100 mpmath one-liners for pi

Since it's pi day today, I thought I'd share a list of mpmath one-liners for computing the value of pi to high precision using various representations in terms of special functions, infinite series, integrals, etc. Most of them can already be found as doctest examples in some form in the mpmath documentation.

A few of the formulas explicitly involve pi. Using those to calculate pi is rather circular (!), though a few of them could still be used for computing pi using numerical root-finding. In any case, most of the formulas are circular even when pi doesn't appear explicitly since mpmath is likely using its value internally. In any further case, the majority of the formulas are not efficient for computing pi to very high precision (at least as written). Still, ~50 digits is no problem. Enjoy!

from mpmath import *
mp.dps = 50; mp.pretty = True

+pi
180*degree
4*atan(1)
16*acot(5)-4*acot(239)
48*acot(49)+128*acot(57)-20*acot(239)+48*acot(110443)
chop(2*j*log((1-j)/(1+j)))
chop(-2j*asinh(1j))
chop(ci(-inf)/1j)
gamma(0.5)**2
beta(0.5,0.5)
(2/diff(erf, 0))**2
findroot(sin, 3)
findroot(cos, 1)*2
chop(-2j*lambertw(-pi/2))
besseljzero(0.5,1)
3*sqrt(3)/2/hyp2f1((-1,3),(1,3),1,1)
8/(hyp2f1(0.5,0.5,1,0.5)*gamma(0.75)/gamma(1.25))**2
4*(hyp1f2(1,1.5,1,1) / struvel(-0.5, 2))**2
1/meijerg([[],[]], [[0],[0.5]], 0)**2
(meijerg([[],[2]], [[1,1.5],[]], 1, 0.5) / erfc(1))**2
(1-e) / meijerg([[1],[0.5]], [[1],[0.5,0]], 1)
sqrt(psi(1,0.25)-8*catalan)
elliprc(1,2)*4
elliprg(0,1,1)*4
2*agm(1,0.5)*ellipk(0.75)
(gamma(0.75)*jtheta(3,0,exp(-pi)))**4
cbrt(gamma(0.25)**4*agm(1,sqrt(2))**2/8)
sqrt(6*zeta(2))
sqrt(6*(zeta(2,3)+5./4))
sqrt(zeta(2,(3,4))+8*catalan)
exp(-2*zeta(0,1,1))/2
sqrt(12*altzeta(2))
4*dirichlet(1,[0,1,0,-1])
2*catalan/dirichlet(-1,[0,1,0,-1],1)
exp(-dirichlet(0,[0,1,0,-1],1))*gamma(0.25)**2/(2*sqrt(2))
sqrt(7*zeta(3)/(4*diff(lerchphi, (-1,-2,1), (0,1,0))))
sqrt(-12*polylog(2,-1))
sqrt(6*log(2)**2+12*polylog(2,0.5))
chop(root(-81j*(polylog(3,root(1,3,1))+4*zeta(3)/9)/2,3))
2*clsin(1,1)+1
(3+sqrt(3)*sqrt(1+8*clcos(2,1)))/2
root(2,6)*sqrt(e)/(glaisher**6*barnesg(0.5)**4)
nsum(lambda k: 4*(-1)**(k+1)/(2*k-1), [1,inf])
nsum(lambda k: (3**k-1)/4**k*zeta(k+1), [1,inf])
nsum(lambda k: 8/(2*k-1)**2, [1,inf])**0.5
nsum(lambda k: 2*fac(k)/fac2(2*k+1), [0,inf])
nsum(lambda k: fac(k)**2/fac(2*k+1), [0,inf])*3*sqrt(3)/2
nsum(lambda k: fac(k)**2/(phi**(2*k+1)*fac(2*k+1)), [0,inf])*(5*sqrt(phi+2))/2
nsum(lambda k: (4/(8*k+1)-2/(8*k+4)-1/(8*k+5)-1/(8*k+6))/16**k, [0,inf])
2/nsum(lambda k: (-1)**k*(4*k+1)*(fac2(2*k-1)/fac2(2*k))**3, [0,inf])
nsum(lambda k: 72/(k*expm1(k*pi))-96/(k*expm1(2*pi*k))+24/(k*expm1(4*pi*k)), [1,inf])
1/nsum(lambda k: binomial(2*k,k)**3*(42*k+5)/2**(12*k+4), [0,inf])
4/nsum(lambda k: (-1)**k*(1123+21460*k)*fac2(2*k-1)*fac2(4*k-1)/(882**(2*k+1)*32**k*fac(k)**3), [0,inf])
