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	<h1>A <a href="https://github.com/czurnieden/Little" title="Little"><span style="font-variant:small-caps;font-weight:bold">Little</span></a> arbitrary precision in JavaScript</h1>
 
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			<header>
				<h2 class="entry-title">Try it out</h2>
			</header>
			<div class="entry-content">
				<p>Try the raw Biginteger and Bigrational libraries, 
                                   short description with some examples below.
                                </p>
<p>
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			<header>
				<h2 class="entry-title" style="margin-left:10em;padding-bottom:3ex;" >Function List</h2>
			</header>
			<div class="entry-content">
<h3 style="margin-left:5em">Some Examples</h3>
<p>
    Most of the functions, especially the basic functions are implemented as prototypes to
    the Bigintgers and Bigrationals respectively, so a simple addition of one and one to get
    two as the expected result (we assume decimal output here) needs the following, ready to
    be put into the textarea above:
</p>
<pre>
var one = new Bigint(1);
var result = one.add(one);
result.toString();
</pre>
<p>
You can give the <code>Bigint</code> constructor small numbers (up to 2<sup>26</sup>) directly,
 larger numbers (up to 2<sup>53</sup>) need to be given indirectly:
</p>
<pre>
var num = 134217728;
var onehundredthirtyfourmilliontwohundredseventeenthousandsevenhundredtwentyeight = num.toBigint();
onehundredthirtyfourmilliontwohundredseventeenthousandsevenhundredtwentyeight.toString();
</pre>
<p>
Even larger numbers need to be given as strings:
</p>
<pre>
var largenumberasastring = "82756729834529384756298356298375629834756298345623894756";
var largenumber = largenumberasastring.toBigint();
largenumber.toString();
</pre>
<p>
The rational number follow the same style with the exception of the second format. You can give
small number directly:
</p>
<pre>
var smallfraction = new Bigrational(234,432);
smallfraction.toString();
</pre>
<p>
Caution: the constructor does no reducing! You have to do it by hand, either in-place
with <code>bigrat.normalize()</code> or on a copy with <code>bigrat.reduce()</code>
</p>
<pre>
var smallfraction = new Bigrational(234,432);
smallfraction.normalize();
smallfraction.toString();
</pre>
<p>
Larger numbers as strings:
</p>
<pre>
var largefractionasastring = "984756398475t3987539/5847698w347349857649569487";
largefraction = largefractionasastring.toBigrational();
largefraction.normalize();
largefraction.toString();
</pre>
<p>
Larger numbers as two Bigintegers:
</p>
<pre>
var numerator   = "984756398475t39875395847698w347349857649569487";
var denominator = "8736452873452873465287346528365823756873465t823746";
numerator = numerator.toBigint();
denominator = denominator.toBigint();
largefraction = new Bigrational(numerator, denominator);
largefraction.normalize();
largefraction.toString();
</pre>
<p>
Bigrational as two Bigints:
<pre>
var numerator   = "-984756398475t39875395847698w347349857649569487";
var denominator = "8736452873452873465287346528365823756873465t823746";
numerator = numerator.toBigint();
denominator = denominator.toBigint();
largefraction = new Bigrational(numerator, denominator);
