Lua by default uses a double precision floating point (double) type as its only numeric type. That's nice and useful. However, I'm working on software that expects to see 64bit integers, for which I don't get around using actual 64bit integers one way or another.
The place where the integer type becomes relevant is for file sizes. Although I don't truly expect to see file sizes beyond what Lua can represent with full "integer" precision using a double, I want to be prepared.
What strategies can you recommend when using a 64bit integer type in parallel with the default numeric type of Lua? I don't really want to throw the default implementation overboard (and I'm not worried of its performance compared to integer arithmetics), but I need some way of representing 64bit integers up to their full precision without too much of a performance penalty.
My problem is that I'm unsure where to modify the behavior. Should I modify the syntax and extend the parser (numbers with appended LL or ULL come to mind, which to my knowledge doesn't exist in default Lua) or should I instead write my own C module and define a userdata type that represents the 64bit integer, along with library functions able to manipulate the values? ...
Note: yes, I am embedding Lua, so I am free to extend it whichever way I please.
As part of LuaJIT's port to ARM CPUs (which often have poor floating-point), LuaJIT implemented a "Dual-number VM", which allows it to switch between integers and floats dynamically as needed. You could use this yourself, just switch between 64-bit integers and doubles instead of 32-bit integers and floats.
It's currently live in builds, so you may want to consider using LuaJIT as your Lua "interpreter." Or you could use it as a way to learn how to do this sort of thing.
However, I do agree with Marcelo; the 53-bit mantissa should be plenty. You shouldn't really need this for a good 10 years or so.
I'd suggest storing your data outside of Lua and use some type of reference to retrieve it when calling your other libraries. You can then push various results onto the Lua stack for the user the see, you can even retrieve the value as a string to be precise, but I would avoid modifying them in Lua and relying on the Lua values when calling your external library.
If you're not going to need floating-point precision at any point in the program, you can just redefine LUA_NUMBER to __int64 (or whatever 64-bit int may be in your environment) in luaconf.h.
Otherwise, you can just bring in another library to handle your integers- for infinite precision, you can use a bignum library such as lhf's lbn.
Related
Is there a node.js module that allows my application to have the same types as MongoDB:
http://docs.mongodb.org/manual/reference/bson-types/
For example, in my node.js app, I want it to have full understanding of the Integer type, but node.js doesn't recognize anything but Number out of the box to my understanding.
node.js doesn't recognize anything but Number out of the box to my understanding.
Well, that's a design decision of JavaScript. JavaScript only has a Number type, which is, if the implementation follows the standard's recommendation, represented as IEEE-754 double precision floating point number. That hides much of the complexity of short vs. long vs. float vs. double and all the other numeric types that are common in most other languages. It's also a good compromise, because they have integer precision up to 2^53 - 1, so you rarely need something outside this range, most of the time, and if you do, a long usually doesn't cut it either...
MongoDB, on the other hand, is written in C++ and is thus much closer to the gritty details of how data is actually stored in memory. Bridging that gap isn't trivial.
Is there a node.js module that allows my application to have the same types as MongoDB?
In a way, you're battling your underlying programming language. If such a module existed, it would also have to somehow implement a whole lot of operations, e.g. long addition (for results > 2^53-1) in a fundamentally different way. For instance, it could do this by converting the digits to a string and adding the individual digits manually, which would be atrociously slow.
Something like this happens in the mongo shell, e.g. for very long integers it uses a syntax like NumberLong("1223"). But it doesn't re-implement the operations, it falls back to regular V8 JavaScript. Proof:
> var foo = NumberLong("36028797018963968") // 2^55
> var foo2 = foo + NumberLong("1")
> foo2
36028797018963970 // should be 36028797018963969
Alternatively, such a module could be a native module that used the existing implementations of a native programming language such as C++. There's a node module called 'edge' that apparently does it for C#. However, that comes with quite a bit of complexity because crossing language borders requires marshalling, which is both expensive and tricky to get right.
In other words: getting a truly different set of types requires another programming language. Making your code roughly stick to your schema is possible, e.g. via mongoose or via coding discipline, but it won't make JS strongly typed.
CLP(FD) allows user to set the domain for every wannabe-integer variable, so it's able to solve equations.