9801/sqrt(8)/nsum(lambda k: fac(4*k)*(1103+26390*k)/(fac(k)**4*396**(4*k)), [0,inf])
426880*sqrt(10005)/nsum(lambda k: (-1)**k*fac(6*k)*(13591409+545140134*k)/(fac(k)**3*fac(3*k)*(640320**3)**k), [0,inf])
4/nsum(lambda k: (6*k+1)*rf(0.5,k)**3/(4**k*fac(k)**3), [0,inf])
(ln(8)+sqrt(48*nsum(lambda m,n: (-1)**(m+n)/(m**2+n**2), [1,inf],[1,inf]) + 9*log(2)**2))/2
-nsum(lambda x,y: (-1)**(x+y)/(x**2+y**2), [-inf,inf], [-inf,inf], ignore=True)/ln2
2*nsum(lambda k: sin(k)/k, [1,inf])+1
quad(lambda x: 2/(x**2+1), [0,inf])
quad(lambda x: exp(-x**2), [-inf,inf])**2
2*quad(lambda x: sqrt(1-x**2), [-1,1])
chop(quad(lambda z: 1/(2j*z), [1,j,-1,-j,1]))
3*(4*log(2+sqrt(3))-quad(lambda x,y: 1/sqrt(1+x**2+y**2), [-1,1],[-1,1]))/2
sqrt(8*quad(lambda x,y: 1/(1-(x*y)**2), [0,1],[0,1]))
sqrt(6*quad(lambda x,y: 1/(1-x*y), [0,1],[0,1]))
sqrt(6*quad(lambda x: x/expm1(x), [0,inf]))
quad(lambda x: (16*x-16)/(x**4-2*x**3+4*x-4), [0,1])
quad(lambda x: sqrt(x-x**2), [0,0.25])*24+3*sqrt(3)/4
mpf(22)/7 - quad(lambda x: x**4*(1-x)**4/(1+x**2), [0,1])
mpf(355)/113 - quad(lambda x: x**8*(1-x)**8*(25+816*x**2)/(1+x**2), [0,1])/3164
2*quadosc(lambda x: sin(x)/x, [0,inf], omega=1)
40*quadosc(lambda x: sin(x)**6/x**6, [0,inf], omega=1)/11
e*quadosc(lambda x: cos(x)/(1+x**2), [-inf,inf], omega=1)
8*quadosc(lambda x: cos(x**2), [0,inf], zeros=lambda n: sqrt(n))**2
2*quadosc(lambda x: sin(exp(x)), [1,inf], zeros=ln)+2*si(e)
exp(2*quad(loggamma, [0,1]))/2
2*nprod(lambda k: sec(pi/2**k), [2,inf])
s=lambda k: sqrt(0.5+s(k-1)/2) if k else 0; 2/nprod(s, [1,inf])
s=lambda k: sqrt(2+s(k-1)) if k else 0; limit(lambda k: sqrt(2-s(k))*2**(k+1), inf)
2*nprod(lambda k: (2*k)**2/((2*k-1)*(2*k+1)), [1,inf])
2*nprod(lambda k: (4*k**2)/(4*k**2-1), [1, inf])
sqrt(6*ln(nprod(lambda k: exp(1/k**2), [1,inf])))
nprod(lambda k: (k**2-1)/(k**2+1), [2,inf])/csch(pi)
nprod(lambda k: (k**2-1)/(k**2+1), [2,inf])*sinh(pi)
nprod(lambda k: (k**4-1)/(k**4+1), [2, inf])*(cosh(sqrt(2)*pi)-cos(sqrt(2)*pi))/sinh(pi)
sinh(pi)/nprod(lambda k: (1-1/k**4), [2, inf])/4
sinh(pi)/nprod(lambda k: (1+1/k**2), [2, inf])/2
(exp(1+euler/2)/nprod(lambda n: (1+1/n)**n * exp(1/(2*n)-1), [1, inf]))**2/2
3*sqrt(2)*cosh(pi*sqrt(3)/2)**2*csch(pi*sqrt(2))/nprod(lambda k: (1+1/k+1/k**2)**2/(1+2/k+3/k**2), [1, inf])
2/e*nprod(lambda k: (1+2/k)**((-1)**(k+1)*k), [1,inf])
limit(lambda k: 16**k/(k*binomial(2*k,k)**2), inf)
limit(lambda x: 4*x*hyp1f2(0.5,1.5,1.5,-x**2), inf)
1/log(limit(lambda n: nprod(lambda k: pi/(2*atan(k)), [n,2*n]), inf),4)
limit(lambda k: 2**(4*k+1)*fac(k)**4/(2*k+1)/fac(2*k)**2, inf)
limit(lambda k: fac(k) / (sqrt(k)*(k/e)**k), inf)**2/2
limit(lambda k: (-(-1)**k*bernoulli(2*k)*2**(2*k-1)/fac(2*k))**(-1/(2*k)), inf)
limit(lambda k: besseljzero(1,k)/k, inf)
1/limit(lambda x: airyai(x)*2*x**0.25*exp(2*x**1.5/3), inf, exp=True)**2
1/limit(lambda x: airybi(x)*x**0.25*exp(-2*x**1.5/3), inf, exp=True)**2

