// pi().
// General includes.
#include "cl_sysdep.h"
// Specification.
#include "cl_F_tran.h"
// Implementation.
#include "cln/lfloat.h"
#include "cl_LF_tran.h"
#include "cl_LF.h"
#include "cln/integer.h"
#include "cl_alloca.h"
namespace cln {
ALL_cl_LF_OPERATIONS_SAME_PRECISION()
// For the next algorithms, I warmly recommend
// [Jörg Arndt: hfloat documentation, august 1996,
// http://www.tu-chemnitz.de/~arndt/ hfdoc.dvi
// But beware of the typos in his formulas! ]
const cl_LF compute_pi_brent_salamin (uintC len)
{
// Methode:
// [Richard P. Brent: Fast multiple-precision evaluation of elementary
// functions. J. ACM 23(1976), 242-251.]
// [Jonathan M. Borwein, Peter B. Borwein: Pi and the AGM.
// Wiley 1987. Algorithm 2.2, p. 48.]
// [Jörg Arndt, formula (4.51)-(4.52).]
// pi = AGM(1,1/sqrt(2))^2 * 2/(1 - sum(k=0..infty, 2^k c_k^2)).
// where the AGM iteration reads
// a_0 := 1, b_0 := 1/sqrt(2).
// a_(k+1) := (a_k + b_k)/2, b_(k+1) := sqrt(a_k*b_k).
// and we set
// c_k^2 := a_k^2 - b_k^2,
// i.e. c_0^2 = 1/2,
// c_(k+1)^2 = a_(k+1)^2 - b_(k+1)^2 = (a_k - b_k)^2/4
// = (a_k - a_(k+1))^2.
// (Actually c_(k+1) is _defined_ as = a_k - a_(k+1) = (a_k - b_k)/2.)
// d=len, n:=intDsize*d. Verwende Long-Floats mit intDsize*(d+1)
// Mantissenbits.
// (let* ((a (coerce 1 'long-float)) ; 1
// (b (sqrt (scale-float a -1))) ; 2^-(1/2)
// (eps (scale-float a (- n))) ; 2^-n
// (t (scale-float a -2)) ; 1/4
// (k 0)
// )
// (loop
// (when (< (- a b) eps) (return))
// (let ((y a))
// (setq a (scale-float (+ a b) -1))
// (setq b (sqrt (* b y)))
// (setq t (- t (scale-float (expt (- a y) 2) k)))
// )
// (incf k)
// )
// (/ (expt a 2) t)
// )
var uintC actuallen = len + 1; // 1 Schutz-Digit
var uintL uexp_limit = LF_exp_mid - intDsize*(uintL)len;
// Ein Long-Float ist genau dann betragsmäßig <2^-n, wenn
// sein Exponent < LF_exp_mid-n = uexp_limit ist.
var cl_LF a = cl_I_to_LF(1,actuallen);
var cl_LF b = sqrt(scale_float(a,-1));
var uintL k = 0;
var cl_LF t = scale_float(a,-2);
until (TheLfloat(a-b)->expo < uexp_limit) {
// |a-b| < 2^-n -> fertig
var cl_LF new_a = scale_float(a+b,-1); // (a+b)/2
b = sqrt(a*b);
var cl_LF a_diff = new_a - a;
t = t - scale_float(square(a_diff),k);
a = new_a;
k++;
}
var cl_LF pires = square(a)/t; // a^2/t
return shorten(pires,len); // verkürzen und fertig
}
// Bit complexity (N := len): O(log(N)*M(N)).
const cl_LF compute_pi_brent_salamin_quartic (uintC len)
{
// See [Borwein, Borwein, section 1.4, exercise 3, p. 17].
// See also [Jörg Arndt], formulas (4.52) and (3.30)[wrong!].
// As above, we are using the formula
// pi = AGM(1,1/sqrt(2))^2 * 2/(1 - sum(k=0..infty, 2^k c_k^2)).
// where the AGM iteration reads
// a_0 := 1, b_0 := 1/sqrt(2).
// a_(k+1) := (a_k + b_k)/2, b_(k+1) := sqrt(a_k*b_k).
// But we keep computing with
// wa_k := sqrt(a_k) and wb_k := sqrt(b_k)
// at the same time and do two iteration steps at once.
