I'm curious about how computers perform calculations for logarithms and trigonometric functions since they don't follow standard arithmetic operations like addition or multiplication. What algorithms or methods do they use to compute these functions?
5 Answers
Computers calculate these functions similarly to calculators by using infinite series. They approximate the results to a certain defined level of accuracy. So, when you input a sine or logarithm, the computer isn't just doing a straightforward calculation; it's summing a series of terms until it reaches an acceptable approximation.
If you're looking to dive deeper, I'd recommend checking out a book on numerical analysis. This field covers how functions are approximated and the nuances of error in those approximations, as well as the computational techniques like loops and recursion that are leveraged during calculations.
I'm not entirely sure how roots are calculated either, or even why raising to the power of 0.5 gives a square root, but I believe it's all approximated in some way I'm not aware of!
When it comes to trigonometric functions, many algorithms, like the Newton-Raphson method, are also employed for approximations. In extreme cases, especially for hardware-optimized functions like floating point division, lookup tables are used for speed and efficiency, which rely on basic interpolation between known values.
There's a deep connection between these functions and binary code. Essentially, mathematics can be broken down into binary operations, with basic addition being the simplest form. From there, operations like multiplication build upon addition. So, even complex calculus can boil down to simpler operations through clever algorithmic design.
Right! And a lot of modern implementations also deal with it through approximation algorithms, with techniques like CORDIC or polynomial approximations. The mathematics library in languages like C handles this micro-optimally using things like Taylor series with range reduction.
Absolutely! Some of the basic trig functions might even be implemented at the hardware level, using specific CPU instructions designed for calculating angles or complex numbers efficiently.