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Fo(\))c(log)q(\()p Fl(b)p Fo(\)\)\))515 872 y(\(the)28 b(sup)r(erlinearit)n(y)e(of)i Fl(M)9 b Fo(\()p Fl(d)p Fo(\))28 b(should)f(b)r(e)h(used)g(:)37 b(2)p Fl(M)9 b Fo(\()p Fl(d=)p Fo(2\))22 b Fg(\024)h Fl(M)9 b Fo(\()p Fl(d)p Fo(\)\).)639 972 y(Finally)-7 b(,)34 b(w)n(e)f(ha)n(v)n(e)e(pro) n(v)n(ed)g(that)i(computing)g Fl(n)p Fo(!)g(can)f(b)r(e)h(done)g(in)g (time)g Fl(C)6 b Fo(\(0)p Fl(;)14 b(n)p Fo(\))31 b(=)515 1072 y Fl(O)r Fo(\(log)q(\()p Fl(n)p Fo(\))p Fl(M)9 b Fo(\()p Fl(n)28 b Fo(log\()p Fl(n)p Fo(\)\).)49 b(This)32 b(is)f(alw)n(a)n(ys)f(b)r(etter)i(than)f(the)h(classical)e(approac)n(h) g(when)515 1171 y(pro)r(ducts)d(on)f(large)g(in)n(tegers)g(are)g(done)h (with)h(a)f(fast)g(metho)r(d.)37 b(When)28 b(FFT)f(is)g(used)h(for)515 1271 y(large)c(m)n(ultiplication)h(\(see)h Ff(FFT)i(b)l(ase)l(d)h (multiplic)l(ation)g(of)f(lar)l(ge)h(numb)l(ers)p Fo(\),)c(it)h(leads)f (to)515 1371 y(the)j(b)r(ound)1518 1470 y Fl(C)6 b Fo(\(0)p Fl(;)14 b(n)p Fo(\))23 b(=)f Fl(O)r Fo(\()p Fl(n)29 b 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Fo(,)j(whic)n(h)e(o)r (ccurs)g(for)515 3078 y Fl(k)26 b Fo(=)c Fl(K)29 b Fg(')22 b Fl(d=)14 b Fo(log\()p Fl(d)p Fo(\).)38 b(W)-7 b(e)28 b(mak)n(e)f(use)g(of)h(the)g(notations)1425 3252 y Fl(Q)p Fo(\()p Fl(a;)14 b(b)p Fo(\))23 b(=)f(\()p Fl(a)d Fo(+)f(1\)\()p Fl(a)h Fo(+)f(2\))c Fg(\001)g(\001)g(\001)f Fl(b;)622 3427 y(P)f Fo(\()p Fl(a;)i(b)p Fo(\))23 b(=)g Fl(b)p Fo(\()p Fl(b)18 b Fg(\000)g Fo(1\))c Fg(\001)g(\001)g(\001)f Fo(\()p Fl(a)19 b Fo(+)f(2\))g(+)g Fl(b)p Fo(\()p Fl(b)g Fg(\000)g Fo(1\))c Fg(\001)g(\001)g(\001)f Fo(\()p Fl(a)19 b Fo(+)f(3\))g(+)g Fg(\001)c(\001)g(\001)19 b Fo(+)f(\()p Fl(b)g Fg(\000)g Fo(1\))p Fl(b)g Fo(+)g Fl(b)g Fo(+)g(1)p Fl(:)515 3572 y(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))27 b(and)h Fl(Q)p Fo(\()p Fl(a;)14 b(b)p Fo(\))27 b(are)g(in)n(tegers)f(whic)n(h)i (satisfy)850 3738 y Fl(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))p 850 3775 247 4 v 850 3851 a Fl(Q)p Fo(\()p Fl(a;)g(b)p Fo(\))1129 3794 y(=)1300 3738 y(1)p 1227 3775 187 4 v 1227 3851 a Fl(a)k Fo(+)g(1)1442 3794 y(+)1766 3738 y(1)p 1535 3775 503 4 v 1535 3851 a(\()p Fl(a)h Fo(+)f(1\)\()p Fl(a)g