1. Essential Mathematics for Games Programmers (Fixed/Float Tutorial)
Lars Bishop

2. Essential Math for Games
Number Spaces
Cardinal – Positive numbers, no fractions
Integer – Pos., neg., zero, no fractions
Rational – Fractions
Irrational – Non-repeating decimals (,e)
Real – Rationals+irrationals
Complex – Real + multiple of -1
a+bi
Essential Math for Games

3. Numerical Representations
In graphics, we often deal with Real numbers ( , 1.5, etc)
Unlike integers, Real numbers have fractional components
There are several common ways of representing these on a computer
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4. Essential Math for Games
Approximating Reals
When we write a Real number on paper, we generally write only a few digits past the decimal point: 1.5, 3.45
Most reals cannot be represented exactly by a few fractional digits
Any written number has inherent precision
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5. Finite Representations
Any number in a computer has finite representation
As a result, any such representation cannot represent every number exactly
When representing numbers, we will always be coping with these limitations
We need to understand and limit error
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6. Essential Math for Games
Fixed Point Numbers
Use integer-like representation
Assume that the least-significant bit is some negative power of 2 (⅛, ¼, etc)
The “binary point” is in the middle of the number, not after the least-significant bit
Bit 7 6 5 4 • 3 2 1
Value 8
1/2
1/4
1/8
1/16
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7. Fixed-Point Nomenclature
Applications can adjust their precision and range by moving the “binary point”
If a fixed point number has
M bits of integral precision
N bits of fractional precision,
It is called an M.N number
This is pronounced “M dot N”
A 32-bit integer would be “32 dot 0”
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8. Essential Math for Games
Fixed Point Benefits
Can represent fractional values with only integer arithmetic
Simple to use and understand
Addition and Subtraction are the same as for integers
Multiplication and Division are only slightly different from their integer siblings
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9. Fixed-point vs. Floating-point
A 32-bit fixed-point actually has more inherent precision than a 32-bit Floating-point number!
Floating-point trades accuracy for range by using some bits for the exponent
“Floating point is for the lazy”
- John von Neumann (paraphrased)
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10. Floating-point ↔ Fixed-point
Assuming an M.N fixed-point system:
IntToFix(i) = i << N
FloatToFix(f) = (int)(f * 2.0N)
Look out for overflow on these!
FixToInt(i) = i >> N
FixToFloat(i) = ((float)i) * 2.0-N
Conversion to int can lose precision
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11. Basic Fixed-Point Math
For A and B, both M.N fixed point values,
A+B = (int)A + (int)B
A-B = (int)A – (int)B
where (int)A is simply the fixed point treated bitwise as an integer
This works because the binary points line up when A and B are both M.N
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12. Multiplication – Basic Idea
Multiplication is a bit more complex, but it is analogous to the grade-school base 10 trick:
We multiply the numbers as integers, and then slide the decimal point to the left by 1+1=2
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13. Essential Math for Games
Multiplication
We want to compute 0.5f x 1.0f = 0.5f.
In 4.4 fixed-point, this is: 1 1 X
After the integer multiply, we get (8x16=128): 1
Then, we need to shift right by 4 (not 8 – we don’t want an integer result, we want a 4.4 result, not 8.0) 1
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14. Challenges with Fixed Point
Range (Overflow)
In the previous multiplication example, if we compute 1.0x1.0=1.0 using the given method, the intermediate value overflows
Precision (Underflow)
If we pre-shift numbers down to avoid overflow, we can end up shifting to 0
Extra shifting required for mul/div
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15. Essential Math for Games
Fixed Point Help
Some CPUs have special instructions
ARM9 (common in handhelds)
Has a 32-bit X 32-bit → 64-bit multiply
Also has similar x 32 → 64
Can avoid overflow or underflow in intermediate values
Allows a free shift in every ALU operation
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16. Essential Math for Games
Fixed Point Summary
Allows fractional math to be done quickly with integer hardware
Requires careful range and precision analysis
Is the only option on many embedded and handheld devices, which don’t have FPU hardware
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17. Floating Point Numbers
Used to represent general non-integers
Often thought of as the set of Reals
This is far from the truth, as you’ll see
Most of this discussion will be about
IEEE bit single-precision
Known in C/C++ as float
Mention of doubles later
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18. FP and Scientific Notation
FP is analogous to scientific notation
Scientific notation is:
± D.DDDD x 10E, where
D.DDDD has a nonzero leading digit
D.DDDD has fixed # of fractional digits
E is a signed integer value
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19. FP/Sci Notation Range/Precision
Precision is not fixed:
Precision of 1.000x10-2 is 100 times more fine-grained than that of 1.000x100
Precision and range are related
The larger the number, the less precise
32-bit int is not a subset of float
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20. FP/Sci Notation Components
Sign
Mantissa
Normalized – has a nonzero integer digit
Limited precision – fixed number of digits
Exponent
Chosen so that the mantissa is normalized
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21. Essential Math for Games
Sign
FP numbers have an explicit sign bit
Float has both 0.0f and -0.0f
By the standard, 0.0f = -0.0f
But the bits are different
Do not memcmp floats!
