Comment by noosphr
8 hours ago
How is going from two functions with one variable to three functions with a variable and a constant a simplification?
8 hours ago
How is going from two functions with one variable to three functions with a variable and a constant a simplification?
If you can't recognize how much simpler the simplified version is, I'm not sure exactly what to tell you. But let's think about it in terms of assembly steps:
1. Multiply the input by itself
2. Add 1
3. Take the square root. There is often a fast square root function available.
The above is a fairly simply sequence of SIMD instructions. You can even do it without SIMD if you want.
Compare this to sinh being (e^x - e^-x) / 2 (you can reduce this to one exponentiation in terms of e^2x but I digress) and arccosh being ln(x + sqrt(s^2 - 1)) and you have an exponentiation, subtraction, division, logarithm, addition, square root and a subtraction. Computers generally implement e^2 and logarithm using numerical method approximations (eg of a Taylor's series expansion).
This is sometimes helpful. But more often it has very little overlap with what I need when I "simplify" some math.
"Simplify" is a very old term (>50y) in computer algebra. Its meaning has become kind of layered in that time.
If simplify means make it fast for a computer to run we might as well make division illegal.