So far so good.
However you can't do the same in CLP(R) or similar languages (where you can do only simple inferences). And it's not hard to understand why: the fractional part of a number may have an almost infinite region, putted down by an implementation limit. This mean the search space will be too large to make any practical use for a solver which deals with floats like with integers. So it's the user task to write generator in CLP(R) and set constraint guards where needed to get variables instantiated with numbers (if simple inference is not possible).
So my question here: is there any CLP(FD)-like language over reals? I think it could be implemented by use of number rounding, searching and following incremental approximation.
There are at least some major CLP(FD) solvers that support real (decision) variables:
Gecode
JaCoP
ECLiPSe CLP (ic library)
Choco (using Ibex)
(The first three also support var float in MiniZinc.)
The answser to your question is yes. There is Constraint-based Solvers dedicated for floating numbers. I do not have a list of solvers but I know that that ibex http://www.ibex-lib.org is a library allowing the use of floats. You should also have a look at SMT-Solvers implementing the Real-Theory (http://smtlib.cs.uiowa.edu/solvers.shtml).
I would like to know whether i can assume that same operations on same 64-bit floating point numbers gives exactly the same results on any modern PC and in most common programming languages? (C++, Java, C#, etc.). We can assume, that we are operating on numbers and result is also a number (no NaNs, INFs and so on).
I know there are two very simmilar standards of computation using floating point numbers (IEEE 854-1987 and IEEE 754-2008). However I don't know how it is in practice.
Modern processors that implement 64-bit floating-point typically implement something that is close to the IEEE 754-1985 standard, recently superseded by the 754-2008 standard.
The 754 standard specifies what result you should get from certain basic operations, notably addition, subtraction, multiplication, division, square root, and negation. In most cases, the numeric result is specified precisely: The result must be the representable number that is closest to the exact mathematical result in the direction specified by the rounding mode (to nearest, toward infinity, toward zero, or toward negative infinity). In "to nearest" mode, the standard also specifies how ties are broken.
Because of this, operations that do not involve exception conditions such as overflow will get the same results on different processors that conform to the standard.
However, there are several issues that interfere with getting identical results on different processors. One of them is that the compiler is often free to implement sequences of floating-point operations in a variety of ways. For example, if you write "a = bc + d" in C, where all variables are declared double, the compiler is free to compute "bc" in either double-precision arithmetic or something with more range or precision. If, for example, the processor has registers capable of holding extended-precision floating-point numbers and doing arithmetic with extended-precision does not take any more CPU time than doing arithmetic with double-precision, a compiler is likely to generate code using extended-precision. On such a processor, you might not get the same results as you would on another processor. Even if the compiler does this regularly, it might not in some circumstances because the registers are full during a complicated sequence, so it stores the intermediate results in memory temporarily. When it does that, it might write just the 64-bit double rather than the extended-precision number. So a routine containing floating-point arithmetic might give different results just because it was compiled with different code, perhaps inlined in one place, and the compiler needed registers for something else.
Some processors have instructions to compute a multiply and an add in one instruction, so "bc + d" might be computed with no intermediate rounding and get a more accurate result than on a processor that first computes bc and then adds d.
Your compiler might have switches to control behavior like this.
There are some places where the 754-1985 standard does not require a unique result. For example, when determining whether underflow has occurred (a result is too small to be represented accurately), the standard allows an implementation to make the determination either before or after it rounds the significand (the fraction bits) to the target precision. So some implementations will tell you underflow has occurred when other implementations will not.
A common feature in processors is to have an "almost IEEE 754" mode that eliminates the difficulty of dealing with underflow by substituting zero instead of returning the very small number that the standard requires. Naturally, you will get different numbers when executing in such a mode than when executing in the more compliant mode. The non-compliant mode may be the default set by your compiler and/or operating system, for reasons of performance.
Note that an IEEE 754 implementation is typically not provided just by hardware but by a combination of hardware and software. The processor may do the bulk of the work but rely on the software to handle certain exceptions, set certain modes, and so on.