A FLINT example: Lambert W function power series

Two days ago, a new version of the Fast Library for Number Theory (FLINT) was released. I contributed a lot of new code to this release, including linear algebra speed improvements and new functionality for fast power series arithmetic and computation of special numbers and polynomials (see the release announcement and some of my benchmarking results).

In this blog post I'll demonstrate how to do power series arithmetic with FLINT, using its fmpq_poly module which implements polynomials over the rational numbers Q. Standard operations (addition, multiplication and division) were available before; the functions I've added include square root, log, exp, sin, tan, atan, etc. (all the usual elementary functions). The same functions are also available for power series over a finite field Z/nZ (with word-size n). Everything is asymptotically fast (the running time is linear in the size of the output, up to logarithmic factors).

Of course, transcendental functions are a bit restricted when considered over Q or Z/nZ, since it's only possible to obtain power series expansions at specific rational points (in most cases just x = 0). So at present, some very interesting numerical applications of fast power series arithmetic are not supported. But some time in the future, we'll probably add support for numerical power series over the reals and complexes as well.

As today's example, let us implement the Lambert W function for the power series ring Q[[x]]. The Lambert W function is defined implicitly by the equation x = W(x) exp(W(z)), which can be solved using Newton iteration with the update step w = w - (w exp(w) - x) / ((w+1) exp(w)).

Power series Newton iteration is just like numerical Newton iteration, except that the convergence behavior is much simpler: starting with a correct first-order expansion, each iteration at least doubles the number of correct coefficients.

A simple recursive implementation with asymptotically optimal performance (up to constant factors) looks as follows:

#include <stdio.h>
#include "flint.h"
#include "fmpq_poly.h"

void lambertw(fmpq_poly_t w, fmpq_poly_t x, long n)
{
    if (n == 1)
    {
        fmpq_poly_zero(w);
    }
    else
    {
        fmpq_poly_t t, u, v;

        lambertw(w, x, (n + 1) / 2);

        fmpq_poly_init(t);
        fmpq_poly_init(u);
        fmpq_poly_init(v);

        fmpq_poly_exp_series(t, w, n);
        fmpq_poly_mullow(u, t, w, n);
        fmpq_poly_sub(v, u, x);
        fmpq_poly_add(t, u, t);
        fmpq_poly_div_series(u, v, t, n);
        fmpq_poly_sub(w, w, u);

        fmpq_poly_clear(t);
        fmpq_poly_clear(u);
        fmpq_poly_clear(v);
    }
}

Beyond the base case W(x) = 0 + O(x), the function just computes w to accuracy ceil(n/2), and then extends it to accuracy n using a single Newton step. As we can see, C code directly using the FLINT library interface gets a bit verbose, but this style has the advantage of giving precise control over temporary memory allocation, polynomial lengths, etc. (it is very similar to the interface of GMP/MPIR).