This is only one of many reasons
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22. Essential Math for Games
Exponent
Like the exponent in scientific notation
But, the exponent’s base is 2, not 10
Stored as a biased number:
ExponentBits = Exponent – Bias
For float, Bias = 127
For float, -125 ≤ Exponent ≤ 128
Exponent term is
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23. Essential Math for Games
Mantissa
Represented as 1-dot-23 fixed point
Store the 23 fractional bits explicitly
Integer bit is implied (“hidden bit”)
Generally, mantissa is normalized
In other words, mantissa is 1.MantissaBits
Similar to the scientific notation standard
In the smallest numbers, the integer bit is assumed to be 0
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24. Binary Representation
Put together, the representation is:
S=Sign bit
E=Exponent bits (8)
M=Fractional mantissa bits (23)
We will write as A = (SA,EA,MA) S
EEEEEEEE
MMMMMMMMMMMMMMMMMMMMMMMM
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25. Essential Math for Games
Special Values
0: S=0, E=All 0s, M=All 0s
-0: S=1, E=All 0s, M=All 0s
+∞: S=0, E=All 1s, M=All 0s
Ex: 1.0f / 0.0f = +∞
−∞: S=1, E=All 1s, M=All 0s
Ex: -1.0f / 0.0f = -∞
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26. Essential Math for Games
Not a Number
Represents undefined results
0.0f / 0.0f = NaN
ACOS(2.0f) = Nan
Two kinds – Quiet and Signaling
Quiet can be passed on to other ops
Signaling traps the code
NaN: E=All 1s, M=Not all 0s
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27. Essential Math for Games
Very Small Numbers
What happens when we run out of smaller and smaller exponents?
Could flush to zero
But, this can lead to the following problem
X-Y=0 does not imply X=Y!
Need to gradually underflow to zero
FP does this by allowing denormals
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28. Essential Math for Games
Denormals
A denormal is an FP number whose hidden mantissa bit is 0, not 1
Indicated by E=0, M=Not all 0s
A denormal is equal to
(-1)S x x 0.MantissaBits
This allows precision to gradually roll off to zero.
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29. Floating-point Add Basics
To add two positive floating point numbers A (EA, MA) and B (EB, MB):
Swap as needed so A has the greater exponent, i.e. EB ≤ EA
Shift MB to the right by EA-EB bits
Add MA+MB and use as the new mantissa
Adjust the new exponent up or down to re-normalize the result.
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30. Essential Math for Games
FP Add Notes
Not a simple process (even for pos #’s)
If A>>B, then B can be shifted to 0
Repeatedly adding small numbers to an accumulator (i.e. A+=B) can gradually lead to huge error
At some point, A stops growing, no matter how many times B is added!
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31. Fun with Floats – The Real World
Can’t discuss every FP issue here: this is an example of why you should care
3D engine saw a spike in basic FP code
Code (SLERP) was +,-,* only
No loops, no complex functions
What could be the problem?
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32. Breaking Down the Problem
Input values looked valid (no NaN)
After a “while”, in a demo, the spike hit
We saved the values, along with timing
In a small app, ran the slow and fast cases in tight loops
The slow cases all had some tiny numbers (~1.0x10-43)
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33. Essential Math for Games
Tiny ≠ 0.0f
Slow cases were denormals
We assumed these numbers to be 0
But FPU was taking care to be accurate
Denormal ops seemed to be slow
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34. Essential Math for Games
Denormal Performance
Did some more timing tests
Even loading a denormal was slow!
Pentium takes a big hit on denormals
True even with exceptions masked!
FPU pipeline gets flushed on denormals
Little things matter
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35. “Don’t Doubles Solve this?”
Doubles do help with most range and many precision problems. But:
Need twice the memory of floats (duh)
Frequently, significantly slower than floats
Some platforms don’t support them
Avoid switching to them without tracking down the problem first
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36. Floating Point Wrap-up
Floats ≠ Reals
Understand the limits of FP
Analyze your FP issues
Don’t just jump to doubles at the first sign of trouble
Be willing to rework your math functions to be FP-friendly
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