When you move beyond the basic arithmetic operations to things like sine and cosine, you are very dependent on the library you use. Transcendental functions are generally calculated with carefully engineered approximations. The implementations are developed independently by various engineers and get different results from each other. On one system, the sin function may give results accurate within an ULP (unit of least precision) for small arguments (less than pi or so) but larger errors for large arguments. On another system, the sin function might give results accurate within several ULP for all arguments. No current math library is known to produce correctly rounded results for all inputs. There is a project, crlibm (Correctly Rounded Libm), that has done some good work toward this goal, and they have developed implementations for significant parts of the math library that are correctly rounded and have good performance, but not all of the math library yet.
In summary, if you have a manageable set of calculations, understand your compiler implementation, and are very careful, you can rely on identical results on different processors. Otherwise, getting completely identical results is not something you can rely on.
If you mean getting exactly the same result, then the answer is no.
You might even get different results for debug (non-optimized) builds vs. release builds (optimized) on the same machine in some cases, so don't even assume that the results might be always identical on different machines.
(This can happen e.g. on a computer with an Intel processor, if the optimizer keeps a variable for an intermediate result in a register, that is stored in memory in the unoptimized build. Since Intel FPU registers are 80 bit, and double variables are 64 bit, the intermediate result will be stored with greater precision in the optimized build, causing different values in later results.).
In practice, however, you may often get the same results, but you shouldn't rely on it.
Modern FPUs all implement IEEE754 floats in single and double formats, and some in extended format. A certain set of operations are supported (pretty much anything in math.h), with some special instructions floating around out there.
assuming you are talking about applying multiple operations, I do not think you will get exact numbers. CPU architecture, compiler use, optimization settings will change the results of your computations.
if you mean the exact order of operations (at the assembly level), I think you will still get variations.for example Intel chips use extended precision (80 bits) internally, which may not be the case for other CPUs. (I do not think extended precision is mandated)
The same C# program can bring out different numerical results on the same PC, once compiled in debug mode without optimization, second time compiled in release mode with optimization enabled. That's my personal experience. We did not regard this when we set up an automatic regression test suite for one of our programs for the first time, and were completely surprised that a lot of our tests failed without any apparent reason.
For C# on x86, 80-bit FP registers are used.
The C# standard says that the processor must operate at the same precision as, or greater than, the type itself (i.e. 64-bit in the case of a 'double'). Promotions are allowed, except for storage. That means that locals and parameters could be at greater than 64-bit precision.
In other words, assigning a member variable to a local variable could (and in fact will under certain circumstances) be enough to give an inequality.
See also: Float/double precision in debug/release modes
For the 64-bit data type, I only know of "double precision" / "binary64" from the IEEE 754 (1985 and 2008 don't differ much here for common cases) being used.
Note: The radix types defined in IEEE 854-1987 are superseded by IEEE 754-2008 anyways.
I'm currently teaching myself Haskell, and I'm wondering what the best practices are when working with strings in Haskell.
The default string implementation in Haskell is a list of Char. This is inefficient for file input-output, according to Real World Haskell, since each character is separately allocated (I assume that this means that a String is basically a linked list in Haskell, but I'm not sure.)
But if the default string implementation is inefficient for file i/o, is it also inefficient for working with Strings in memory? Why or why not? C uses an array of char to represent a String, and I assumed that this would be the default way of doing things in most languages.
As I see it, the list implementation of String will take up more memory, since each character will require overhead, and also more time to iterate over, because a pointer dereferencing will be required to get to the next char. But I've liked playing with Haskell so far, so I want to believe that the default implementation is efficient.
Apart from String/ByteString there is now the Text library which combines the best of both worlds—it works with Unicode while being ByteString-based internally, so you get fast, correct strings.
Best practices for working with strings performantly in Haskell are basically: Use Data.ByteString/Data.ByteString.Lazy.
http://hackage.haskell.org/packages/archive/bytestring/latest/doc/html/
As far as the efficiency of the default string implementation goes in Haskell, it's not. Each Char represents a Unicode codepoint which means it needs at least 21bits per Char.
Since a String is just [Char], that is a linked list of Char, it means Strings have poor locality of reference, and again means that Strings are fairly large in memory, at a minimum it's N * (21bits + Mbits) where N is the length of the string and M is the size of a pointer (32, 64, what have you) and unlike many other places where Haskell uses lists where other languages might use different structures (I'm thinking specifically of control flow here), Strings are much less likely to be able to be optimized to loops, etc. by the compiler.
