Every now and then, you’ll see issues that can be described as “choppy” animations, or perhaps your code doesn’t work when comparing various variables.

Sometimes, it could be related to something called “floating point precision”.

It stems from the fact that on a computer, real numbers have to be approximated in some way.

And as you might imagine, the approximation result might not always be what you intended. (See table at the end)

## TL;DR:

Floating point values…

- … does not have infinite range
- … does not have infinite precision
- … exist in most modern engines.
- … have better precision when closer to 0 than FLT_MAX (3.40282e+38)
- … should (most often) not be compared using the
**==**or**!=**operators - … can be tricky to use, even with operators
**<**and**>**due to imprecision

## In (only slightly) more detail:

You use floating point representations in code (Lua, C++, shaders etc) and data (textures, heightmaps etc). That’s why it’s very good to know about the general workings of them, in order to anticipate any issues that might arise.

What we’re using in our C++ engine are 32 bit floating point numbers. Lua itself uses 64bit floating point numbers.

In OpenGL ES, you can set lowp / mediump / highp precision on your variables.

The OpenGL ES standard sets a minumum requirement of the precision so it might still differ between graphics drivers.

## No infinite precision

If you think about it, a 32 bit integer can represent the values -2147483647 to 2147483647, that’s it. Then all 32 bits are used. But, if you would also like decimals, how could you fit all those numbers, as well as decimals, in only 32 bits?

The answer is, of course, you cannot. You have to make sacrifices.

In the single floating point representation (usually refers to the standard IEEE 754) case, this means that we sacrifice precision the larger the number is.

Since the precision varies depending how large your number is, the operators might not work as you’d expect.

## Accumulation

Another thing to consider, is how the precision “error” of calculations accumulate.

It’s very easy to calculate something that has less precision than you’d like, and use that value in your next calculation, in yet another manner that might introduce even more precision errors.

## How to deal with it?

A common thing is to keep your data and computations closer to 0 (or Origin) than in world space.

- E.g. Moving camera and objects towards Origin (0,0,0). Used for open world games, infinite runners etc.

Another way is to increase the number of bits to represent the value

- E.g 64 bit precision on CPU, or
*highp*in shaders

Each way will come with a drawback, you have to decide which one is the best for you.

## Addendum

To give you an idea of how much the precision changes, the larger your numbers get:

