIOUS bars are needed to calculate the value for today, and ‘forward ref’ tells how many FUTURE bars are needed to calculate value for today.

If these numbers are known, during execution of given formula AmiBroker takes FIRST visible (or in-range) bar and subtracts ‘backward ref” and takes LAST visible (or in-range) bar and adds ‘forward ref’ to calculate first and last bar needed for calculation of the formula.

Now, how does AmiBroker know a correct “backward ref” and “forward ref” for the entire formula?
Well, every AmiBroker’s built-in function is written so that it knows its own requirements and adds them to global “backward ref” and “forward ref” each time given function is called from your formula.

The whole process starts with setting initial BackwardRef to 30 and ForwardRef to zero. These initial values are used to give “safety margin” for simple loops/scripts.

Next, when parser scans the formula like this:

Buy Ref MAC40 ), -);

it analyses it and “sees” the MA with parameter 40. It knows that simple moving average of period 40 requires 40 past bars and zero future bars to calculate correctly so it does the following (all internally):

BackwardRef BackwardRef 40;
ForwardRef ForwardRef 0;

So now, the value of BackwardRef will be 70 (40+30(initial)), and ForwardRef will be zero.

Next the parser sees Ref( .., -1 );

It knows that Ref with shift of -1 requires 1 past bar and zero future bars so it “accumulates” requirements this way:

BackwardRef BackwardRef 1;
ForwardRef ForwardRef 0;

So it ends up with:

BackwardRef 71;
ForwardRef 0;

The BackwardRef and ForwardRef numbers are displayed by AFL Editor’s Tools->Check and Profile as well as on charts when “Display chart timing” is selected in the preferences.

If you use Check and Profile tool, it will tell you that the formula

Buy Ref MAC40 ), -);

requires 71 past bars and 0 future bars.

You can modify it to

Buy Ref MAC50 ), -);

and it will tell you that it requires 82 past bars (30+50+2) and zero future bars.

If you modify it to

Buy Ref MAC50 ), );

It will tell you that it needs 80 past bars (30+50) and ONE future bar (from Ref).

Thanks to that your formula will use 80 bars prior to first visible (or in-range) bar leading to correct calculation result, while improving the speed of execution by not using bars preceding required ones.

IMPORTANT NOTES

It is very important to understand, that the above estimate requirements while fairly conservative,
and working fine in majority of cases, may NOT give you identical results with QuickAFL enabled, if your formulas use:
a) JScript/VBScript scripting
b) for/while/do loops using more than 30 past bars
c) any functions from external indicator DLLs
d) certain functions that use recursive calculation such as very long exponential averages or TimeFrame functions with much higher intervals than base interval

In these cases, you may need to use SetBarsRequired() function to set initial requirements to value higher than default 30. For example, by placing

SetBarsRequired1000);

at the TOP of your formula you are telling AmiBroker to add 1000 bars PRIOR to first visible (or in-range) bar to ensure more data to stabilise indicators.

You can also effectively turn OFF QuickAFL by adding:

SetBarsRequiredsbrAllsbrAll );

at the top of your formula. It tells AmiBroker to use ALL bars all the time.

It is also worth noting that certain functions like cumulative sum (Cum()) by default request ALL past bars to guarantee the same results when QuickAFL is enabled. But when using such a function, you may or may NOT want to use all bars. So SetBarsRequired() gives you also ability to DECREASE the requirements of formula. This is done by placing SetBarsRequired at the END of your formula, as any call to SetBarsRequired effectively overwrites previously calculated estimate. So
if you write

Cum);
SetBarsRequired1000); // use 1000 past bars DESPITE using Cum()

You may force AmiBroker to use only 1000 bars prior first visible even though Cum() by itself would require all bars.

It is also worth noting that when QuickAFL is used, BarIndex() function does NOT represent elements of the AFL array, but rather the indexes of ENTIRE quotation array. With QuickAFL turned on, an AFL array is usually shorter than quotation array, as illustrated in this picture:

QuickAFL, BarIndex and BarCount

SPECIAL CASE: AddToComposite function

Since AddToComposite creates artificial stock data it is desirable that it works the same regardless of how many ‘visible’ bars there are or how many bars are needed by other parts of the formula.

For this reason internally AddToComposite does this:

SetBarsRequiredsbrAllsbrAll );

which effectivelly means “use all available bars” for the formula. AddToComposite function simply tells the AFL engine to use all available bars (from the very first to the very last) regardless of how formula looks like. This is to ensure that AddToComposite updates ALL bars of the composite

The side-effect is that “Check And Profile” feature will see that it needs to reference future bars and display a warning even though this is false alert because AddToComposite itself has no impact on trading system at all.

Now why this shows only when flag atcFlagEnableInBacktest is on ??
It is simple: this is so because it means that AddToComposite is ACTIVE in BACKTEST.
http://www.amibroker.com/guide/afl/afl_view.php?name=ADDTOCOMPOSITE

Since “Check And Profile” uses “BACKTEST” state you get such result.

If atcFlagEnableInBacktest is not specified AddToComposite is not enabled in Backtest and hence does not affect calculation of BackwardRef and ForwardRef during “Check And Profile”.

BACKWARD COMPATIBILITY NOTES

a) QuickAFL is available in Automatic Analysis in version 5.14.0 or higher
b) sbrAll constant is available in Automatic Analysis in version 5.14.0 or higher. If you are using older versions you should use numeric constant of: 1000000 instead.

New keywords in AFL and possible conflict with user-defined variables

AmiBroker 4.91.0 BETA introduced the following new keywords:

switch, case, break, continue, default

You have to make sure that your formulas do not use them as variable names. The above words are now reserved AFL keywords and if you use them for your own variables you need to replace this identifiers with names that do not conflict with the reserved keywords.

This article shows how to perform multiple-file text replace very quickly. (more…)

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