largefraction.normalize();
numerator2 = largefraction.num;
denominator2 = largefraction.den;
sign = largefraction.sign;
alert(numerator2.toString() + "\r\n" + denominator2.toString() + "\r\n" + sign);
</pre>
</p>
<p>
Chaining is possible but does not work between the Bigints and Bigrationals automatically;
these are the raw libraries. You need to do it manually as described above.<br>
Otherwise:
</p>
<pre>
var a = new Bigint(123);
var b = new Bigint(243);
/* c = a * a + b * b */
var c = a.mul(a).add(b.mul(b));
/* c = (a * a + b) * b */
var d = a.mul(a).add(b).mul(b);
"c = " + c.toString() + "\r\nd = " + d.toString();
</pre>
<p>
Finaly, compute the Riemann Zeta function at s = 3 (Apéry's constant) to 100
decimal digits precision.<br>
Please be patient, this may last a while. Most of the time is spent in the
function <code>D()</code> b.t.w. which is to the highest extend unoptimised.<br>
Borwein, Peter. <i>An efficient algorithm for the Riemann zeta function.</i>
Canadian Mathematical Society Conference Proceedings. Vol. 27. 2000
</p>
<pre>
function D(n) {
    var ret, sum, i;
    var ret = new Array(n + 1);
    var sum = new Bigrational(0, 1);
    var N = new Bigint(n);
    var num, den;
    for (i = 0; i <= n; i++) {
        num = factorial(n + i - 1).lShift(2 * i);
        den = factorial(n - i).mul(factorial(2 * i));
        num = new Bigrational(num, den);
        num.normalize();
        sum = sum.add(num)
        ret[i] = N.mul(sum.num).div(sum.den);
    }
    return ret;
}
function zeta(s, n) {
    var Ds, sum, k;
    Ds = D(n);
    sum = new Bigrational(0, 1);
    var sign = 1;
    var t1, t2, t3, t4, sm1;
    for (k = 0; k &lt; n; k++) {
        t1 = Ds[k].sub(Ds[n]);
        t1.sign = (sign > 0) ? MP_ZPOS : MP_NEG;
        sign = -sign;
        // here is the reason why large s do need more time
        t2 = (k + 1).bigpow(s);
        t3 = new Bigrational(t1, t2);
        t3.normalize();
        sum = sum.add(t3);
    }
    sm1 = 1 - s;
    if (sm1 &lt; 0) {
        t3 = new Bigint(1);
        t3.lShiftInplace(-sm1);
        t4 = new Bigint(1)
        t3 = new Bigrational(t4, t3);
        t2 = (new Bigrational(1, 1)).sub(t3)
    } else {
        //not yet implemented although Bernoulli numbers are
        return (new Bigrational()).setNaN();
    }
    t3 = new Bigrational(Ds[n], new Bigint(1))
    t1 = t3.mul(t2)
    t1.inverse();
    sum = sum.mul(t1);
    return sum;
}
var ret = zeta(3,Math.ceil(100*1.3));
var multiplier = (10).bigpow(100);
var t = ret.num.mul(multiplier);
var rs = t.div(ret.den).toString();
rs.slice(0,1)+"."+rs.slice(1,rs.length)
</pre>
<p>
Yes, the last two lines do not really fit, but if we had a working implementation of
an arbitray precision floating point number, we wouldn't have to do it so complicated ;-)
</p>
<h3 style="margin-left:5em;padding-top:3ex;">Common Functions</h3>
Same name for both Bigint and Bigrational (do not mix, they are just the same names).
Some do work for JavaScript's native numbers, too, if indicated with <code>Number</code>.<br>
Despite of their similar names they might work differently. The square root for example does an
integer (truncating) square root for Bigintegers but a normal square root with a fractional part
to arbitrary but preset precision for Bigrationals.