Table contains: integer, the next 32 bit floating point value after the integer, the difference between the two previous values

```
f: 0.00 next: 1.4012984643248170709e-45 diff: 1.4012984643248170709237e-45
f: 1.00 next: 1.0000001192092895508 diff: 1.1920928955078125e-07
f: 2.00 next: 2.0000002384185791016 diff: 2.384185791015625e-07
f: 3.00 next: 3.0000002384185791016 diff: 2.384185791015625e-07
f: 4.00 next: 4.0000004768371582031 diff: 4.76837158203125e-07
f: 5.00 next: 5.0000004768371582031 diff: 4.76837158203125e-07
f: 6.00 next: 6.0000004768371582031 diff: 4.76837158203125e-07
f: 7.00 next: 7.0000004768371582031 diff: 4.76837158203125e-07
f: 8.00 next: 8.0000009536743164062 diff: 9.5367431640625e-07
f: 9.00 next: 9.0000009536743164062 diff: 9.5367431640625e-07
f: 10.00 next: 10.000000953674316406 diff: 9.5367431640625e-07
f: 20.00 next: 20.000001907348632812 diff: 1.9073486328125e-06
f: 30.00 next: 30.000001907348632812 diff: 1.9073486328125e-06
f: 40.00 next: 40.000003814697265625 diff: 3.814697265625e-06
f: 50.00 next: 50.000003814697265625 diff: 3.814697265625e-06
f: 60.00 next: 60.000003814697265625 diff: 3.814697265625e-06
f: 70.00 next: 70.00000762939453125 diff: 7.62939453125e-06
f: 80.00 next: 80.00000762939453125 diff: 7.62939453125e-06
f: 90.00 next: 90.00000762939453125 diff: 7.62939453125e-06
f: 100.00 next: 100.00000762939453125 diff: 7.62939453125e-06
f: 200.00 next: 200.0000152587890625 diff: 1.52587890625e-05
f: 300.00 next: 300.000030517578125 diff: 3.0517578125e-05
f: 400.00 next: 400.000030517578125 diff: 3.0517578125e-05
f: 500.00 next: 500.000030517578125 diff: 3.0517578125e-05
f: 600.00 next: 600.00006103515625 diff: 6.103515625e-05
f: 700.00 next: 700.00006103515625 diff: 6.103515625e-05
f: 800.00 next: 800.00006103515625 diff: 6.103515625e-05
f: 900.00 next: 900.00006103515625 diff: 6.103515625e-05
f: 1000.00 next: 1000.00006103515625 diff: 6.103515625e-05
f: 2000.00 next: 2000.0001220703125 diff: 0.0001220703125
f: 3000.00 next: 3000.000244140625 diff: 0.000244140625
f: 4000.00 next: 4000.000244140625 diff: 0.000244140625
f: 5000.00 next: 5000.00048828125 diff: 0.00048828125
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f: 9e+09 next: 9000000512 diff: 1024
f: 1e+10 next: 10000001024 diff: 1024
f: 2e+10 next: 20000002048 diff: 2048
f: 3e+10 next: 30000003072 diff: 2048
f: 4e+10 next: 40000004096 diff: 4096
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f: 1e+12 next: 1000000061440 diff: 65536
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f: 5e+12 next: 5000000438272 diff: 524288
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f: 1e+16 next: 10000001346306048 diff: 1.07374e+09
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f: 1e+17 next: 100000007020609536 diff: 8.58993e+09
f: 2e+17 next: 200000014041219072 diff: 1.71799e+10
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f: 6e+17 next: 600000059303526400 diff: 6.87195e+10
f: 7e+17 next: 700000092093939712 diff: 6.87195e+10
f: 8e+17 next: 800000056164876288 diff: 6.87195e+10
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f: 1e+18 next: 1000000053026226176 diff: 6.87195e+10
f: 2e+18 next: 2000000106052452352 diff: 1.37439e+11
f: 3e+18 next: 3000000159078678528 diff: 2.74878e+11
f: 4e+18 next: 4000000212104904704 diff: 2.74878e+11
f: 5e+18 next: 5000000540009037824 diff: 5.49756e+11
f: 6e+18 next: 6000000318157357056 diff: 5.49756e+11
f: 7e+18 next: 7000000646061490176 diff: 5.49756e+11
f: 8e+18 next: 8000000424209809408 diff: 5.49756e+11
f: 9e+18 next: 9000000752113942528 diff: 5.49756e+11
f: 1e+19 next: 10000001080018075648 diff: 1.09951e+12
f: 2e+19 next: 20000002160036151296 diff: 2.19902e+12
f: 3e+19 next: 30000003240054226944 diff: 2.19902e+12
f: 4e+19 next: 40000004320072302592 diff: 4.39805e+12
f: 5e+19 next: 50000005400090378240 diff: 4.39805e+12
f: 6e+19 next: 60000006480108453888 diff: 4.39805e+12
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f: 2e+20 next: 2.0000002160036151296e+20 diff: 1.75922e+13
f: 3e+20 next: 3.0000004119663529165e+20 diff: 3.51844e+13
f: 4e+20 next: 4.0000004320072302592e+20 diff: 3.51844e+13
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f: 2e+21 next: 2.0000001808192430408e+21 diff: 1.40737e+14
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f: 2e+31 next: 2.0000000905412234205e+31 diff: 1.20893e+24