// The iteration takes now takes the form
// wa_0 := 1, wb_0 := 2^-1/4,
// (wa_k, wb_k, a_k, b_k)
// --> ((wa_k^2 + wb_k^2)/2, wa_k*wb_k)
// --> (((wa_k + wb_k)/2)^2, sqrt(wa_k*wb_k*(wa_k^2 + wb_k^2)/2)),
// i.e. wa_(k+2) = (wa_k + wb_k)/2 and
// wb_(k+2) = sqrt(sqrt(wa_k*wb_k*(wa_k^2 + wb_k^2)/2)).
// For the sum, we can group two successive items together:
// 2^k * c_k^2 + 2^(k+1) * c_(k+1)^2
// = 2^k * [(a_k^2 - b_k^2) + 2*((a_k - b_k)/2)^2]
// = 2^k * [3/2*a_k^2 - a_k*b_k - 1/2*b_k^2]
// = 2^k * [2*a_k^2 - 1/2*(a_k+b_k)^2]
// = 2^(k+1) * [a_k^2 - ((a_k+b_k)/2)^2]
// = 2^(k+1) * [wa_k^4 - ((wa_k^2+wb_k^2)/2)^2].
// Hence,
// pi = AGM(1,1/sqrt(2))^2 * 1/(1/2 - sum(k even, 2^k*[....])).
var uintC actuallen = len + 1; // 1 Schutz-Digit
var uintL uexp_limit = LF_exp_mid - intDsize*(uintL)len;
var cl_LF one = cl_I_to_LF(1,actuallen);
var cl_LF a = one;
var cl_LF wa = one;
var cl_LF b = sqrt(scale_float(one,-1));
var cl_LF wb = sqrt(b);
// We keep a = wa^2, b = wb^2.
var uintL k = 0;
var cl_LF t = scale_float(one,-1);
until (TheLfloat(wa-wb)->expo < uexp_limit) {
// |wa-wb| < 2^-n -> fertig
var cl_LF wawb = wa*wb;
var cl_LF new_wa = scale_float(wa+wb,-1);
var cl_LF a_b = scale_float(a+b,-1);
var cl_LF new_a = scale_float(a_b+wawb,-1);
var cl_LF new_b = sqrt(wawb*a_b);
var cl_LF new_wb = sqrt(new_b);
t = t - scale_float((a - a_b)*(a + a_b),k);
a = new_a; wa = new_wa;
b = new_b; wb = new_wb;
k += 2;
}
var cl_LF pires = square(a)/t;
return shorten(pires,len); // verkürzen und fertig
}
// Bit complexity (N := len): O(log(N)*M(N)).
const cl_LF compute_pi_ramanujan_163 (uintC len)
{
// 1/pi = 1/sqrt(-1728 J)
// * sum(n=0..infty, (6n)! (A+nB) / 12^(3n) (3n)! n!^3 J^n)
// mit J = -53360^3 = - (2^4 5 23 29)^3
// A = 163096908 = 2^2 3 13 1045493
// B = 6541681608 = 2^3 3^3 7 11 19 127 163
// See [Jörg Arndt], formulas (4.27)-(4.30).
// This is also the formula used in Pari.
// The absolute value of the n-th summand is approximately
// |J|^-n * n^(-1/2) * B/(2*pi^(3/2)),
// hence every summand gives more than 14 new decimal digits
// in precision.
// The sum is best evaluated using fixed-point arithmetic,
// so that the precision is reduced for the later summands.
var uintL actuallen = len + 4; // 4 Schutz-Digits
var sintL scale = intDsize*actuallen;
static const cl_I A = "163096908";
static const cl_I B = "6541681608";
//static const cl_I J1 = "10939058860032000"; // 72*abs(J)
static const cl_I J2 = "333833583375"; // odd part of J1
var cl_I sum = 0;
var cl_I n = 0;
var cl_I factor = ash(1,scale);
while (!zerop(factor)) {
sum = sum + factor * (A+n*B);
factor = factor * ((6*n+1)*(2*n+1)*(6*n+5));
n = n+1;
factor = truncate1(factor,n*n*n*J2);
// Finally divide by 2^15 and change sign.
if (minusp(factor))
factor = (-factor) >> 15;
else
factor = -(factor >> 15);
}
var cl_LF fsum = scale_float(cl_I_to_LF(sum,actuallen),-scale);
static const cl_I J3 = "262537412640768000"; // -1728*J
var cl_LF pires = sqrt(cl_I_to_LF(J3,actuallen)) / fsum;
return shorten(pires,len); // verkürzen und fertig
}
// Bit complexity (N := len): O(N^2).
#if defined(__mips__) && !defined(__GNUC__) // workaround SGI CC bug
#define A A_fast
#define B B_fast
#define J3 J3_fast
#endif
const cl_LF compute_pi_ramanujan_163_fast (uintC len)
{
// Same formula as above, using a binary splitting evaluation.