And while a Char corresponds to a codepoint, the Haskell 98 report doesn't specify anything about the encoding used when doing file IO, not even a default much less a way to change it. In practice GHC provides an extensions to do e.g. binary IO, but you're going off the reservation at that point anyway.
Even with operations like prepending to front of the string it's unlikely that a String will beat a ByteString in practice.
The answer is a bit more complex than just "use lazy bytestrings".
Byte strings only store 8 bits per value, whereas String holds real Unicode characters. So if you want to work with Unicode then you have to convert to and from UTF-8 or UTF-16 all the time, which is more expensive than just using strings. Don't make the mistake of assuming that your program will only need ASCII. Unless its just throwaway code then one day someone will need to put in a Euro symbol (U+20AC) or accented characters, and your nice fast bytestring implementation will be irretrievably broken.
Byte strings make some things, like prepending to the start of a string, more expensive.
That said, if you need performance and you can represent your data purely in bytestrings, then do so.
The basic answer given, use ByteString, is correct. That said, all of the three answers before mine have inaccuracies.
Regarding UTF-8: whether this will be an issue or not depends entirely on what sort of processing you do with your strings. If you're simply treating them as single chunks of data (which includes operations such as concatenation, though not splitting), or doing certain limited byte-based operations (e.g., finding the length of the string in bytes, rather than the length in characters), you won't have any issues. If you are using I18N, there are enough other issues that simply using String rather than ByteString will start to fix only a very few of the problems you'll encounter.
Prepending single bytes to the front of a ByteString is probably more expensive than doing the same for a String. However, if you're doing a lot of this, it's probably possible to find ways of dealing with your particular problem that are cheaper.
But the end result would be, for the poster of the original question: yes, Strings are inefficient in Haskell, though rather handy. If you're worried about efficiency, use ByteStrings, and view them as either arrays of Char8 or Word8, depending on your purpose (ASCII/ISO-8859-1 vs Unicode of some sort, or just arbitrary binary data). Generally, use Lazy ByteStrings (where prepending to the start of a string is actually a very fast operation) unless you know why you want non-lazy ones (which is usually wrapped up in an appreciation of the performance aspects of lazy evaluation).
For what it's worth, I am building an automated trading system entirely in Haskell, and one of the things we need to do is very quickly parse a market data feed we receive over a network connection. I can handle reading and parsing 300 messages per second with a negligable amount of CPU; as far as handling this data goes, GHC-compiled Haskell performs close enough to C that it's nowhere near entering my list of notable issues.
What programming languages support arbitrary precision arithmetic and could you give a short example of how to print an arbitrary number of digits?
Some languages have this support built in. For example, take a look at java.math.BigDecimal in Java, or decimal.Decimal in Python.
Other languages frequently have a library available to provide this feature. For example, in C you could use GMP or other options.
The "Arbitrary-precision software" section of this article gives a good rundown of your options.
Mathematica.
N[Pi, 100]
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
Not only does mathematica have arbitrary precision but by default it has infinite precision. It keeps things like 1/3 as rationals and even expressions involving things like Sqrt[2] it maintains symbolically until you ask for a numeric approximation, which you can have to any number of decimal places.
In Common Lisp,
(format t "~D~%" (expt 7 77))
"~D~%" in printf format would be "%d\n". Arbitrary precision arithmetic is built into Common Lisp.
Smalltalk supports arbitrary precision Integers and Fractions from the beginning.
Note that gnu Smalltalk implementation does use GMP under the hood.
I'm also developping ArbitraryPrecisionFloat for various dialects (Squeak/Pharo Visualworks and Dolphin), see http://www.squeaksource.com/ArbitraryPrecisionFl.html
Python has such ability. There is an excellent example here.