We add a simple test main routine:

int main()
{
   fmpq_poly_t x;
   fmpq_poly_t w;

   fmpq_poly_init(x);
   fmpq_poly_init(w);

   fmpq_poly_set_coeff_ui(x, 1, 1);
   lambertw(w, x, 10);

   fmpq_poly_print_pretty(w, "x");
   printf("\n");

   fmpq_poly_clear(x);
   fmpq_poly_clear(w);
}

The output of the program is:

531441/4480*x^9 - 16384/315*x^8 + 16807/720*x^7 - 54/5*x^6 + 125/24*x^5 - 8/3*x^4 + 3/2*x^3 - 1*x^2 + 1*x

It is well known that the coefficients in this series are given in closed form by (-k)^k-1 / k!, so we can check that the output is correct.

Computing 1000 terms takes just a few seconds. If this sounds like much, remember that the coefficients grow rapidly: together, the computed numerators and denominators have over 2 million digits!

So far this is perhaps not so interesting, as we could compute the coefficients faster using a direct formula. But the nice thing is that arbitrary compositions are allowed, i.e we can compute W(f(x)) for any given power series f, and this will still be just as fast.

Let's consider a nontrivial example: the infinite "power tower" T(z) = z^zzz... A moment's reflection shows that this is an analytic function with a rational power series expansion around z = 1. In fact, we have explicitly T(z) = W(-log(z))/(-log(z)). We can compute this series expansion (in the shifted variable x = z - 1) as follows:

int main()
{
    fmpq_poly_t x;
    fmpq_poly_t w;

    long n = 10;

    fmpq_poly_init(x);
    fmpq_poly_init(w);

    fmpq_poly_set_coeff_ui(x, 0, 1);
    fmpq_poly_set_coeff_ui(x, 1, 1);
    fmpq_poly_log_series(x, x, n + 1);
    fmpq_poly_neg(x, x);
    lambertw(w, x, n + 1);
    fmpq_poly_shift_right(w, w, 1);
    fmpq_poly_shift_right(x, x, 1);
    fmpq_poly_div_series(w, w, x, n);

    fmpq_poly_print_pretty(w, "x");
    printf("\n");

    fmpq_poly_clear(x);
    fmpq_poly_clear(w);
}

The only complication is that fmpq_poly_div_series requires a nonzero leading coefficient in the denominator, so we must shift both series down one power.

The program outputs:

118001/2520*x^9 + 123101/5040*x^8 + 4681/360*x^7 + 283/40*x^6 + 4*x^5 + 7/3*x^4 + 3/2*x^3 + 1*x^2 + 1*x + 1

To make things nicer, we assume that the coefficients have the form a_k / k! (i.e. that T(z) is the exponential generating function for a_k) and change the output code to something like the following:

    long k;
    mpq_t t;
    mpz_t u;
    mpq_init(t);
    mpz_init(u);

    for (k = 0; k < n; k++)
    {
        fmpq_poly_get_coeff_mpq(t, w, k);
        mpz_fac_ui(u, k);
        mpz_mul(mpq_numref(t), mpq_numref(t), u);
        mpq_canonicalize(t);
        gmp_printf("%Qd ", t);
    }

    mpq_clear(t);
    mpz_clear(u);

This indeed gives us an integer sequence:

1 1 2 9 56 480 5094 65534 984808 16992144

Now what is the value of the 1000th coefficient (to be precise, a₁₀₀₀, the initial one being the 0th!) in this sequence? After a simple modification of the program, 2.9 seconds of computation gives:

