# Fixed Point Math Exposed

If you are used to writing software for modern machines, you probably don’t think much about computing something like one divided by three. Modern computers handle floating point quite well. However, in constrained systems, there is a trap you should be aware of. While modern compilers are happy to let you use and abuse floating point numbers, the hardware is often woefully slow. It also tends to eat up lots of resources. So what do you do? Well, as [Low Byte Productions] explains, you can opt for fixed-point math.

In theory, the idea is simple. Just put an arbitrary decimal point in your integers. So, for example, if we have two numbers, say 123 and 456, we could remember that we really mean 1.23 and 4.56. Adding, then, becomes trivial since 123+456=579, which is, of course, 5.79.

# Embed With Elliot: Keeping It Integral

If there’s one thing that a lot of small microcontrollers hate (and that includes the AVR-based Arduini), it’s floating-point numbers. And if there’s another thing they hate it’s division. For instance, dividing 72.3 by 12.9 on an Arduino UNO takes around 32 microseconds and 500 bytes, while dividing 72 by 13 takes 14 microseconds and 86 bytes. Multiplying 72 by 12 takes a bit under 2.2 microseconds. So roughly speaking, dividing floats is twice as slow as dividing (16-bit) integers, and dividing at all is five to seven times slower than multiplying.

There’s a whole lot of the time that you just don’t care about speed. For instance, if you’re doing a calculation that only runs infrequently, it doesn’t matter if you’re using floats or slow division routines. But if you ever find yourself in a tight loop that’s using floating-point math and/or doing division, and you need to get a bit more speed, I’ve got some tips for you.

Some of these tips (in particular the integer division tricks at the end) are arcane wizardry — only to be used when the situation really calls for it. But if you’re doing the same calculations repeatedly, you can often gain a lot just by giving the microcontroller numbers in the format it natively understands. Have a little sympathy for the poor little silicon beasties trapped inside!

# A Detailed Tutorial On Speeding Up AVR Division

[Alan Burlison] is working on an Arduino project with an accelerometer and a few LEDs. Having the LEDs light up as his board is tilted to one side or another is an easy enough project a computer cowboy could whip out in an hour, but [Alan] – ever the perfectionist – decided to optimize his code so his accelerometer-controlled LEDs don’t jitter. The result is a spectacular blog post chronicling the pitfalls of floating point math and division on an AVR.

To remove the jitter from his LEDs, [Alan] used a smoothing algorithm known as an exponential moving average. This algorithm uses multiplication and is usually implemented using floating point arithmetic. Unfortunately, AVRs don’t have floating point arithmetic so [Alan] used fixed point arithmetic – a system similar to balancing your checkbook in cents rather than dollars.

With a clever use of bit shifting to calculate the average with scaling, [Alan] was able to make the fixed point version nearly six times faster than  the floating point algorithm implementation. After digging into the assembly of his fixed point algorithm, he was able to speed it up to 10 times faster than floating point arithmetic.

The takeaway from [Alan]’s adventures in arithmetic is that division on an AVR is slow. Not very surprising after you realize the AVR doesn’t have a division instruction. Of course, sometimes you can’t get around having to divide so multiplying by the reciprocal and using fixed point arithmetic is the way to go if speed is an issue.

Sure, squeezing every last cycle out of an 8 bit microcontroller is a bit excessive if you’re just using an Arduino as a switch. If you’re doing something with graphics or need very fast response times, [Alan] gives a lot of really useful tips.