I would like to note before hand I am trying to improve my math by making a calculator application.
Anyways, let's say I would like to compute $2^{2.534}$.
Beforehand, I know that:
- $$a^x=e^{xln(a)}$$
I already have a working algorithm on finding $ln(a)$, but ultimately in the end this still gives me a rational number and without actually knowing how to compute it, I arrive to my original problem again.
At this point, I figure I can store a table of already computed values for $e^{x}$, and then just interpolate values? However, I feel that my accuracy would greatly depend on how many values I store and it makes me come to hte question of how did anyone create these tables in the first place?
The logarithm and trigonometric tables were tools I still used in the early 70s when I attended high school and calculators were too expensive ans/or not allowed in classroom. The mathematicians of precomputer era calculated tables by hand. They had polynomial expansion like Taylor or MacLaurin formulae.
Logarithm standard formula
$$\log(1+x)=x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6+x^7/7-x^8/8+\ldots$$
is too slow, and soon were invented accelerated methods for alternating series like this.
Essentially the calculation were done by hand
Example. For small $x$
$\sin x\approx x-\frac{x^3}{6}+\frac{x^5}{120}$
Try $x=0.15$
$\sin 0.15 \approx 0.15 -\frac{0.15^3}{6}+\frac{0.15}{120}=\frac{3}{20}-\frac{27}{8000}+\frac{243}{3200000}= \color{red}{0.149438132}8125$
My calculator gives $\color{red}{0.149438132}4736$
Impressive, isn't it? Only three terms, such a precision... but caution if we look for values "far" from zero, like $x=2$ we get a bad result
$\sin 2\approx 2-8/3+32/120\approx 0.93$
while $\sin 1\approx 0.909297$
Hope this can be useful