Fo(+)g(2\))2067 3794 y(+)g Fg(\001)c(\001)g(\001)k Fo(+)2669 3738 y(1)p 2358 3775 664 4 v 2358 3851 a(\()p Fl(a)g Fo(+)h(1\)\()p Fl(a)f Fo(+)g(2\))c Fg(\001)g(\001)g(\001)f Fl(b)3031 3794 y(:)515 4018 y Fo(In)28 b(particular,)e(1)18 b(+)g Fl(P)12 b Fo(\(0)p Fl(;)i(K)6 b Fo(\))p Fl(=Q)p Fo(\(0)p Fl(;)14 b(K)6 b Fo(\))25 b(are)i(the)h(\014rst)f Fl(K)33 b Fo(terms)28 b(of)f(the)h(series)f(\(*\).)639 4117 y(T)-7 b(o)27 b(compute)h Fl(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))27 b(and)g Fl(Q)p Fo(\()p Fl(a;)14 b(b)p Fo(\))27 b(b)n(y)h(the)f(binary)g(splitting)h(metho)r(d,)g(w)n(e)f(pro)r(ceed) 515 4217 y(as)g(follo)n(ws.)36 b(W)-7 b(e)28 b(set)f Fl(m)h Fo(to)f(the)h(half)g(of)f Fl(a)19 b Fo(+)f Fl(b)p Fo(,)1407 4437 y Fl(m)23 b Fo(=)1590 4320 y Fe(\024)1644 4381 y Fl(a)c Fo(+)f Fl(b)p 1644 4418 181 4 v 1714 4494 a Fo(2)1835 4320 y Fe(\025)1976 4437 y Fo(\()p Fl(integ)s(er)r(par)r(t) p Fo(\))p Fl(:)515 4657 y Fo(Then)28 b(w)n(e)f(ha)n(v)n(e)730 4832 y Fl(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))23 b(=)f Fl(P)12 b Fo(\()p Fl(a;)i(m)p Fo(\))p Fl(Q)p Fo(\()p Fl(m;)g(b)p Fo(\))k(+)g Fl(P)12 b Fo(\()p Fl(m;)i(b)p Fo(\))p Fl(;)180 b(Q)p Fo(\()p Fl(a;)14 b(b)p Fo(\))23 b(=)f Fl(Q)p Fo(\()p Fl(a;)14 b(m)p Fo(\))p Fl(Q)p Fo(\()p Fl(m;)g(b)p Fo(\))p Fl(:)515 5006 y Fo(These)27 b(op)r(erations)f(are)h(p)r(erformed)g (recursiv)n(ely)-7 b(.)p eop %%Page: 3 3 3 2 bop 515 232 a Fp(Mathematical)27 b(constan)n(ts)f(and)i (computation)1308 b Fo(3)639 523 y(The)36 b(cost)e(of)i(the)f(pro)r (cess)f(to)i(compute)f Fl(P)12 b Fo(\(0)p Fl(;)i(K)6 b Fo(\))34 b(and)i Fl(Q)p Fo(\(0)p Fl(;)14 b(K)6 b Fo(\))34 b(is)h(easily)g(pro)n(v)n(ed)515 623 y(to)d(b)r(e)g Fl(O)r Fo(\(log)q(\()p Fl(K)6 b Fo(\))p Fl(M)j Fo(\()p Fl(K)19 b Fo(log)q(\()p Fl(K)6 b Fo(\)\)\).)50 b(A)33 b(\014nal)f(division)f (of)h Fl(P)12 b Fo(\(0)p Fl(;)i(K)6 b Fo(\))32 b(b)n(y)f Fl(Q)p Fo(\(0)p Fl(;)14 b(K)6 b Fo(\))32 b(is)g(then)515 722 y(required)h(to)h(compute)h(the)g(\014rst)f Fl(K)40 b Fo(terms)34 b(of)g(the)h(series)e(\(its)i(cost)f(is)g Fl(O)r Fo(\()p Fl(M)9 b Fo(\()p Fl(n)p Fo(\)\),)37 b(see)515 822 y Ff(Inverse)31 b(and)h(n-th)e(r)l(o)l(ots)h(of)h(lar)l(ge)g(numb)l (ers)p Fo(\).)40 b(Since)30 b Fl(K)g Fg(')25 b Fl(n=)14 