11608872341636087705816513947297167830568265588875720061704
01832880235304267566817912141661469362953389062004053809005
65797054717998071778437757582562676432270594729770831984037
17901116787718293231769568392734610884078152929278291961741
58010228897635319982035567487202368704727403137478203768363
54056589570878404139562784693762331122998711070595645913436
44753733499423283972136827590268687580725109528808039530647
10910254098110789162443473433433758060122558659258182027755
69656436509351036076228649393400187670469063215003559774586
49501015173633083100668758800804388616363320813332492596835
40185987183963214465225072970422690115905543500507650640978
08856685726892919091844572545581642428942983342505179168857
61923031601434642410137173087273453449219217659949560840949
29145910407919393564145312029717057693032572341514569188719
42207889248610196901459400077483577940763454422516589494589
38697976290832628091067571489853751119661925805775760182956
07151657547554699411688610841404991952520564137242651305186
19966880917401902668151574186675809680229260294868082194497
63338464294487320831362657576767926588975644587806316363928
21662453081804476234328933125206970873131871382852201414093
31942812710129867491990841736391939490562342870154316209797
95555638177793757660689621198594912024704112203014400855204
04879191040818216462884689447945725483793082854991264186114
00713712447555062853630274495412279277142852027491666742488
18689076794537156576609645279481454870296442864829766014978
76385015229773871193575960430599394232421616401025152808967
97542967829629757402705726445239053261557399630212654678115
91948563122399554735529747742515102962530483866618795187470
92568029262248891738821070847168914030430887617489382116571
31479578425767585519331805968937010542495567221591600504522
70151935685333213987251220404383044513120115761331175072544
91881860724844683157343078083901966247367831930705346651165
57731933519958498663270193078704185994119446629783305199163
25824443621182783667024174595493553934149891052564101562124
66082538519787858297949190033471879555319648142879656530503
22140399695072998272983889906823049155302053273484019653833
08158019685729676988160041114485564188896445502120959889736
26684734069125268167350474483728161637188322446040542612820
83620649731423678182582137133666912162187578149277916758677
65932622140692260754343559763758688544180440952477345437585
88260535486569816885029406514351482276962081562798684604230
27051552771077659399889469617306015354335528530235916712574
33756257973655927835185354982512983428012895270181767297060
61394636504681554763302758450669487653360858511886083023090
56603401440047692698200295529572915618836122163118770906896
63441094011689868848158568518095899683719854486361541380832
18026233272569661209672552513531416295218659379214599386577
71439492527626159018195922050167504883881038997644963556212
95634222871269535245013411241216112695705600000000000000000
0000000000000000000000000 

In fact, if we look up the first 10 coefficients in the On-Line Encyclopedia of Integer Sequences, we find http://oeis.org/A033917. This OEIS entry lists the representation

a(n) = Sum_{k=0..n} Stirling1(n, k)*(k+1)^(k-1)

Since FLINT supports fast vector computation of Stirling numbers, this formula can be implemented efficiently:

#include "fmpz.h"
#include "fmpz_vec.h"
#include "arith.h"

void coefficient(fmpz_t a, long n)
{
    long k;
    fmpz * s;
    fmpz_t t;

    s = _fmpz_vec_init(n + 1);
    fmpz_stirling1_vec(s, n, n + 1);

    fmpz_init(t);
    fmpz_zero(a);
    for (k = 1; k <= n; k++)
    {
        fmpz_set_ui(t, k + 1);
        fmpz_pow_ui(t, t, k - 1);
        fmpz_addmul(a, s + k, t);
    }

    _fmpz_vec_clear(s, n + 1);
    fmpz_clear(t);
}

int main()
{
    fmpz_t a;
    fmpz_init(a);
    coefficient(a, 1000);
    fmpz_print(a);
    printf("\n");
    fmpz_clear(a);
}

And indeed, the output turns out to be the same!

This program is faster, taking only 0.1 seconds to run. But of course, it only gives us a single coefficient, and would be slower for computing a range of values by making repeated calls.

Similar ideas to those presented here (basically, reducing a problem to fast polynomial multiplication using generating functions, Newton iteration, etc.) are used internally by FLINT for computation of the standard elementary functions themselves as well as various special numbers and polynomials (Bernoulli numbers and polynomials, partitions, Stirling numbers, Bell numbers, etc). The internal code uses a lot of tricks to reduce overhead and handle special cases faster, however. (See the previous blog post Fast combinatorial and number-theoretic functions with FLINT 2, and for more recent information the release announcement and benchmarks page linked at the top of this post.)

In other news, I haven't written a lot of code for mpmath or Sage recently. Of course, my hope is that FLINT (2) will make it into Sage in the not too distant future. The fast polynomial and power series arithmetic support in FLINT will also be very useful for future special functions applications (in mpmath and elsewhere).