<dl>
<dt>add</dt><dd>Addition</dd>
<dt>addInt</dt><dd>Addition with second argument a small integer</dd>
<dt>sub</dt><dd>Subtraction</dd>
<dt>subInt</dt><dd>Subtraction with second argument a small integer</dd>
<dt>mul</dt><dd>Multiplication</dd>
<dt>mulInt</dt><dd>Multiplication with second argument a small integer</dd>
<dt>div</dt><dd>Division</dd>
<dt>divInt</dt><dd>Division with second argument a small integer</dd>
<dt> sqr </dt><dd> square </dd>
<dt> sqrt </dt><dd> square root  </dd>
<dt> nthroot </dt><dd> n<sup>th</sup>-root (Newton)</dd>
<dt>abs </dt><dd>return absolute value</dd>
<dt>copy </dt><dd> returns a deep copy </dd>
<dt>free</dt><dd>An offering to the garbage-collector. To free it completely it needs an extra
 <code>oldbignumber = null;</code></dd>
<dt> cmp </dt><dd> compare. returns MP_GT if the second one is greater, MP_LT if it is smaller
 and MP_EQ if both are equal </dd>
<dt>isNaN</dt><dd>Is the Number not a number? <code>Number</code></dd>
<dt>setNaN</dt><dd>The Number is not a number</dd>
<dt>isFinite </dt><dd> is it finite? </dd>
<dt>isInf </dt><dd> is it not finite? </dd>
<dt>setInf </dt><dd> tell that it not finite? </dd>
<dt>isEven</dt><dd>Is the Number even? <code>Number</code></dd>
<dt>isOdd</dt><dd>Is the Number odd? <code>Number</code></dd>
<dt>isOne </dt><dd> is it 1 (one)? </dd>
<dt>isUnity</dt><dd> is it one or minus one? </dd>
<dt> isZero </dt><dd> is it zero? </dd>
<dt> neg </dt><dd> change sign of copy and return it </dd>
<dt>sign</dt><dd>Sign of the number <code>Number</code></dd>
<dt>toString</dt><dd>return value as a string <code>Number</code><br>
The <code>Bigint</code> function <code>toString</code> and hence the <code>Bigrational</code>
one, too, accepts more bases (2-62) than the native one of the <code>Number</code> object (2-36).
</dd>
</dl>
There is also a <a href="https://github.com/czurnieden/primesieve/">primesieve</a> available.
This program offers ten functions which are in alphabetical order:

<dl>
<dt><code>fill(amount)</code> </dt>
<dd>
Build a prime sieve up to the amount <code>amount</code>. Allows for skipping of the
automated adapting of the sieve-size if <code>amount</code> is chosen high enough.
<br>
Returns: <code>undefined</code> in case of an error
<br>
See also: <code>grow(amount)</code>
</dd>
<dt><code>grow(amount)</code></dt>
<dd>
Grows the prime sieve by the amount <code>amount</code>. It just builds a new prime sieve with
a different limit (if higher then the already existing limit) for now. Mostly
used internally.
<br>
Returns:  <code>undefined</code> in case of an error
<br>
See also: <code>fill(amount)</code>
</dd>
<dt><code>isSmallPrime(number)</code></dt>
<dd>
Tests if <code>number</code> is a prime. Automatically builds a sieve if <code>number</code> is larger then the existing sieve.
<br>
Returns: <code>true</code> or <code>false</code> if <code>number</code> is a prime or not, respectively or <code>undefined</code> in case of an error.
<br>
See also: <code>Primesieve.raiseLimit(limit)</code>
</dd>
<dt><code>nextPrime(number)</code> </dt>
<dd>
Searches for the nearest prime larger than <code>number</code>. Automatically builds a
sieve if <code>number</code> is larger then the existing sieve.
<br>
Returns: <code>Number</code> or <code>undefined</code> in case of an error
</dd>

<dt><code>precPrime(number)</code></dt>
<dd>
Searches for the nearest prime smaller than <code>number</code>. Does <strong>not</strong> automatically build a sieve if <code>number</code> is larger then the existing sieve. Should it?
<br>
Returns: <code>Number</code> or <code>undefined</code> in case of an error
</dd>
<dt><code>primePiApprox(number)</code> </dt>
<dd>
The approximated result (upper bound) of the prime counting function for
<code>number</code>
<br>
Returns: <code>Number</code> or <code>undefined</code> in case of an error
</dd>

<dt><code>primePi(number)</code></dt>
<dd>
The exact result of the prime counting function for <code>number</code>. Automatically
builds a sieve if <code>number</code> is larger then the existing sieve.
