How to make a power function for fractional powers in C++?

Question

I know that C++ has cmath and math.h libraries which provide pow() function.

However, I want to build my own power function which can operate with fractional powers. I mean I can write a loop to find 2^50 or something with integral powers, but I need a function that can do fractional powers like 5^0.3 as well.
What approach do I use here?

I think Taylor series could be the solution. So this is what I thought:

a^b = e ^ b ln(a)

We know that Taylor expansion of e^x is:
e^x = 1 + x + x^2/2! + x^3/3! ....

but there is no standard expansion for ln(x)...
[There is an expansion for ln (1+x) but it only works if |x|<1 , which means I could only find powers of numbers < 1 with it]

How do I solve this problem?

Answer 1

let a = e*a'. then:

ln a =
ln (e*a') =
ln e + ln a' =
1 + ln a'

by the logarithm rules.

So you could use the expansion you described, you just need to divide by `e` a couple times until your number falls into the -1<a<1 range.

The taylor expansion for a^b maps really nicely to a for loop btw, see if you can figure it out :P

https://code.sololearn.com/WsE1NAsJHFKJ/?ref=app

Answer 2

ok. it's a good practice project.

Updated my code.
Read the comments inside the code for various explorations I made on what the program do. Tweak and experiment.

I tried to translate Schindlabua 's javascript to c++.
The results are different at about the 14th decimal place.
Which one is more accurate? I don't know, perhaps someone could explain...

It was a learning experience.
😁
Good topic.

included link so you don't have to scroll back.

https://code.sololearn.com/cRQQmm3ACZMC/?ref=app

Answer 3

@Scindlabua Thanks This is great. Got it

Answer 4

Bob_Li I mentioned it on the first line of my question right? I DO want to reinvent the wheel..

Answer 5

There is a series expansion of a^x

a^x = 1+x(lna)+(xlna)^2/2!+(xlna)^3/3!+(xlna)^4/4!+⋯

This is called Maclaurin Series.

got the idea from this link https://www.emathzone.com/tutorials/calculus/maclaurin-series-of-ax.html

Answer 6

ROOKIE I know that one. It uses ln a, which is the main problem we are trying to solve.

Answer 7

you can do that with cmath, why reinvent the wheel?

https://code.sololearn.com/cRQQmm3ACZMC/?ref=app