b Fo(log\()p Fl(n)p Fo(\),)30 b(w)n(e)e(ha)n(v)n(e)g(\014nally)515 922 y(obtained)d(a)g(pro)r(cess)f(to)i(compute)f Fl(e)g Fo(in)h(time)g Fl(O)r Fo(\(log)q(\()p Fl(n)p Fo(\))p Fl(M)9 b Fo(\()p Fl(n)p Fo(\)\).)37 b(An)26 b(FFT)g(based)f(m)n(ulti-) 515 1021 y(plication)g(satis\014es)f Fl(M)9 b Fo(\()p Fl(n)p Fo(\))23 b(=)f Fl(O)r Fo(\()p Fl(n)29 b Fo(log\()p Fl(n)p Fo(\)\))d(and)f(leads)f(to)h(a)g(total)g(cost)f(of)h Fl(O)r Fo(\()p Fl(n)k Fo(log\()p Fl(n)p Fo(\))3309 991 y Fm(2)3347 1021 y Fo(\))515 1121 y(to)e(compute)h Fl(n)f Fo(digits)h(of)f Fl(e)p Fo(.)639 1220 y(An)k(easy)e(co)r(de)h (illustrating)f(the)h(binary)g(splitting)g(metho)r(d)g(to)g(compute)g Fl(e)g Fo(can)g(b)r(e)515 1320 y(found)e(in)g Ff(Easy)i(pr)l(o)l(gr)l (ams)h(for)f(c)l(onstants)f(c)l(omputation)p Fo(.)515 1536 y Fa(Practical)k(implemen)m(tation)515 1689 y Fo(The)i (computation)g(of)h(the)f(exp)r(onen)n(tial)g(series)f(thanks)h(to)h (the)f(binary)g(splitting)h(ap-)515 1789 y(proac)n(h)e(is)h(quite)h (easy)-7 b(.)60 b(In)36 b(the)g(practice,)h(one)e(should)g(stop)h(the)g (splitting)g(recursion)515 1888 y(when)28 b Fl(b)19 b Fg(\000)g Fl(a)28 b Fo(b)r(ecomes)g(small)h(\(the)g(threshold)f(should) g(b)r(e)h(an)f(optimized)h(v)-5 b(alue\))29 b(to)f(use)515 1988 y(a)33 b(classical)f(approac)n(h.)52 b(An)34 b(sligh)n(t)e(impro)n (v)n(emen)n(t)h(can)f(also)h(b)r(e)g(obtained)g(b)n(y)h(using)e(a)515 2088 y(classical)25 b(m)n(ultiplication)i(when)g Fl(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))26 b(and)h Fl(Q)p Fo(\()p Fl(a;)14 b(b)p Fo(\))26 b(ha)n(v)n(e)g(a)g(small)g(n)n(um)n(b)r(er)h (of)f(digits.)639 2187 y(Thanks)k(to)g(the)g(binary)g(splitting)g(and)g (using)g(the)h(FFT)f(co)r(de)g(of)g Ff(PiF)-6 b(ast)p Fo(,)32 b(the)e(\014rst)515 2287 y(million)d(digits)g(of)g Fl(e)g Fo(w)n(as)f(computed)h(in)h(less)e(than)h(10)g(seconds)f(\(p)r (en)n(tium)i(I)r(I)f(350\).)36 b(The)515 2386 y(binary)20 b(splitting)h(metho)r(d)g(to)g(compute)f Fl(e)h Fo(is)f(b)r(etter)i (than)e(an)n(y)g(other)h(approac)n(hes)d(\(m)n(uc)n(h)515 2486 y(b)r(etter)28 b(than)f(the)h(A)n(GM)g(based)f(approac)n(h,)f(see) h Ff(The)k(c)l(onstant)e Fl(e)p Fo(\).)639 2586 y(It)j(m)n(ust)g(b)r(e) h(p)r(oin)n(ted)f(out)g(that)g(binary)f(splitting)h(needs)g(more)f (memory)g(than)h(the)515 2685 y(classical)37 b(metho)r(d)i(\(this)g(is) g(nearly)e(alw)n(a)n(ys)g(the)i(case)f(:)59 b(fast)38 b(metho)r(ds)h(are)f(memory)515 2785 y(consuming\).)e(Netherv)n(eless,) 27 b(memory)g(is)g(quite)h(easy)f(to)g(manage)g(with)h(this)g(pro)r (cess.)515 3060 y Fk(3)134 b(Binary)45 b(splitting)i(on)d(the)i(arctan) f(series)515 3241 y Fo(W)-7 b(e)35 b(no)n(w)g(presen)n(t)g(the)g (binary)g(splitting)h(metho)r(d)f(to)h(compute)f(digits)g(of)g(the)h (arctan)515 3341 y(series)916 3476 y(arctan)1161 3359 y Fe(\022)1232 3420 y Fo(1)p 1232 3457 42 4 v 1233 3533 a Fl(q)1283 3359 y Fe(\023)1368 3476 y Fo(=)1465 3420 y(1)p 1465 3457 V 1466 3533 a Fl(q)1535 3476 y Fg(\000)1667 3420 y Fo(1)p 1628 3457 119 4 v 1628 3533 a(3)p Fl(q)1710 3509 y Fm(3)1775 3476 y Fo(+)1907 3420 y(1)p 1868 3457 V 1868 3533 a(5)p Fl(q)1950 3509 y Fm(5)2016 3476 y Fo(+)18 b Fg(\001)c(\001)g(\001)22 b Fo(=)2307 3397 y Fe(X)2306 3576 y Fi(k)q Fh(\025)p Fm(0)2592 3420 y Fo(\()p Fg(\000)p Fo(1\))2763 3390 y Fi(k)p 2452 3457 493 4 v 2452 3533 a Fo(\(2)p Fl(k)f Fo(+)d(1\))p Fl(q)2787 3509 y Fm(2)p Fi(k)q Fm(+1)2954 3476 y Fl(:)515 3711 y Fo(The)36 b(tec)n(hnique)h(is) f(sligh)n(tly)g(more)f(complicated)h(than)h(with)f(the)h(exp)r(onen)n (tial)f(series.)515 3810 y(W)-7 b(e)28 b(need)f(to)h(in)n(tro)r(duce)f (three)h(in)n(tegers)1260 3993 y Fl(R)q Fo(\()p Fl(a;)14 b(b)p Fo(\))23 b(=)f(\(2)p Fl(a)d Fo(+)f(3\)\(2)p Fl(a)g Fo(+)g(5\))c Fg(\001)g(\001)g(\001)f Fo(\(2)p Fl(b)18 b Fo(+)g(1\))p Fl(;)1136 4176 y(Q)p Fo(\()p Fl(a;)c(b)p Fo(\))23 b(=)f(\(2)p Fl(a)c Fo(+)g(3\)\(2)p Fl(a)g Fo(+)g(5\))c Fg(\001)g(\001)g(\001)g Fo(\(2)p Fl(b)k Fo(+)g(1\))p Fl(q)2529 4141 y Fm(2\()p Fi(b)p Fh(\000)p Fi(a)p Fm(\))2735 4176 y Fl(;)515 4373 y(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))23 b(=)f(\()p Fg(\000)p Fo(1\))1042 4339 y Fi(a)p Fm(+1)1176 4317 y Fl(R)q Fo(\()p Fl(a;)14 b(b)p Fo(\))p 1176 4354 245 4 v 1184 4430 a(2)p Fl(a)k Fo(+)g(3)1431 4373 y Fl(q)1471 4339 y Fm(2\()p Fi(b)p Fh(\000)p Fi(a)p Fh(\000)p Fm(1\))1762 4373 y Fo(+\()p Fg(\000)p Fo(1\))1998 4339 y Fi(a)p Fm(+2)2131 4317 y Fl(R)q Fo(\()p Fl(a;)c(b)p Fo(\))p 2131 4354 V 2139 4430 a(2)p Fl(a)k Fo(+)g(5)2386 4373 y Fl(q)2426 4339 y Fm(2\()p Fi(b)p