What is a hacker, if not somebody who comes up with solutions that other just don’t see? All the pieces may be in place, but it takes that one special person to view the pieces as greater than the sum of their parts. As [Chris Staecker] explains in the video below the break, Henri Genaille was one such person.
When French mathematician Edouard Lucas (himself well known for calculating the longest prime number found by hand) posed a mathematical problem at the French Academy, a French railway engineer named Henri Genaille developed the rods we’re discussing now.
Genaille’s Rods are designed to perform multiplication. But rather than require computation by the user, the rods would simply need to be laid out in the correct order. The solution could readily be found by just following the lines in the correct pattern. This might sound a lot like cheating, and that’s exactly what it is. No manual math needed to be done. Genaille also created rods for doing long division, which we’re sure were every bit as enthralling as the multiplication rods. Demonstrations of both are included in the video below.
While Genaille’s Rods have gone the way of the slide rule, we can’t help but wonder how many engineers and scientists carried around a set of marked up wooden sticks in their pocket protector.
If designing and building manual mathematical machines is something that you think really adds up to a good time, check out this post on how to design and build your own circular slide rule!
Continue reading “Genaille’s Rods: When Paint Sticks Do Math”
[Paul Curtis] over at Segger has an interesting series of blog posts about calculating division. This used to be a hotter topic, but nowadays many computers or computer languages have support for multiplication and division built-in. But some processors lack the instructions and a library to do it might be less than ideal. Knowing how to roll your own might allow you to optimize for speed or space. The current installment covers using Newton’s algorithm to do division.
Steve Martin had a famous bit about how to be a millionaire and never pay taxes. He started out by saying, “First… get a million dollar. Then…” This method is a bit like that since you first have to know how to multiply before you can divide. The basic premise is twofold: Newton’s method let you refine an estimate of a reciprocal by successive multiplications and then multiplying a number a reciprocal is the same as dividing. In other words, if we need to divide 34 by 6, you could rewrite 34/6 to 34 * 1/6 and the answer is the same.
Continue reading “Apple Falling Division”
Dividing by zero — the fundamental no-can-do of arithmetic. It is somewhat surrounded by mystery, and is a constant source for internet humor, whether it involves exploding microcontrollers, the collapse of the universe, or crashing your own world by having Siri tell you that you have no friends.
It’s also one of the few things
gcc will warn you about by default, which caused a rather vivid discussion with interesting insights when I recently wrote about compiler warnings. And if you’re running a modern operating system, it might even send you a signal that something’s gone wrong and let you handle it in your code. Dividing by zero is more than theoretical, and serves as a great introduction to signals, so let’s have a closer look at it.
Chances are, the first time you heard about division itself back in elementary school, it was taught that dividing by zero is strictly forbidden — and obviously you didn’t want your teacher call the cops on you, so you obeyed and refrained from it. But as with many other things in life, the older you get, the less restrictive they become, and dividing by zero eventually turned from forbidden into simply being impossible and yielding an undefined result.
And indeed, if a = b/0, it would mean in reverse that a×0 = b. If b itself was zero, the equation would be true for every single number there is, making it impossible to define a concrete value for a. And if b was any other value, no single value multiplied by zero could result in anything non-zero. Once we move into the realms of calculus, we will learn that infinity appears to be the answer, but that’s in the end just replacing one abstract, mind-boggling concept with another one. And it won’t answer one question: how does all this play out in a processor? Continue reading “Creating Black Holes: Division By Zero In Practice”
[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.