<br>
Returns: <code>Number</code> or <code>undefined</code> in case of an error
</dd>

<dt><code>primeRange(low,high)</code></dt>
<dd>
Calculates the range of primes between <code>low</code> and <code>high</code>. It will detect if <code>low</code> and <code>high</code> are reversed and reruns itself with the arguments reversed.
Automatically builds a sieve if <code>number</code> is larger then the existing sieve.
<br>
Returns: <code>Array</code> or <code>undefined</code> in case of an error
</dd>
<dt><code>primes(number)</code></dt>
<dd>
Calculates primes up to and including <code>number</code>. Automatically builds a sieve if
<code>number</code> is larger then the existing sieve.
<br>
Returns: <code>Array</code> or <code>undefined</code> in case of an error
</dd>
<dt><code>raiseLimit(number)</code></dt>
<dd>
This module <code>primesieve</code> has a limit for the maximum size for the automatic
sieve building. It is pre-defined at <code>0x800000</code> which is one megabyte. This
function can raise it to the manager.
<br>
Returns: <code>undefined</code> in case of an error
</dd>
<dt><code>sieve()</code></dt>
<dd>
Get the raw prime sieve. This function is not implemented in the CLI version.
<br>
Returns: The raw prime sieve, either an `UInt32Array` or a normal `Array`
</dd>
<dt><code>sterror()</code></dt>
<dd>
An error number to string conversion. The following errors are used:
<ol>
<li>"Success"</li>
<li>"Argument not an integer"</li>
<li>"Argument too low"</li>
<li>"Argument too high"</li>
<li>"Prime wanted is higher than the limit"</li>
<li>"Unknown error"</li>
</ol>
The numbers of the list correspond to the error numbers.
<br>
Returns: <code>String</code>
<br>
See also: <code>error</code>
</dd>
<dt><code>error</code></dt>
<dd>
Variable holding the error number. For a table of errors see <code>strerror()</code>
</dd>
</dl>


<h3 style="margin-left:5em">Biginteger</h3>
<dl>
<dt> and </dt><dd> binary arithmetic <code>c = a&b</code>  </dd>
<dt> clamp</dt><dd> strip leading zeros </dd>
<dt> clearBit </dt><dd> clear bit (set to zero) of <code>this</code> </dd>
<dt> clear</dt><dd> An offering to the garbage-collector together with a set-to-zero of the <code>Bigint</code> for later use </dd>
<dt> decr </dt><dd> subtract one from <code>this</code> (like <code>i--;</code>) in-place  </dd>
<dt> digits </dt><dd> number of decimal digits (return one for zero)</dd>
<dt> divmod </dt><dd> returns a tuple with the quotient and the modulus respectively, rounding to -inf (like in PARI/GP for example). Differs from <code>divrem</code> when some of the arguments are negative</dd>
<dt> divrem </dt><dd> returns a tuple with the quotient and the reminder respectively</dd>
<dt> divremInt  </dt><dd> like <code>divrem</code> but the second argument can be a small integer </dd>
<dt> dlShift </dt><dd> multiply with 2<sup>bigbase</sup>, return copy </dd>
<dt> dlShiftInplace </dt><dd> multiply with 2<sup>bigbase</sup> in-place </dd>
<dt> drShift </dt><dd> divide by 2<sup>bigbase</sup>, return copy </dd>
<dt> drShiftInplace </dt><dd> divide by 2<sup>bigbase</sup> in-place</dd>
<dt> egcd </dt><dd> extended gcd </dd>
<dt> exactDiv3 </dt><dd> works if the number is divisible by three without a reminder </dd>
<dt> flipBit </dt><dd> flip bit of <code>this</code> </dd>
<dt> gcd </dt><dd> compute the greatest common denominator </dd>
<dt> getBit </dt><dd> get bit of <code>this</code> </dd>
<dt> grow </dt><dd> grows the digits array without changing <code>this.used</code>. Returns the new length of the digit array  </dd>
<dt> highBit </dt><dd> highest bit set (zero based) <code>Number</code> </dd>
<dt> ilog2 </dt><dd> integer logarith base 2 (actually just <code>highBit() + 1</code>) </dd>
