calc VS arb

http://www.isthe.com/chongo/tech/comp/calc/index.html not sure why you chose exactly the same name as the original calc. Even if you plan to create a drop in replacement. It's not good practise to use exactly the same name than another active project in the same problem domain

my benchmarks againts qalc and c-calc shows that c-calc takes about 0.8ms mean to run a calculation,qalc takes about 96.4ms to run a calculation,and mine takes 0.5ms to run a calculation, of course this is pretty much just the startup time however i cant measure the runtime speed againts c-calc because it does not allow multiple arguments like mine does.

I like calc myself, works quite well. Doesn't have e preloaded as a constant, to my knowledge at least, but otherwise it is very good.

calc: a command line calculator with arbitrary precision. GNU bc is good too, but calc has more built-in commands (combinatorial, number theory functions for example). You can use it interactively, or simply to provide the results of a calculation. Try this: in your terminal, enter

There's also another calc: https://github.com/lcn2/calc

You can instead list your criteria for good number format and look at alternatives with those lenses. Floating point is designed for a good balance between dynamic range and precision, and IEEE 754 binary formats can be seen as a FP standard particularly optimized for numerical calculation.

There are several other FP formats. The most popular one is IEEE 754 minus subnormal numbers, followed by bfloat16, IEEE 754 decimal formats (formerly IEEE 854) and posits. Only first two have good hardware supports. The lack of subnormal number means that `a <=> b` can't be no longer rewritten to `a - b <=> 0` among others but is widely believed to be faster. (I don't fully agree, but it's indeed true for existing contemporary hardwares.) IEEE 754 decimal formats are notable for lack of normalization guarantee. Posits are, in some sense, what IEEE 754 would have been if designed today, and in fact aren't that fundamentally different from IEEE 754 in my opinion.

Fixed-point formats share pros and cons of finitely sized integer numbers and you should have no difficulty to analyze them. In short, they offer a smaller dynamic range compared to FP, but its truncation model is much simpler to reason. In turn you will get a varying precision and out-of-bound issues.

Rational number formats look very promising at the beginning, but they are much harder to implement efficiently. You will need a fast GCD algorithm (not Euclidean) and also have to handle out-of-bound numerators and denumerators. In fact, many rational number formats rely on arbitrary-precision integers precisely for avoiding those issues, and inherit the same set of issues---unbounded memory usage and computational overhead. Approximate rational number formats are much rarer, and I'm only aware of the Inigo Quilez's floating-bar experiment [1] in this space.

[1] https://iquilezles.org/articles/floatingbar/

Interval/ball/affine arithmetics and others are means to automatically approximate an error analysis. They have a good property of being never incorrect, but it is still really easy for them to throw up and give a correct but useless answer like [-inf, inf]. Also they are somewhat awkward in a typical procedural paradigm because comparisons will return a tri-state boolean (true, false, unsure). Nevertheless they are often useful when correctly used. Fredrik Johansson's Arb [2] is a good starting point in my opinion.

[2] https://arblib.org/

Finally you can model a number as a function that returns a successively accurate approximation. This is called the constructive or exact real number, and simultaneously most expensive and most correct. One of the most glaring problems is that an equality is not always decidable, and practical applications tend to have various heuristics to get around this fact. Amazingly enough, Android's built-in calculator is one of the most used applications that use this model [3].

[3] https://dl.acm.org/doi/pdf/10.1145/2911981

https://github.com/JuliaMath/Bessels.jl/blob/master/src/bess...

Thanks! I love it, so easy to understand and follow.

My favourite work on the subject is Fredrik Johansson's:

https://github.com/fredrik-johansson/arb

Whenever I feel down and without energy I just read something in there

Probably the most popular package for dealing with error propagation and arbitrary precision arithmetic in Python is mpmath, more specifically the mp.iv module. For more serious applications I'd take a look at MPFR and Arb, both in C. And there are tons of ball arithmetic and interval arithmetic libraries in Fortran.

You're in good company too. Using intervals to bound error is the entire idea behind the arb library.

The error bounds approach is probably what you’re looking for. A better search term for that is “interval arithmetic.” There are many good software packages for interval arithmetic, like Arb.

I do agree with your list, so that is something! I will add, balls are underrated, ditto intervals (nominally more efficient, but on x86 switching rounding modes is 20-30 cycles...)

Ball arithmetic looks interesting. As far as I know, arb is the primary implementation.