Does there exist a good reference on the algorithms used by calculators, especially on the trigonometric and transcendental functions?
I would still like to know how Casio generates its random numbers. I still wonder if they are any good.
Algorithms – How Calculator Algorithms Work
algorithmscalculator
Related Solutions
Modern Computer Arithmetic suggests using an arithmetic-geometric mean algorithm. I'm not sure if this approach is meant for the low amount of precision one typically works with or if its meant for calculation in very high precision.
Another approach is to observe that the Taylor series for $\ln(x)$ is efficient if $x$ is very close to $1$. We can use algebraic identities to reduce the general case to this special case.
One method is to use the identity
$$ \ln(x) = 2 \ln(\sqrt{x})$$
to reduce the calculation of $\ln(x)$ to that of an argument closer to 1. We could use a similar identity for more general radicals if we can compute those efficiently.
By iteratively taking roots until we get an argument very close to $1$, we can reduce to
$$ \ln(x) = m \ln(\sqrt[m]{x})$$
which can be computed by the Taylor series.
If you store numbers in mantissa-exponent form in base 10, an easy identity to exploit is
$$ \ln(m \cdot 10^e) = e \ln(10) + \ln(m)$$
so the plan is to precompute the value of $\ln(10)$, and then use another method to obtain $\ln(m)$, where $m$ is not large or small.
A similar identity holds in base 2, which a computer is likely to use.
A way to use lookup tables to accelerate the calculation of $\ln(x)$ when $x$ is not large or small is to observe that
$$ \ln(x) = \ln(k) + \ln(x/k) $$
The idea here is that you store a table of $\ln(k)$ for enough values of $k$ so that you can choose the $k$ nearest $x$ to make $x/k$ very near $1$, and then all that's left is to compute $\ln(x/k)$.
On my Casio fx-85GT PLUS, the comma key (i.e. SHIFT + right parenthesis) can be used:
To convert between rectangular and polar cooordinates. For example: $$ \text{Pol}\left(\sqrt{2}\color{red},\sqrt{2}\right) \\ r=2,\theta=0.7853981 \\ \text{Rec}\left(\sqrt{2}\color{red},\pi\div4\right) \\ X=1,Y=1 $$
To generate random integers, e.g. in the range $1$ to $6$: $$ \text{RanInt#}(1\color{red},6) $$ Then keep pressing the '=' key.
To calculate logarithms to an arbitrary base, such as: $$ \log(2\color{red},16) \\ 4 $$
These examples are in the booklet that came with the calculator. There may be other things you can do with it.
Best Answer
I would recommend reading Gerald Rising's Inside your Calculator (which has a supplementary website); there is a nice discussion of the methods used by some calculators that is suitable at the undergraduate level.
Otherwise, to really figure out what methods they are using, it might help to search the technical notes of the manufacturer's websites. For instance, Texas Instruments has notes like this one on their "knowledge base" that discuss "what's under the hood", though not in detail of course. (Sometimes, hobbyist sites like this one also discuss calculator algorithms.)