<dt> ilogb(base) </dt><dd> integer logarithm of base <code>base</code>  </dd>
<dt> incr </dt><dd> add one to <code>this</code> (like <code>i++;</code>) in-place </dd>
<dt> isNeg </dt><dd>  is the Bigint smaller than zero?</dd>
<dt> isOk </dt><dd> is a number and is finite  </dd>
<dt> isPos </dt><dd>is the Bigint greater than zero?  </dd>
<dt> isPow2 </dt><dd> is it a power of two? <code>Number</code> </dd>
<dt> jacobi </dt><dd> Jacobi function  </dd>
<dt> lcm </dt><dd> lowest common multiplicator </dd>
<dt> lowBit </dt><dd>  lowest bit set (zero based) <code>Number</code>  </dd>
<dt> lShift </dt><dd> multiply with 2<sup>n</sup>, return copy  </dd>
<dt> lShiftInplace </dt><dd> multiply with 2<sup>n</sup> in-place  </dd>
<dt> mask </dt><dd> create a bitmask <code>c = (1&lt;&lt;n)-1</code> </dd>
<dt> mod2d </dt><dd> returns reminder of division by  2<sup>n</sup></dd>
<dt> modInv </dt><dd> modular inverse <code>Number</code></dd>
<dt> mulInt </dt><dd> multiply with a small integer (&lt; 2<sup>bigbase</sup>)  </dd>
<dt> not </dt><dd> binary arithmetic <code>c = ~a</code>  </dd>
<dt> notInplace </dt><dd> binary arithmetic <code>a = ~a</code>  </dd>
<dt> nthrootold </dt><dd>  integer n<sup>th</sup>-root (bracketing) </dd>
<dt> numSetBits </dt><dd> Hamming weight </dd>
<dt> or </dt><dd> binary arithmetic <code>c = a|b</code>  </dd>
<dt> pow </dt><dd> returns <code>this</code> to the power of a small integer or a Bigint</dd>
<dt> random (bits,seed)</dt><dd> A random Bigint of <code>bits</code> length in bits and optional small integer as the seed to the underlying PRNG (Bob Jenkins' small PRNG (with the extra round))</dd>
<dt> rShift </dt><dd> divide by 2<sup>n</sup>, return copy  </dd>
<dt> rShiftInplace </dt><dd> divide by 2<sup>n</sup> in-place  </dd>
<dt> rShiftRounded </dt><dd> divide by 2<sup>n</sup>, round to +inf, return copy  </dd>
<dt> setBit </dt><dd> set bit (set to one) of <code>this</code> </dd>
<dt> swap </dt><dd> swap two Bigints (with deep copy) </dd>
<dt> toNumber </dt><dd> Bigint to JavaScript Number </dd>
<dt> xor </dt><dd> binary arithmetic <code>c = a^b</code>  </dd>

</dl>

<h3 style="margin-left:5em">Bigrational</h3>
<dl>
<dt>bernoulli</dt><dd> Bernoulli numbers  </dd>
<dt>fastround</dt><dd> Faster rounding method but with larger results </dd>
<dt>fitsBitPrecision</dt><dd> If number fits a given bit-precision </dd>
<dt>fitsDecPrecision</dt><dd>  If number fits a given decimal-precision   </dd>
<dt>inverse</dt><dd>  Inverses the fraction  in-place</dd>
<dt>isBitPrecision</dt><dd> If number is of given bit-precision  </dd>
<dt>isDecPrecision</dt><dd> If number is of given decimal-precision   </dd>
<dt>isEpsGreaterZero</dt><dd> greater than zero plus EPS  </dd>
<dt>isEpsLowerZero</dt><dd> lower than zero minus EPS   </dd>
<dt>isEpsZero</dt><dd> number inside range zero plus/minus EPS  </dd>
<dt>isInt</dt><dd> is the number an integer, that is, is the denominator one  </dd>
<dt>isLowerOne</dt><dd> Is the absolute value between zero and one  </dd>
<dt>normalize</dt><dd>  Reduce the fraction in-place </dd>
<dt>parts</dt><dd> returns the parts of the fraction splitted into the integer part and the fraction part  </dd>
<dt>powInt</dt><dd>  to the power of a small integer </dd>
<dt>reciprocal</dt><dd> returns reciprocal of the fraction </dd>
<dt>reduce</dt><dd>  returns the reduced fraction   </dd>
<dt>round</dt><dd> Median rounding  </dd>
</dl>

<h3 style="margin-left:5em">Integer Functions</h3>
These are normal JavaScript functions, no prototypes.<br>
There are some tricks used with the factors of factorials, the functions can be
used themself but they were not meant for that and hence are tricky to use. Please
see the source for more information until proper documentation has been written.