f: 3e+31 next: 3.0000001962581261115e+31 diff: 2.41785e+24
f: 4e+31 next: 4.000000181082446841e+31 diff: 2.41785e+24
f: 5e+31 next: 5.0000006494770954163e+31 diff: 4.8357e+24
f: 6e+31 next: 6.0000003925162522229e+31 diff: 4.8357e+24
f: 7e+31 next: 7.0000006191257368754e+31 diff: 4.8357e+24
f: 8e+31 next: 8.0000003621648936819e+31 diff: 4.8357e+24
f: 9e+31 next: 9.0000010723447061802e+31 diff: 9.67141e+24
f: 1e+32 next: 1.0000001298954190833e+32 diff: 9.67141e+24
f: 2e+32 next: 2.0000002597908381665e+32 diff: 1.93428e+25
f: 3e+32 next: 3.0000000995440605423e+32 diff: 1.93428e+25
f: 4e+32 next: 4.0000005195816763331e+32 diff: 3.86856e+25
f: 5e+32 next: 5.0000003593348987088e+32 diff: 3.86856e+25
f: 6e+32 next: 6.0000001990881210846e+32 diff: 3.86856e+25
f: 7e+32 next: 7.0000008125538680137e+32 diff: 7.73713e+25
f: 8e+32 next: 8.0000010391633526661e+32 diff: 7.73713e+25
f: 9e+32 next: 9.0000004920603127652e+32 diff: 7.73713e+25
f: 1e+33 next: 1.0000000718669797418e+33 diff: 7.73713e+25
f: 2e+33 next: 2.0000001437339594835e+33 diff: 1.54743e+26
f: 3e+33 next: 3.000000370343444136e+33 diff: 3.09485e+26
f: 4e+33 next: 4.0000002874679189671e+33 diff: 3.09485e+26
f: 5e+33 next: 5.0000002045923937981e+33 diff: 3.09485e+26
f: 6e+33 next: 6.0000007406868882719e+33 diff: 6.1897e+26
f: 7e+33 next: 7.000000657811363103e+33 diff: 6.1897e+26
f: 8e+33 next: 8.0000005749358379341e+33 diff: 6.1897e+26
f: 9e+33 next: 9.0000004920603127652e+33 diff: 6.1897e+26
f: 1e+34 next: 1.0000000409184787596e+34 diff: 6.1897e+26
f: 2e+34 next: 2.0000000818369575193e+34 diff: 1.23794e+27
f: 3e+34 next: 3.000000370343444136e+34 diff: 2.47588e+27
f: 4e+34 next: 4.0000001636739150385e+34 diff: 2.47588e+27
f: 5e+34 next: 5.0000006997684095123e+34 diff: 4.95176e+27
f: 6e+34 next: 6.0000007406868882719e+34 diff: 4.95176e+27
f: 7e+34 next: 7.0000002864293513174e+34 diff: 4.95176e+27
f: 8e+34 next: 8.000000327347830077e+34 diff: 4.95176e+27
f: 9e+34 next: 9.000001358618340265e+34 diff: 9.90352e+27
f: 1e+35 next: 1.0000001399536819025e+35 diff: 9.90352e+27
f: 2e+35 next: 2.0000002799073638049e+35 diff: 1.9807e+28
f: 3e+35 next: 3.0000001227554362789e+35 diff: 1.9807e+28
f: 4e+35 next: 4.0000005598147276098e+35 diff: 3.96141e+28
f: 5e+35 next: 5.0000002045923937981e+35 diff: 3.96141e+28
f: 6e+35 next: 6.0000002455108725578e+35 diff: 3.96141e+28
f: 7e+35 next: 7.0000006825701638887e+35 diff: 7.92282e+28
f: 8e+35 next: 8.0000011196294552197e+35 diff: 7.92282e+28
f: 9e+35 next: 9.000000764407121408e+35 diff: 7.92282e+28
f: 1e+36 next: 1.0000000409184787596e+36 diff: 7.92282e+28
f: 2e+36 next: 2.0000000818369575193e+36 diff: 1.58456e+29
f: 3e+36 next: 3.0000003604399238217e+36 diff: 3.16913e+29
f: 4e+36 next: 4.0000001636739150385e+36 diff: 3.16913e+29
f: 5e+36 next: 5.0000002838205563124e+36 diff: 3.16913e+29
f: 6e+36 next: 6.0000007208798476434e+36 diff: 6.33825e+29
f: 7e+36 next: 7.0000008410264889173e+36 diff: 6.33825e+29
f: 8e+36 next: 8.000000327347830077e+36 diff: 6.33825e+29
f: 9e+36 next: 9.0000004474944713509e+36 diff: 6.33825e+29
f: 1e+37 next: 1.0000000567641112625e+37 diff: 6.33825e+29
f: 2e+37 next: 2.000000113528222525e+37 diff: 1.26765e+30
f: 3e+37 next: 3.0000003604399238217e+37 diff: 2.5353e+30
f: 4e+37 next: 4.0000002270564450499e+37 diff: 2.5353e+30
f: 5e+37 next: 5.0000003472030863238e+37 diff: 5.0706e+30
f: 6e+37 next: 6.0000007208798476434e+37 diff: 5.0706e+30
f: 7e+37 next: 7.0000005874963688716e+37 diff: 5.0706e+30
f: 8e+37 next: 8.0000004541128900999e+37 diff: 5.0706e+30
f: 9e+37 next: 9.0000008277896514194e+37 diff: 1.01412e+31
f: 1e+38 next: 1.0000000694406172648e+38 diff: 1.01412e+31
f: 2e+38 next: 2.0000001388812345295e+38 diff: 2.02824e+31
f: 3e+38 next: 3.0000002083218517943e+38 diff: 2.02824e+31
```

# Conclusion

I hope I at least shed *some* light on the topic, and you at least know that there is such a thing as “floating point precision errors”.

And, now that you’re aware of the topic, you know what to google for

Here are some reading materials:

- http://gamedevs.org/uploads/numerical-robustness-geometric-calculations.pdf
- https://en.wikipedia.org/wiki/Single-precision_floating-point_format
- http://fabiensanglard.net/floating_point_visually_explained (Added 1/10/2017)

*Please let me know if this topic contains any errors, and I’ll try to correct them asap!*