Fh(\000)p Fi(a)p Fh(\000)p Fm(2\))2717 4373 y Fo(+)p Fg(\001)c(\001)g(\001)p Fo(+\()p Fg(\000)p Fo(1\))3115 4339 y Fi(b)p Fh(\000)p Fm(1)3242 4317 y Fl(R)q Fo(\()p Fl(a;)g(b)p Fo(\))p 3242 4354 V 3254 4430 a(2)p Fl(b)k Fg(\000)g Fo(1)3497 4373 y Fl(q)3537 4339 y Fm(2)3574 4373 y Fo(+\()p Fg(\000)p Fo(1\))3810 4339 y Fi(b)3853 4317 y Fl(R)q Fo(\()p Fl(a;)c(b)p Fo(\))p 3853 4354 V 3865 4430 a(2)p Fl(b)k Fo(+)g(1)4107 4373 y Fl(:)639 4553 y Fo(Notice)28 b(that)1263 4657 y Fl(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))p 1263 4694 247 4 v 1263 4770 a Fl(Q)p Fo(\()p Fl(a;)g(b)p Fo(\))1542 4713 y(=)1719 4609 y Fi(b)1674 4634 y Fe(X)1630 4813 y Fi(k)q Fm(=)p Fi(a)p Fm(+1)1838 4713 y Fo(\()p Fg(\000)p Fo(1\))2009 4679 y Fi(k)2313 4657 y Fo(1)p 2059 4694 549 4 v 2059 4772 a(\(2)p Fl(k)21 b Fo(+)d(1\))p Fl(q)2394 4748 y Fm(2\()p Fi(k)q Fh(\000)p Fi(a)p Fm(\))2618 4713 y Fl(:)p eop %%Page: 4 4 4 3 bop 515 232 a Fp(Mathematical)27 b(constan)n(ts)f(and)i (computation)1308 b Fo(4)515 523 y(With)28 b(the)g(particular)e(case)h Fl(a)c Fo(=)g(0,)k Fl(b)c Fo(=)f Fl(K)6 b Fo(,)28 b(the)g(expression) 1634 693 y(1)p 1634 730 42 4 v 1635 806 a Fl(q)1699 632 y Fe(\022)1761 749 y Fo(1)17 b(+)1914 693 y Fl(P)12 b Fo(\(0)p Fl(;)i(K)6 b Fo(\))p 1913 730 286 4 v 1913 806 a Fl(Q)p Fo(\(0)p Fl(;)14 b(K)6 b Fo(\))2208 632 y Fe(\023)515 975 y Fo(is)27 b(equal)g(to)h(the)g(\014rst)f Fl(K)33 b Fo(terms)28 b(of)f(the)h(arctan)f(series.)639 1075 y(The)34 b(splitting)h(pro)r(cess)e(consists)g(in)h(recursiv)n(ely)e(p) r(erforming)i(the)g(op)r(erations)f(\()p Fl(m)515 1174 y Fo(denotes)27 b(the)h(half)g(of)f Fl(a)19 b Fo(+)f Fl(b)p Fo(\))515 1355 y Fl(P)12 b Fo(\()p Fl(a;)i(b)p Fo(\))23 b(=)f Fl(Q)p Fo(\()p Fl(m;)14 b(b)p Fo(\))p Fl(P)e Fo(\()p Fl(a;)i(m)p Fo(\)+)p Fl(R)q Fo(\()p Fl(a;)g(m)p Fo(\))p Fl(P)e Fo(\()p Fl(m;)i(b)p Fo(\))p Fl(;)96 b(Q)p Fo(\()p Fl(a;)14 b(b)p Fo(\))23 b(=)g Fl(Q)p Fo(\()p Fl(a;)14 b(m)p Fo(\))p Fl(Q)p Fo(\()p Fl(m;)g(b)p Fo(\))p Fl(;)96 b(R)q Fo(\()p Fl(a;)14 b(b)p Fo(\))23 b(=)f Fl(R)q Fo(\()p Fl(a;)14 b(m)p Fo(\))p Fl(R)q Fo(\()p Fl(m;)g(b)p Fo(\))p Fl(:)639 1536 y Fo(No)n(w,)33 b(to)g(compute)f Fl(n)h Fo(decimal)f(digits)g(of)g(arctan)o(\(1)p Fl(=q)s Fo(\),)i(one)e(m)n(ust)g(stop)g(the)h(sum-)515 