<dl>
<dt>add_factored_factorial</dt><dd> </dd>
<dt>compute_factored_factorial</dt><dd> </dd>
<dt>compute_signed_factored_factorial</dt><dd> </dd>
<dt>factor_factorial</dt><dd> </dd>
<dt>negate_factored_factorials</dt><dd> </dd>
<dt>power_factored_factorials</dt><dd> </dd>
<dt>subtract_factored_factorials( subtrahend,</dt><dd> </dd>
<dl>
The normal integer functions are:
<dt>bell</dt><dd><a href="http://en.wikipedia.org/wiki/Bell_number">Bell numbers</a></dd>
<dt>binomial</dt><dd><a href="http://en.wikipedia.org/wiki/Binomial_coefficient">Binomial coefficients</a> </dd>
<dt>catalan</dt><dd><a href="http://en.wikipedia.org/wiki/Catalan_number">Catalan numbers</a></dd>
<dt>doublefactorial</dt><dd><a href="http://en.wikipedia.org/wiki/Double_factorial">Doublefactorial</a></dd>
<dt>euler</dt><dd><a href="http://en.wikipedia.org/wiki/Euler_number">Eulerian (zig) numbers</a></dd>
<dt>factorial</dt><dd><a href="http://en.wikipedia.org/wiki/Factorial">Factorial</a></dd>
<dt>fallingfactorial</dt><dd><a href="http://en.wikipedia.org/wiki/Pochhammer_symbol">Falling factorial</a></dd>
<dt>fibonacci</dt><dd><a href="http://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci numbers (fast algorithm using matrix exponentiation)</a> </dd>
<dt>free_stirling1cache</dt><dd>Make offerings to the garbage-collector </dd>
<dt>free_stirling2cache</dt><dd>Make offerings to the garbage-collector </dd>
<dt>prime_divisors</dt><dd>How often does the prime p divide the factorial of the number n </dd>
<dt>risingfactorial</dt><dd><a href="http://en.wikipedia.org/wiki/Pochhammer_symbol">Rising factorial<</a>/dd>
<dt>stirling1</dt><dd><a href="http://en.wikipedia.org/wiki/Stirling_number">Stirling numbers of the first kind (cached)</a> </dd>
<dt>stirling2</dt><dd><a href="http://en.wikipedia.org/wiki/Stirling_number">Stirling numbers of the second kind (series, faster for small numbers) </a></dd>
<dt>stirling2caching</dt><dd> <a href="http://en.wikipedia.org/wiki/Stirling_number">Stirling numbers of the second kind (cached)</a></dd>
<dt>subfactorial</dt><dd><a href="http://en.wikipedia.org/wiki/Derangement">Sub-factorial (derangement) </a></dd>
<dt>superfactorial</dt><dd><a href="http://mathworld.wolfram.com/Superfactorial.html">Superfactorial (as defined by Sloane and Plouffe)</a></dd>
</dl>
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