// See [Borwein, Borwein, section 10.2.3].
var uintL actuallen = len + 4; // 4 Schutz-Digits
static const cl_I A = "163096908";
static const cl_I B = "6541681608";
static const cl_I J1 = "10939058860032000"; // 72*abs(J)
// Evaluate a sum(0 <= n < N, a(n)/b(n) * (p(0)...p(n))/(q(0)...q(n)))
// with appropriate N, and
// a(n) = A+n*B, b(n) = 1,
// p(n) = -(6n-5)(2n-1)(6n-1) for n>0,
// q(n) = 72*|J|*n^3 for n>0.
var const uintL n_slope = (uintL)(intDsize*32*0.02122673)+1;
// n_slope >= 32*intDsize*log(2)/log(|J|), normally n_slope = 22.
var uintL N = (n_slope*actuallen)/32 + 1;
// N > intDsize*log(2)/log(|J|) * actuallen, hence
// |J|^-N < 2^(-intDsize*actuallen).
CL_ALLOCA_STACK;
var cl_I* av = (cl_I*) cl_alloca(N*sizeof(cl_I));
var cl_I* pv = (cl_I*) cl_alloca(N*sizeof(cl_I));
var cl_I* qv = (cl_I*) cl_alloca(N*sizeof(cl_I));
var uintL* qsv = (uintL*) cl_alloca(N*sizeof(uintL));
var uintL n;
for (n = 0; n < N; n++) {
init1(cl_I, av[n]) (A+n*B);
if (n==0) {
init1(cl_I, pv[n]) (1);
init1(cl_I, qv[n]) (1);
} else {
init1(cl_I, pv[n]) (-((cl_I)(6*n-5)*(cl_I)(2*n-1)*(cl_I)(6*n-1)));
init1(cl_I, qv[n]) ((cl_I)n*(cl_I)n*(cl_I)n*J1);
}
}
var cl_pqa_series series;
series.av = av;
series.pv = pv; series.qv = qv;
series.qsv = (len >= 35 ? qsv : 0); // 5% speedup for large len's
var cl_LF fsum = eval_rational_series(N,series,actuallen);
for (n = 0; n < N; n++) {
av[n].~cl_I();
pv[n].~cl_I();
qv[n].~cl_I();
}
static const cl_I J3 = "262537412640768000"; // -1728*J
var cl_LF pires = sqrt(cl_I_to_LF(J3,actuallen)) / fsum;
return shorten(pires,len); // verkürzen und fertig
}
// Bit complexity (N := len): O(log(N)^2*M(N)).
// Timings of the above algorithms, on an i486 33 MHz, running Linux.
// N Brent Brent4 R 163 R 163 fast
// 10 0.0079 0.0079 0.0052 0.0042
// 25 0.026 0.026 0.014 0.012
// 50 0.085 0.090 0.037 0.033
// 100 0.29 0.29 0.113 0.098
// 250 1.55 1.63 0.60 0.49
// 500 5.7 5.7 2.24 1.71
// 1000 21.6 22.9 8.5 5.5
// 2500 89 95 49 19.6
// 5000 217 218 188 49
// 10000 509 540 747 117
// 25000 1304 1310 4912 343
// We see that
// - "Brent4" isn't worth it: No speed improvement over "Brent".
// - "R 163" is pretty fast at the beginning, but it is an O(N^2)
// algorithm, hence it loses in the end,
// - "R 163 fast", which uses the same formula as "R 163", but evaluates
// it using binary splitting, is an O(log N * M(N)) algorithm, and
// outperforms all of the others.
const cl_LF pi (uintC len)
{
var uintC oldlen = TheLfloat(cl_LF_pi)->len; // vorhandene Länge
if (len < oldlen)
return shorten(cl_LF_pi,len);
if (len == oldlen)
return cl_LF_pi;
// TheLfloat(cl_LF_pi)->len um mindestens einen konstanten Faktor
// > 1 wachsen lassen, damit es nicht zu häufig nachberechnet wird:
var uintC newlen = len;
oldlen += floor(oldlen,2); // oldlen * 3/2
if (newlen < oldlen)
newlen = oldlen;
// gewünschte > vorhandene Länge -> muß nachberechnen:
cl_LF_pi = compute_pi_ramanujan_163_fast(newlen);
return (len < newlen ? shorten(cl_LF_pi,len) : cl_LF_pi);
}
} // namespace cln
syntax highlighted by Code2HTML, v. 0.9.1