Coping With Fixed Point-Bry

7/28/2019 Coping With Fixed Point-Bry

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Copyright Khronos Group, 2004 - Page 1

Coping with Fixed PointCoping with Fixed Point

Mik BRY CEO

[email protected]


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Overviewverview Fixed Point theory and history

Floating Point to Fixed Point

Maintain accuracy avoid overflows and optimization tricks

Effective use of Fixed Point in OpenGL ES


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Fixed Point theory and historyixed Point theory and history Widely used in software 3D

Fixed Form

Fast and simple to implement


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Widely used in software 3Didely used in software 3DFixed Point number is represented by a real number in an integer formatwith an imaginary radix point separating the integer and fractional part.

Prior to HW 3D, fixed point was widely used since the beginning of realtime3D.

But in small handheld devices, we need to go back to a constrained worldwhere we have to take care of limited memory, tiny screen. And also oncertain targets avoids floating computings.


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Fixed Form used inixed Form used in OpenGL ESpenGL ES Signed format : s15.16 used in

OpenGL ES

GLFixed = number*2^16

GLFixed = number


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Fast and simple to implementast and simple to implementBasic math operations

Adding/Sub : same as int opts

FNUM a = INT2FNUM(1);FNUM b = INT2FNUM(2);

a += b;

Multiply/div (avoid it see later)

#define FMUL(x,y) (x*y) >>16

#define FDIV(x,y)(x/y)>>16

Comparisons same as int excepting zero compare


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Floating Point to Fixed Pointloating Point to Fixed Point Floats format : standardized

Maths errors support

Convert Fixed and Float numbers


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So :

Float = mantissa*2^(exponent-127)

Fixed = number*2^16


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Representation errors supportepresentation errors support Floats IEEE Standard has a full range of exception handling

Floats support for NaN and infinite

Overflow and Underflow are checked


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Convert Fixed and float numbersonvert Fixed and float numbersIn preprocessing and constants vars it is transparent

#define FNUM_PI (int)(3.14f*65536) // 2^16=65536

Simple implementation, not fast at all:

Float to Fixed:

fixedNum = (int)(floatNum*65536

Fixed to Float:

floatNum = ((float)fixedNum)/65536

Without Floating support at all in C compiler it is a little bit more tricky


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Effective use of Fixed Point inffective use of Fixed Point in OpenGL ESpenGL ES Why Fixed Points in OpenGL ES CL?

ARM chipset : no floating point support

Easy to implements in embedded systems

But not as perfect as Floating math so 2 OpenGL ES profi les


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ARM architectureRM architectureDesigned for low power consumption so :

No floating point support in mainstream ARM7 ARM9 cores

No divide support in ARM7

Small cache memory L1 and no L2


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Easy to implementsasy to implements So

Fixed Points fi t perfectly to ARM cores

No needs for DSP or FPU

Caution of Divide support

And cache memory


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OpenGL ESpenGL ES support for Fixed Mathupport for Fixed MathCommon Lite Profile is a strict f ixed point implementation

GLFixed : a 32 bits integerClampx :

GL Commands mapping using fixed math: x instead of f

glClearColorx(0, 0, 0, 0);

OES_Fixed_Point extension


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Maintain accuracy and overflows andaintain accuracy and overflows andoptimization tricksptimization tricksAccuracy and range

Overflow underflow

Avoiding pitfalls

Optimization tricks


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Accuracy and Rangeccuracy and RangeA fixed point number has a limited integer range. It is not possible to

represent very large and very small numbers.

integer range in GLFixed : -32768 < integer part < 32768 (2^15)

Fractionnal accuracy : 0,0000152587890625 // 1/(2^16)

A fixed point number has limited accuracy.

You must choose small numbers and inputs

a normalize data as possible.


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Overflow underflowverflow underflowOverflow

- An "overflow" wi ll occur when the result of a arithmetic operation is too large to f it intothe fixed representation of a fixed point number

a = 0x7FFF


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Optimization tricksptimization tricks Trigonometric operations

Using LookupTable best with 1024 bytes size first quadrant and s8:24 foravoiding underflow.

Square operationsUsing logarithm functions LUT are too big for small cache

Use other fixed formats range s24:8 for larger numbers and convert to

GLFixed:

Fixed24_8Num = glFixedNum>>8 // You lost some precision but higherinteger range

If (Fixed24_8Num&0x7F000000 != 0) error // to big number

Else glFixedNum = Fixed24_8Num


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Avoiding pitfallsvoiding pitfallsOverflow

Integer Range

Also divide errors (try to not use it)


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Fixed Point inixed Point in OpenGL ESpenGL ESIn all Profi le

Easy to implements in using same methods as floating ones and usingGLFixed

Take care of pitfalls and use small numbers

Perfect for limited devices with small memory and screen and a simpleARM processor.


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Questions?uestions?

Mik BRY [email protected]

Coping With Fixed Point-Bry

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