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POThe best cross platform (portable) arbitrary precision math library
 

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  1. POThe best cross platform (portable) arbitrary precision math library
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    <p>I'm looking for a good arbitrary precision math library in C or C++. Could you please give me some advices / suggestions?</p> <p>The primary requirements:</p> <ol> <li>It <b>MUST</b> handle arbitrarily big integers (my primary interest is on integers). In case that you don't know what the word arbitrarily big means, imagine something like 100000! (the factorial of 100000).</li> <li>The precision <b>MUST NOT NEED</b> to be specified during library initialization / object creation. The precision should <b>ONLY</b> be constrained by the available resources of the system.</li> <li>It <b>SHOULD</b> utilize the full power of the platform, and should handle "small" numbers natively. That means on a 64-bit platform, calculating 2^33 + 2^32 should use the available 64-bit CPU instructions. The library <b>SHOULD NOT</b> calculate this in the same way as it does with 2^66 + 2^65 on the same platform.</li> <li>It <b>MUST</b> handle addition (+), subtraction (-), multiplication (*), integer division (/), remainder (%), power (**), increment (++), decrement (--), gcd(), factorial(), and other common integer arithmetic calculations efficiently. Ability to handle functions like sqrt() (square root), log() (logarithm) that do not produce integer results is a plus. Ability to handle <a href="http://en.wikipedia.org/wiki/Symbolic_computation" rel="noreferrer">symbolic computations</a> is even better. </ol> <p>Here are what I found so far:</p> <ol> <li><b>Java</b>'s <b><a href="http://java.sun.com/javase/6/docs/api/java/math/BigInteger.html" rel="noreferrer">BigInteger</a></b> and <b><a href="http://java.sun.com/javase/6/docs/api/java/math/BigDecimal.html" rel="noreferrer">BigDecimal</a></b> class: I have been using these so far. I have read the source code, but I don't understand the math underneath. It may be based on theories / algorithms that I have never learnt.</li> <li>The built-in integer type or in core libraries of <b><a href="http://www.gnu.org/software/bc/manual/html_mono/bc.html" rel="noreferrer">bc</a></b> / <b><a href="http://docs.python.org/release/3.1.2/reference/lexical_analysis.html#integer-literals" rel="noreferrer">Python</a></b> / <b>Ruby</b> / <b>Haskell</b> / <b>Lisp</b> / <b>Erlang</b> / <b>OCaml</b> / <b>PHP</b> / some other languages: I have ever used some of these, but I have no idea on which library they are using, or which kind of implementation they are using.</li> </ol> <p>What I have already known:</p> <ol> <li>Using a <b><i>char</i></b> as a decimal digit, and a <b><i>char*</i></b> as a decimal string and do calculations on the digits using a for-loop.</li> <li>Using an <b><i>int</i></b> (or a <b><i>long int</i></b>, or a <b><i>long long</i></b>) as a basic "unit" and an array of it as an arbitrary long integer, and do calculations on the elements using a for-loop.</li> <li>Using an integer type to store a decimal digit (or a few digits) as <b><a href="http://en.wikipedia.org/wiki/Binary-coded_decimal" rel="noreferrer">BCD (Binary-coded decimal)</a></b>.</li> <li><b><a href="http://en.wikipedia.org/wiki/Booth%27s_multiplication_algorithm" rel="noreferrer">Booth's multiplication algorithm</a></b></li> </ol> <p>What I don't know:</p> <ol> <li> Printing the binary array mentioned above in decimal without using naive methods. Example of a naive method: (1) add the bits from the lowest to the highest: 1, 2, 4, 8, 16, 32, ... (2) use a <b><i>char*</i></b> string mentioned above to store the intermediate decimal results). </li> </ol> <p>What I appreciate:</p> <ol> <li>Good comparisons on <b><a href="http://gmplib.org/" rel="noreferrer">GMP</a></b>, <b><a href="http://www.mpfr.org/" rel="noreferrer">MPFR</a></b>, <b><a href="http://speleotrove.com/decimal/decnumber.html" rel="noreferrer">decNumber</a></b> (or other libraries that are good in your opinion).</li> <li>Good suggestions on books / articles that I should read. For example, an illustration with figures on how an <b>un-naive</b> binary to decimal conversion algorithm works is good. The article <b><a href="http://www.cs.uiowa.edu/~jones/bcd/decimal.html" rel="noreferrer">"Binary to Decimal Conversion in Limited Precision"</a></b> by Douglas W. Jones is an example of a good article.</li> <li>Any help.</li> </ol> <p>Please <b>DO NOT</b> answer this question if:</p> <ol> <li>you think using a <b><i>double</i></b> (or a <b><i>long double</i></b>, or a <b><i>long long double</i></b>) can solve this problem easily. If you do think so, it means that you don't understand the issue under discussion.</li> </ol>
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    1. COAs far as I can see, GMP seems to be a good library. What I wonder is why there's a need for the contributors of Python / Haskell / Erlang / etc to re-invent the wheel. If GMP is so good, why don't rely on it? The GPL / LGPL license may be one of the issues, but despite of this (and also the rounding mode issue), are there any other disadvantages of GMP? Is the built-in integer of Python / Haskell / Erlang / any cryptography library faster than GMP? If so, I would like to extract and use it, if license permits.
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    2. COI found a nice article at http://www.cs.uiowa.edu/~jones/bcd/decimal.html by Douglas W. Jones. The article describes a method to convert a 16-bit integer to decimal representation using only 8-bit integer arithmetic. The idea is to break the 16-bit number into 4 nibbles, each representing a base-16 "digit". So, digit-0 (n0) represents x1, n1 => x16, n2 => x256, n3 => x4096. Then, it is obvious that digit-0 of the decimal number (d0) is digit-0 of the result of n0 * 1 + n1 * 6 + n2 * 6 + n3 * 6. By handling the carry properly, d1 to d4 can also be computed.
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    3. COHowever, as far as I could imagine, Douglas's idea above could not be extended to handle arbitrarily long binary integers. It is because the numbers 1 (16^0), 16 (16^1), 256 (16^2) and 4096 (16^3) are pre-calculated. The problem then becomes how to represent 16^n in decimal for arbitrarily large n.
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