From the article:
from math import log as _flog
from decimal import getcontext, Decimal
def log(x):
if x < 0:
return Decimal("NaN")
if x == 0:
return Decimal("-inf")
getcontext().prec += 3
eps = Decimal("10")**(-getcontext().prec+2)
# A good initial estimate is needed
r = Decimal(repr(_flog(float(x))))
while 1:
r2 = r - 1 + x/exp(r)
if abs(r2-r) < eps:
break
else:
r = r2
getcontext().prec -= 3
return +r
Also, the python quick start tutorial discusses the arbitrary precision: http://docs.python.org/lib/decimal-tutorial.html
and describes getcontext:
the getcontext() function accesses the
current context and allows the
settings to be changed.
Edit: Added clarification on getcontext.
Many people recommended Python's decimal module, but I would recommend using mpmath over decimal for any serious numeric uses.
COBOL
77 VALUE PIC S9(4)V9(4).
a signed variable witch 4 decimals.
PL/1
DCL VALUE DEC FIXED (4,4);
:-) I can't remember the other old stuff...
Jokes apart, as my example show, I think you shouldn't choose a programming language depending on a single feature. Virtually all decent and recent language support fixed precision in some dedicated classes.
Scheme (a variation of lisp) has a capability called 'bignum'. there are many good scheme implementations available both full language environments and embeddable scripting options.
a few I can vouch for
MitScheme (also referred to as gnu scheme)
PLTScheme
Chezscheme
Guile (also a gnu project)
Scheme 48
Ruby whole numbers and floating point numbers (mathematically speaking: rational numbers) are by default not strictly tied to the classical CPU related limits. In Ruby the integers and floats are automatically, transparently, switched to some "bignum types", if the size exceeds the maximum of the classical sizes.
One probably wants to use some reasonably optimized and "complete", multifarious, math library that uses the "bignums". This is where the Mathematica-like software truly shines with its capabilities.
As of 2011 the Mathematica is extremely expensive and terribly restricted from hacking and reshipping point of view, specially, if one wants to ship the math software as a component of a small, low price end, web application or an open source project. If one needs to do only raw number crunching, where visualizations are not required, then there exists a very viable alternative to the Mathematica and Maple. The alternative is the REDUCE Computer Algebra System, which is Lisp based, open source and mature (for decades) and under active development (in 2011). Like Mathematica, the REDUCE uses symbolic calculation.
For the recognition of the Mathematica I say that as of 2011 it seems to me that the Mathematica is the best at interactive visualizations, but I think that from programming point of view there are more convenient alternatives even if Mathematica were an open source project. To me it seems that the Mahtematica is also a bit slow and not suitable for working with huge data sets. It seems to me that the niche of the Mathematica is theoretical math, not real-life number crunching. On the other hand the publisher of the Mathematica, the Wolfram Research, is hosting and maintaining one of the most high quality, if not THE most high quality, free to use, math reference sites on planet Earth: the http://mathworld.wolfram.com/
The online documentation system that comes bundled with the Mathematica is also truly good.
When talking about speed, then it's worth to mention that REDUCE is said to run even on a Linux router. The REDUCE itself is written in Lisp, but it comes with 2 of its very own, specific, Lisp implementations. One of the Lisps is implemented in Java and the other is implemented in C. Both of them work decently, at least from math point of view. The REDUCE has 2 modes: the traditional "math mode" and a "programmers mode" that allows full access to all of the internals by the language that the REDUCE is self written in: Lisp.
So, my opinion is that if one looks at the amount of work that it takes to write math routines, not to mention all of the symbolic calculations that are all MATURE in the REDUCE, then one can save enormous amount of time (decades, literally) by doing most of the math part in REDUCE, specially given that it has been tested and debugged by professional mathematicians over a long period of time, used for doing symbolic calculations on old-era supercomputers for real professional tasks and works wonderfully, truly fast, on modern low end computers. Neither has it crashed on me, unlike at least one commercial package that I don't want to name here.
http://www.reduce-algebra.com/
To illustrate, where the symbolic calculation is essential in practice, I bring an example of solving a system of linear equations by matrix inversion. To invert a matrix, one needs to find determinants. The rounding that takes place with the directly CPU supported floating point types, can render a matrix that theoretically has an inverse, to a matrix that does not have an inverse. This in turn introduces a situation, where most of the time the software might work just fine, but if the data is a bit "unfortunate" then the application crashes, despite the fact that algorithmically there's nothing wrong in the software, other than the rounding of floating point numbers.
The absolute precision rational numbers do have a serious limitation. The more computations is performed with them, the more memory they consume. As of 2011 I don't know any solutions to that problem other than just being careful and keeping track of the number of operations that has been performed with the numbers and then rounding the numbers to save memory, but one has to do the rounding at a very precise stage of the calculations to avoid the aforementioned problems. If possible, then the rounding should be done at the very end of the calculations as the very last operation.
In PHP you have BCMath. You not need to load any dll or compile any module.
Supports numbers of any size and precision, represented as string
<?php
$a = '1.234';
$b = '5';
echo bcadd($a, $b); // 6
echo bcadd($a, $b, 4); // 6.2340
?>
Apparently Tcl also has them, from version 8.5, courtesy of LibTomMath:
http://wiki.tcl.tk/5193
http://www.tcl.tk/cgi-bin/tct/tip/237.html
http://math.libtomcrypt.com/
There are several Javascript libraries that handle arbitrary-precision arithmetic.
For example, using my big.js library:
Big.DP = 20; // Decimal Places
var pi = Big(355).div(113)
console.log( pi.toString() ); // '3.14159292035398230088'
In R you can use the Rmpfr package:
library(Rmpfr)
exp(mpfr(1, 120))
## 1 'mpfr' number of precision 120 bits
## [1] 2.7182818284590452353602874713526624979
You can find the vignette here: Arbitrarily Accurate Computation with R:
The Rmpfr Package
Java natively can do bignum operations with BigDecimal. GMP is the defacto standard library for bignum with C/C++.
If you want to work in the .NET world you can use still use the java.math.BigDecimal class. Just add a reference to vjslib (in the framework) and then you can use the java classes.
The great thing is, they can be used fron any .NET language. For example in C#:
using java.math;
namespace MyNamespace
{
class Program
{
static void Main(string[] args)
{
BigDecimal bd = new BigDecimal("12345678901234567890.1234567890123456789");
Console.WriteLine(bd.ToString());
}
}
}
The (free) basic program x11 basic ( http://x11-basic.sourceforge.net/ ) has arbitrary precision for integers. (and some useful commands as well, e.g. nextprime( abcd...pqrs))
IBM's interpreted scripting language Rexx, provides custom precision setting with Numeric. https://www.ibm.com/docs/en/zos/2.1.0?topic=instructions-numeric.
The language runs on mainframes and pc operating systems and has very powerful parsing and variable handling as well as extension packages. Object Rexx is the most recent implementation. Links from https://en.wikipedia.org/wiki/Rexx
Haskell has excellent support for arbitrary-precision arithmetic built in, and using it is the default behavior. At the REPL, with no imports or setup required:
Prelude> 2 ^ 2 ^ 12
1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190336
(try this yourself at https://tryhaskell.org/)
If you're writing code stored in a file and you want to print a number, you have to convert it to a string first. The show function does that.
module Test where
main = do
let x = 2 ^ 2 ^ 12
let xStr = show x
putStrLn xStr
(try this yourself at code.world: https://www.code.world/haskell#Pb_gPCQuqY7r77v1IHH_vWg)
What's more, Haskell's Num abstraction lets you defer deciding what type to use as long as possible.
-- Define a function to make big numbers. The (inferred) type is generic.
Prelude> superbig n = 2 ^ 2 ^ n
-- We can call this function with different concrete types and get different results.
Prelude> superbig 5 :: Int
4294967296
Prelude> superbig 5 :: Float
4.2949673e9
-- The `Int` type is not arbitrary precision, and we might overflow.
Prelude> superbig 6 :: Int
0
-- `Double` can hold bigger numbers.
Prelude> superbig 6 :: Double
1.8446744073709552e19
Prelude> superbig 9 :: Double
1.3407807929942597e154
-- But it is also not arbitrary precision, and can still overflow.
Prelude> superbig 10 :: Double
Infinity
-- The Integer type is arbitrary-precision though, and can go as big as we have memory for and patience to wait for the result.
Prelude> superbig 12 :: Integer
1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190336
-- If we don't specify a type, Haskell will infer one with arbitrary precision.
Prelude> superbig 12
1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190336