1636 y(mation)d(at)h Fl(k)g Fo(=)c Fl(K)37 b Fo(as)29 b(so)r(on)h(as)g(10)1666 1606 y Fi(n)1739 1636 y Fl(<)d(q)1871 1606 y Fm(2)p Fi(K)t Fm(+1)2053 1636 y Fo(,)k(whic)n(h)g(giv)n(es)e Fl(K)34 b Fg(')28 b Fl(n)f Fo(log\(10\))p Fl(=)p Fo(2.)45 b(The)515 1735 y(total)27 b(cost)g(of)h(the)g(pro)r(cess)e(to)i(compute)f(the)h (\014rst)g Fl(n)f Fo(digits)h(of)f(the)h(series)f(is)1042 1916 y Fl(O)r Fo(\(log)q(\()p Fl(K)6 b Fo(\))p Fl(M)j Fo(\()p Fl(K)19 b Fo(log\()p Fl(K)6 b Fo(\)\)\))24 b(=)f Fl(O)r Fo(\(log)q(\()p Fl(n)p Fo(\))p Fl(M)9 b Fo(\()p Fl(n)28 b Fo(log\()p Fl(n)p Fo(\)\)\))p Fl(:)515 2097 y Fo(With)g(FFT)g(based)f(m)n(ultiplication,)h(this)g(leads)f(to)g(the) h(b)r(ound)g Fl(O)r Fo(\()p Fl(n)h Fo(log\()p Fl(n)p Fo(\))2979 2067 y Fm(3)3017 2097 y Fo(\).)639 2197 y(Notice)41 b(that)g(the)g(pro)r(cess)e(is)i(more)e(complicated)h(and)h(exp)r (ensiv)n(e)f(than)h(for)f(the)515 2296 y(exp)r(onen)n(tial)27 b(series)f(for)h(sev)n(eral)f(reasons)g(:)639 2461 y Fg(\017)41 b Fo(An)28 b(auxilliary)f(large)f(in)n(teger)g Fl(R)q Fo(\()p 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(computation)1308 b Fo(5)515 523 y(Borw)n(ein,)39 b(Bradley)-7 b(,)40 b(Crandall)d Ff("Computational)j(Str)l(ate)l(gies)f(for)h(the)g (R)n(iemann)f(Zeta)515 623 y(F)-6 b(unction")p Fo(\).)36 b(The)27 b(FEE)f(metho)r(d)h(of)f(E.)h(A.)g(Karatsuba)e(can)h(also)g(b) r(e)h(used)f(to)h(compute)515 722 y(other)19 b(constan)n(ts)f(whic)n(h) i(do)f(not)g(write)h(in)g(a)f(geometrically)e(con)n(v)n(ergen)n(t)h(h)n (yp)r(ergeometric)515 822 y(series)25 b(:)37 b(examples)25 b(include)i Fl(\020)6 b Fo(\()p Fl(n)p Fo(\))28 b(for)e(in)n(teger)f(v) -5 b(alues)26 b(of)h Fl(n)f Fo([6],)h(Hurwitz)f(Zeta)g(function)515 922 y(and)h(Diric)n(hlet)h(L-Series)e([7].)515 1196 y Fk(References)515 1378 y Fo([1])41 b(J.)27 b(M.)h(and)g(P)-7 b(.)27 b(B.)g(Borw)n(ein)g Ff(Pi)j(and)g(the)g(A)n(GM)42 b Fo(John)27 b(Wiley)h(and)f(sons,)g(\(1987\))515 1544 y([2])41 b(R.)30 b(P)-7 b(.)29 b(Bren)n(t,)h(The)f(Complexit)n(y)h(of)f (Multiple-Precision)g(Arithmetic,)i Ff(Complexity)644 1644 y(of)36 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(Scienti\014c)d Fo(\(1999\),)e(303-314.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF