File Coverage

blib/lib/Math/AnyNum.pm

Criterion	Covered	Total	%
statement	12	14	85.7
branch			n/a
condition			n/a
subroutine	5	5	100.0
pod			n/a
total	17	19	89.4

line	stmt	sub	time	code
1				package Math::AnyNum;
2
3	1	1	47966	use 5.014;
	1		5
4	1	1	8	use strict;
	1		2
	1		28
5	1	1	7	use warnings;
	1		2
	1		51
6
7	1	1	7	no warnings qw(numeric uninitialized);
	1		3
	1		39
8
9	1	1	164	use Math::MPFR qw();
	0
	0
10				use Math::GMPq qw();
11				use Math::GMPz qw();
12				use Math::MPC qw();
13
14				use POSIX qw(ULONG_MAX LONG_MIN);
15
16				our $VERSION = '0.16';
17				our ($ROUND, $PREC);
18
19				BEGIN {
20				$ROUND = Math::MPFR::MPFR_RNDN();
21				$PREC = 192;
22				}
23
24				use overload
25				'""' => \&stringify,
26				'0+' => \&numify,
27				bool => \&boolify,
28
29				'+' => \&add,
30				'*' => \&mul,
31
32				'==' => \&eq,
33				'!=' => \&ne,
34
35				'&' => \&and,
36				'\|' => \&or,
37				'^' => \&xor,
38				'~' => \¬,
39
40				'>' => sub { $_[2] ? (goto <) : (goto >) },
41				'>=' => sub { $_[2] ? (goto &le) : (goto &ge) },
42				'<' => sub { $_[2] ? (goto >) : (goto <) },
43				'<=' => sub { $_[2] ? (goto &ge) : (goto &le) },
44
45				'<=>' => sub { $_[2] ? -(&cmp($_[0], $_[1]) // return undef) : &cmp($_[0], $_[1]) },
46
47				'>>' => sub { @_ = ($_[1], $_[0]) if $_[2]; goto &rsft },
48				'<<' => sub { @_ = ($_[1], $_[0]) if $_[2]; goto &lsft },
49				'/' => sub { @_ = ($_[1], $_[0]) if $_[2]; goto &div },
50				'-' => sub { @_ = ($_[1], $_[0]) if $_[2]; goto &sub },
51
52				'**' => sub { @_ = $_[2] ? @_[1, 0] : @_[0, 1]; goto &pow },
53				'%' => sub { @_ = $_[2] ? @_[1, 0] : @_[0, 1]; goto &mod },
54
55				atan2 => sub { @_ = $_[2] ? @_[1, 0] : @_[0, 1]; goto &atan2 },
56
57				eq => sub { "$_[0]" eq "$_[1]" },
58				ne => sub { "$_[0]" ne "$_[1]" },
59
60				cmp => sub { $_[2] ? ("$_[1]" cmp $_[0]->stringify) : ($_[0]->stringify cmp "$_[1]") },
61
62				neg => \&neg,
63				sin => \&sin,
64				cos => \&cos,
65				exp => \&exp,
66				log => \&ln,
67				int => \&int,
68				abs => \&abs,
69				sqrt => \&sqrt;
70
71				{
72
73				my %const = ( # prototypes are assigned in import()
74				e => \&e,
75				phi => \&phi,
76				tau => \&tau,
77				pi => \&pi,
78				ln2 => \&ln2,
79				euler => \&euler,
80				i => \&i,
81				catalan => \&catalan,
82				Inf => \&inf,
83				NaN => \&nan,
84				);
85
86				my %trig = (
87				sin => sub (_) { goto &sin }, # built-in function
88				sinh => \&sinh,
89				asin => \&asin,
90				asinh => \&asinh,
91
92				cos => sub (_) { goto &cos }, # built-in function
93				cosh => \&cosh,
94				acos => \&acos,
95				acosh => \&acosh,
96
97				tan => \&tan,
98				tanh => \&tanh,
99				atan => \&atan,
100				atanh => \&atanh,
101
102				cot => \&cot,
103				coth => \&coth,
104				acot => \&acot,
105				acoth => \&acoth,
106
107				sec => \&sec,
108				sech => \&sech,
109				asec => \&asec,
110				asech => \&asech,
111
112				csc => \&csc,
113				csch => \&csch,
114				acsc => \&acsc,
115				acsch => \&acsch,
116
117				atan2 => \&atan2,
118				deg2rad => \°2rad,
119				rad2deg => \&rad2deg,
120				);
121
122				my %special = (
123				beta => \&beta,
124				zeta => \&zeta,
125				eta => \&eta,
126
127				gamma => \&gamma,
128				lgamma => \&lgamma,
129				lngamma => \&lngamma,
130				digamma => \&digamma,
131
132				Ai => \&Ai,
133				Ei => \&Ei,
134				Li => \&Li,
135				Li2 => \&Li2,
136
137				lgrt => \&lgrt,
138				LambertW => \&LambertW,
139
140				BesselJ => \&BesselJ,
141				BesselY => \&BesselY,
142
143				pow => \&pow,
144				sqr => \&sqr,
145				norm => \&norm,
146				sqrt => sub (_) { goto &sqrt }, # built-in function
147				cbrt => \&cbrt,
148				root => \&root,
149
150				exp => sub (_) { goto &exp }, # built-in function
151				exp2 => \&exp2,
152				exp10 => \&exp10,
153
154				ln => sub ($) { goto &ln }, # used in overloading
155				log => \&log, # built-in function
156				log2 => \&log2,
157				log10 => \&log10,
158
159				mod => \&mod,
160				abs => sub (_) { goto &abs }, # built-in function
161
162				erf => \&erf,
163				erfc => \&erfc,
164
165				hypot => \&hypot,
166				agm => \&agm,
167
168				bernreal => \&bernreal,
169				harmreal => \&harmreal,
170
171				polygonal_root => \&polygonal_root,
172				polygonal_root2 => \&polygonal_root2,
173				);
174
175				my %ntheory = (
176				factorial => \&factorial,
177				dfactorial => \&dfactorial,
178				mfactorial => \&mfactorial,
179				primorial => \&primorial,
180				binomial => \&binomial,
181
182				rising_factorial => \&rising_factorial,
183				falling_factorial => \&falling_factorial,
184
185				lucas => \&lucas,
186				fibonacci => \&fibonacci,
187
188				faulhaber_sum => \&faulhaber_sum,
189
190				bernfrac => \&bernfrac,
191				harmfrac => \&harmfrac,
192
193				lcm => \&lcm,
194				gcd => \&gcd,
195				valuation => \&valuation,
196				kronecker => \&kronecker,
197
198				remdiv => \&remdiv,
199				divmod => \&divmod,
200
201				iadd => \&iadd,
202				isub => \&isub,
203				imul => \&imul,
204				idiv => \&idiv,
205				imod => \&imod,
206
207				ipow => \&ipow,
208				ipow2 => \&ipow2,
209				ipow10 => \&ipow10,
210
211				iroot => \&iroot,
212				isqrt => \&isqrt,
213				icbrt => \&icbrt,
214
215				ilog => \&ilog,
216				ilog2 => \&ilog2,
217				ilog10 => \&ilog10,
218
219				isqrtrem => \&isqrtrem,
220				irootrem => \&irootrem,
221
222				polygonal => \&polygonal,
223				ipolygonal_root => \&ipolygonal_root,
224				ipolygonal_root2 => \&ipolygonal_root2,
225
226				powmod => \&powmod,
227				invmod => \&invmod,
228
229				is_power => \&is_power,
230				is_square => \&is_square,
231				is_polygonal => \&is_polygonal,
232				is_polygonal2 => \&is_polygonal2,
233
234				is_prime => \&is_prime,
235				is_coprime => \&is_coprime,
236				next_prime => \&next_prime,
237				);
238
239				my %misc = (
240				rand => \&rand,
241				irand => \&irand,
242
243				seed => \&seed,
244				iseed => \&iseed,
245
246				floor => \&floor,
247				ceil => \&ceil,
248				round => \&round,
249				sgn => \&sgn,
250				acmp => \&acmp,
251
252				popcount => \&popcount,
253
254				neg => sub ($) { goto &neg }, # used in overloading
255				inv => \&inv,
256				conj => \&conj,
257				real => \&real,
258				imag => \&imag,
259				reals => \&reals,
260
261				int => sub (_) { goto &int }, # built-in function
262				rat => \&rat,
263				float => \&float,
264				complex => \&complex,
265
266				numerator => \&numerator,
267				denominator => \&denominator,
268				nude => \&nude,
269
270				digits => \&digits,
271
272				as_bin => \&as_bin,
273				as_hex => \&as_hex,
274				as_oct => \&as_oct,
275				as_int => \&as_int,
276				as_frac => \&as_frac,
277				as_dec => \&as_dec,
278
279				rat_approx => \&rat_approx,
280
281				is_inf => \&is_inf,
282				is_ninf => \&is_ninf,
283				is_neg => \&is_neg,
284				is_pos => \&is_pos,
285				is_nan => \&is_nan,
286				is_rat => \&is_rat,
287				is_real => \&is_real,
288				is_imag => \&is_imag,
289				is_int => \&is_int,
290				is_complex => \&is_complex,
291				is_zero => \&is_zero,
292				is_one => \&is_one,
293				is_mone => \&is_mone,
294
295				is_odd => \&is_odd,
296				is_even => \&is_even,
297				is_div => \&is_div,
298				);
299
300				sub import {
301				shift;
302
303				my $caller = caller(0);
304
305				while (@_) {
306				my $name = shift(@_);
307
308				if ($name eq ':overload') {
309				overload::constant
310				integer => sub { bless \Math::GMPz::Rmpz_init_set_ui($_[0]) },
311				float => sub { bless \_str2obj($_[0]) },
312				binary => sub {
313				my $const = ($_[0] =~ tr/_//dr);
314				my $prefix = substr($const, 0, 2);
315				bless \(
316				$prefix eq '0x' ? Math::GMPz::Rmpz_init_set_str(substr($const, 2) \|\| 0, 16)
317				: $prefix eq '0b' ? Math::GMPz::Rmpz_init_set_str(substr($const, 2) \|\| 0, 2)
318				: Math::GMPz::Rmpz_init_set_str(substr($const, 1) \|\| 0, 8)
319				);
320				};
321
322				# Export 'Inf', 'NaN' and 'i' as constants
323				foreach my $pair (['Inf', inf()], ['NaN', nan()], ['i', i()]) {
324				my $sub = $caller . '::' . $pair->[0];
325				no strict 'refs';
326				no warnings 'redefine';
327				my $value = $pair->[1];
328				*$sub = sub () { $value };
329				}
330				}
331				elsif (exists $const{$name}) {
332				no strict 'refs';
333				no warnings 'redefine';
334				my $caller_sub = $caller . '::' . $name;
335				my $sub = $const{$name};
336				my $value = $sub->();
337				*$caller_sub = sub() { $value }
338				}
339				elsif ( exists($special{$name})
340				or exists($trig{$name})
341				or exists($ntheory{$name})
342				or exists($misc{$name})) {
343				no strict 'refs';
344				no warnings 'redefine';
345				my $caller_sub = $caller . '::' . $name;
346				*$caller_sub = $ntheory{$name} // $special{$name} // $trig{$name} // $misc{$name};
347				}
348				elsif ($name eq ':trig') {
349				push @_, keys(%trig);
350				}
351				elsif ($name eq ':ntheory') {
352				push @_, keys(%ntheory);
353				}
354				elsif ($name eq ':special') {
355				push @_, keys(%special);
356				}
357				elsif ($name eq ':misc') {
358				push @_, keys(%misc);
359				}
360				elsif ($name eq ':all') {
361				push @_, keys(%const), keys(%trig), keys(%special), keys(%ntheory), keys(%misc);
362				}
363				elsif ($name eq 'PREC') {
364				my $prec = CORE::int(shift(@_));
365				if ( $prec < Math::MPFR::RMPFR_PREC_MIN()
366				or $prec > Math::MPFR::RMPFR_PREC_MAX()) {
367				die "invalid value for <>: must be between "
368				. Math::MPFR::RMPFR_PREC_MIN() . " and "
369				. Math::MPFR::RMPFR_PREC_MAX()
370				. ", but got <<$prec>>";
371				}
372				$PREC = $prec;
373				}
374				else {
375				die "unknown import: <<$name>>";
376				}
377				}
378				return;
379				}
380
381				sub unimport {
382				overload::remove_constant('binary', '', 'float', '', 'integer');
383				}
384				}
385
386				# Converts a string into an mpq object
387				sub _str2obj {
388				my ($s) = @_;
389
390				$s \|\| goto &_zero;
391
392				$s = lc($s);
393
394				if ($s eq 'inf' or $s eq '+inf') {
395				goto &_inf;
396				}
397				elsif ($s eq '-inf') {
398				goto &_ninf;
399				}
400				elsif ($s eq 'nan') {
401				goto &_nan;
402				}
403
404				# Remove underscores
405				$s =~ tr/_//d;
406
407				# Performance improvement for Perl integers
408				if (CORE::int($s) eq $s and $s >= LONG_MIN and $s <= ULONG_MAX) {
409				return (
410				$s < 0
411				? Math::GMPz::Rmpz_init_set_si($s)
412				: Math::GMPz::Rmpz_init_set_ui($s)
413				);
414				}
415
416				# Complex number
417				if (substr($s, -1) eq 'i') {
418
419				if ($s eq 'i' or $s eq '+i') {
420				my $r = Math::MPC::Rmpc_init2($PREC);
421				Math::MPC::Rmpc_set_ui_ui($r, 0, 1, $ROUND);
422				return $r;
423				}
424				elsif ($s eq '-i') {
425				my $r = Math::MPC::Rmpc_init2($PREC);
426				Math::MPC::Rmpc_set_si_si($r, 0, -1, $ROUND);
427				return $r;
428				}
429
430				my ($re, $im);
431
432				state $numeric_re = qr/[+-]?+(?=\.?[0-9])[0-9]*+(?:\.[0-9]++)?(?:[Ee](?:[+-]?+[0-9]+))?/;
433				state $unsigned_re = qr/(?=\.?[0-9])[0-9]*+(?:\.[0-9]++)?(?:[Ee](?:[+-]?+[0-9]+))?/;
434
435				if ($s =~ /^($numeric_re)\s([-+])\s($unsigned_re)i\z/o) {
436				($re, $im) = ($1, $3);
437				$im = "-$im" if $2 eq '-';
438				}
439				elsif ($s =~ /^($numeric_re)i\z/o) {
440				($re, $im) = (0, $1);
441				}
442				elsif ($s =~ /^($numeric_re)\s([-+])\si\z/o) {
443				($re, $im) = ($1, 1);
444				$im = -1 if $2 eq '-';
445				}
446
447				if (defined($re) and defined($im)) {
448
449				my $r = Math::MPC::Rmpc_init2($PREC);
450
451				$re = _str2obj($re);
452				$im = _str2obj($im);
453
454				my $sig = join(' ', ref($re), ref($im));
455
456				if ($sig eq q{Math::MPFR Math::MPFR}) {
457				Math::MPC::Rmpc_set_fr_fr($r, $re, $im, $ROUND);
458				}
459				elsif ($sig eq q{Math::GMPz Math::GMPz}) {
460				Math::MPC::Rmpc_set_z_z($r, $re, $im, $ROUND);
461				}
462				elsif ($sig eq q{Math::GMPz Math::MPFR}) {
463				Math::MPC::Rmpc_set_z_fr($r, $re, $im, $ROUND);
464				}
465				elsif ($sig eq q{Math::MPFR Math::GMPz}) {
466				Math::MPC::Rmpc_set_fr_z($r, $re, $im, $ROUND);
467				}
468				else { # this should never happen
469				$re = _any2mpfr($re);
470				$im = _any2mpfr($im);
471				Math::MPC::Rmpc_set_fr_fr($r, $re, $im, $ROUND);
472				}
473
474				return $r;
475				}
476				}
477
478				# Floating point value
479				if ($s =~ tr/e.//) {
480				my $r = Math::MPFR::Rmpfr_init2($PREC);
481				if (Math::MPFR::Rmpfr_set_str($r, $s, 10, $ROUND)) {
482				Math::MPFR::Rmpfr_set_nan($r);
483				}
484				return $r;
485				}
486
487				# Fractional value
488				if (index($s, '/') != -1 and $s =~ m{^\s[-+]?[0-9]+\s/\s[-+]?[1-9]+[0-9]\s*\z}) {
489				my $r = Math::GMPq::Rmpq_init();
490				Math::GMPq::Rmpq_set_str($r, $s, 10);
491				Math::GMPq::Rmpq_canonicalize($r);
492				return $r;
493				}
494
495				$s =~ s/^\+//;
496
497				eval { Math::GMPz::Rmpz_init_set_str($s, 10) } // goto &_nan;
498				}
499
500				# Parse a base-10 string as a base-10 fraction
501				sub _str2frac {
502				my ($str) = @_;
503
504				my $sign = substr($str, 0, 1);
505				if ($sign eq '-') {
506				substr($str, 0, 1, '');
507				$sign = '-';
508				}
509				else {
510				substr($str, 0, 1, '') if ($sign eq '+');
511				$sign = '';
512				}
513
514				my $i;
515				if (($i = index($str, 'e')) != -1) {
516
517				my $exp = substr($str, $i + 1);
518
519				# Handle specially numbers with very big exponents
520				# (not a very good solution, but this will happen very rarely, if ever)
521				if (CORE::abs($exp) >= 1000000) {
522				Math::MPFR::Rmpfr_set_str((my $mpfr = Math::MPFR::Rmpfr_init2($PREC)), "$sign$str", 10, $ROUND);
523				Math::MPFR::Rmpfr_get_q((my $mpq = Math::GMPq::Rmpq_init()), $mpfr);
524				return Math::GMPq::Rmpq_get_str($mpq, 10);
525				}
526
527				my ($before, $after) = split(/\./, substr($str, 0, $i));
528
529				if (!defined($after)) { # return faster for numbers like "13e2"
530				if ($exp >= 0) {
531				return ("$sign$before" . ('0' x $exp));
532				}
533				else {
534				$after = '';
535				}
536				}
537
538				my $numerator = "$before$after";
539				my $denominator = "1";
540
541				if ($exp < 1) {
542				$denominator .= '0' x (CORE::abs($exp) + CORE::length($after));
543				}
544				else {
545				my $diff = ($exp - CORE::length($after));
546				if ($diff >= 0) {
547				$numerator .= '0' x $diff;
548				}
549				else {
550				my $s = "$before$after";
551				substr($s, $exp + CORE::length($before), 0, '.');
552				return _str2frac("$sign$s");
553				}
554				}
555
556				"$sign$numerator/$denominator";
557				}
558				elsif (($i = index($str, '.')) != -1) {
559				my ($before, $after) = (substr($str, 0, $i), substr($str, $i + 1));
560				if (($after =~ tr/0//) == CORE::length($after)) {
561				return "$sign$before";
562				}
563				$sign . ("$before$after/1" =~ s/^0+//r) . ('0' x CORE::length($after));
564				}
565				else {
566				"$sign$str";
567				}
568				}
569
570				#
571				## MPZ
572				#
573				sub _mpz2mpq {
574				my $r = Math::GMPq::Rmpq_init();
575				Math::GMPq::Rmpq_set_z($r, $_[0]);
576				$r;
577				}
578
579				sub _mpz2mpfr {
580				my $r = Math::MPFR::Rmpfr_init2($PREC);
581				Math::MPFR::Rmpfr_set_z($r, $_[0], $ROUND);
582				$r;
583				}
584
585				sub _mpz2mpc {
586				my $r = Math::MPC::Rmpc_init2($PREC);
587				Math::MPC::Rmpc_set_z($r, $_[0], $ROUND);
588				$r;
589				}
590
591				#
592				## MPQ
593				#
594
595				sub _mpq2mpz {
596				my $z = Math::GMPz::Rmpz_init();
597				Math::GMPz::Rmpz_set_q($z, $_[0]);
598				$z;
599				}
600
601				sub _mpq2mpfr {
602				my $r = Math::MPFR::Rmpfr_init2($PREC);
603				Math::MPFR::Rmpfr_set_q($r, $_[0], $ROUND);
604				$r;
605				}
606
607				sub _mpq2mpc {
608				my $r = Math::MPC::Rmpc_init2($PREC);
609				Math::MPC::Rmpc_set_q($r, $_[0], $ROUND);
610				$r;
611				}
612
613				#
614				## MPFR
615				#
616
617				sub _mpfr2mpc {
618				my $r = Math::MPC::Rmpc_init2($PREC);
619				Math::MPC::Rmpc_set_fr($r, $_[0], $ROUND);
620				$r;
621				}
622
623				#
624				## Any
625				#
626
627				sub _any2mpc {
628				my ($x) = @_;
629
630				ref($x) eq 'Math::MPC' && return $x;
631				ref($x) eq 'Math::GMPq' && goto &_mpq2mpc;
632				ref($x) eq 'Math::GMPz' && goto &_mpz2mpc;
633
634				goto &_mpfr2mpc;
635				}
636
637				sub _any2mpfr {
638				my ($x) = @_;
639
640				ref($x) eq 'Math::MPFR' && return $x;
641				ref($x) eq 'Math::GMPq' && goto &_mpq2mpfr;
642				ref($x) eq 'Math::GMPz' && goto &_mpz2mpfr;
643
644				my $fr = Math::MPFR::Rmpfr_init2($PREC);
645				Math::MPC::RMPC_IM($fr, $x);
646
647				Math::MPFR::Rmpfr_zero_p($fr)
648				? Math::MPC::RMPC_RE($fr, $x)
649				: Math::MPFR::Rmpfr_set_nan($fr);
650
651				$fr;
652				}
653
654				sub _any2mpz {
655				my ($x) = @_;
656
657				ref($x) eq 'Math::GMPz' && return $x;
658				ref($x) eq 'Math::GMPq' && goto &_mpq2mpz;
659
660				if (ref($x) eq 'Math::MPFR') {
661				if (Math::MPFR::Rmpfr_number_p($x)) {
662				my $z = Math::GMPz::Rmpz_init();
663				Math::MPFR::Rmpfr_get_z($z, $x, Math::MPFR::MPFR_RNDZ);
664				return $z;
665				}
666				return;
667				}
668
669				(@_) = _any2mpfr($x);
670				goto &_any2mpz;
671				}
672
673				sub _any2mpq {
674				my ($x) = @_;
675
676				ref($x) eq 'Math::GMPq' && return $x;
677				ref($x) eq 'Math::GMPz' && goto &_mpz2mpq;
678
679				if (ref($x) eq 'Math::MPFR') {
680				if (Math::MPFR::Rmpfr_number_p($x)) {
681				my $q = Math::GMPq::Rmpq_init();
682				Math::MPFR::Rmpfr_get_q($q, $x);
683				return $q;
684				}
685				return;
686				}
687
688				(@_) = _any2mpfr($x);
689				goto &_any2mpq;
690				}
691
692				sub _any2ui {
693				my ($x) = @_;
694				goto(ref($x) =~ tr/:/_/rs);
695
696				Math_GMPz: {
697
698				if (Math::GMPz::Rmpz_fits_ulong_p($x)) {
699				goto &Math::GMPz::Rmpz_get_ui;
700				}
701
702				return;
703				}
704
705				Math_GMPq: {
706
707				if (Math::GMPq::Rmpq_integer_p($x)) {
708				@_ = ($x = _mpq2mpz($x));
709				goto Math_GMPz;
710				}
711
712				my $d = CORE::int(Math::GMPq::Rmpq_get_d($x));
713				return (($d < 0 or $d > ULONG_MAX) ? undef : $d);
714				}
715
716				Math_MPFR: {
717
718				if (Math::MPFR::Rmpfr_integer_p($x) and Math::MPFR::Rmpfr_fits_ulong_p($x, $ROUND)) {
719				push @_, $ROUND;
720				goto &Math::MPFR::Rmpfr_get_ui;
721				}
722
723				if (Math::MPFR::Rmpfr_number_p($x)) {
724				my $d = CORE::int(Math::MPFR::Rmpfr_get_d($x, $ROUND));
725				return (($d < 0 or $d > ULONG_MAX) ? undef : $d);
726				}
727
728				return;
729				}
730
731				Math_MPC: {
732				@_ = ($x = _any2mpfr($x));
733				goto Math_MPFR;
734				}
735				}
736
737				sub _any2si {
738				my ($x) = @_;
739				goto(ref($x) =~ tr/:/_/rs);
740
741				Math_GMPz: {
742
743				if (Math::GMPz::Rmpz_fits_slong_p($x)) {
744				goto &Math::GMPz::Rmpz_get_si;
745				}
746
747				if (Math::GMPz::Rmpz_fits_ulong_p($x)) {
748				goto &Math::GMPz::Rmpz_get_ui;
749				}
750
751				return;
752				}
753
754				Math_GMPq: {
755
756				if (Math::GMPq::Rmpq_integer_p($x)) {
757				@_ = ($x = _mpq2mpz($x));
758				goto Math_GMPz;
759				}
760
761				my $d = CORE::int(Math::GMPq::Rmpq_get_d($x));
762				return (($d < LONG_MIN or $d > ULONG_MAX) ? undef : $d);
763				}
764
765				Math_MPFR: {
766
767				if (Math::MPFR::Rmpfr_integer_p($x)) {
768				if (Math::MPFR::Rmpfr_fits_slong_p($x, $ROUND)) {
769				push @_, $ROUND;
770				goto &Math::MPFR::Rmpfr_get_si;
771				}
772
773				if (Math::MPFR::Rmpfr_fits_ulong_p($x, $ROUND)) {
774				push @_, $ROUND;
775				goto &Math::MPFR::Rmpfr_get_ui;
776				}
777				}
778
779				if (Math::MPFR::Rmpfr_number_p($x)) {
780				my $d = CORE::int(Math::MPFR::Rmpfr_get_d($x, $ROUND));
781				return (($d < LONG_MIN or $d > ULONG_MAX) ? undef : $d);
782				}
783
784				return;
785				}
786
787				Math_MPC: {
788				@_ = ($x = _any2mpfr($x));
789				goto Math_MPFR;
790				}
791				}
792
793				#
794				## Anything to MPFR (including strings)
795				#
796				sub _star2mpfr {
797				my ($x) = @_;
798
799				$x =
800				ref($x) eq __PACKAGE__ ? $$x
801				: ref($x) ? _star2obj($x)
802				: _str2obj($x);
803
804				ref($x) eq 'Math::MPFR' and return $x;
805
806				(@_) = $x;
807				ref($x) eq 'Math::GMPz' && goto &_mpz2mpfr;
808				ref($x) eq 'Math::GMPq' && goto &_mpq2mpfr;
809				goto &_any2mpfr;
810				}
811
812				#
813				## Anything to GMPz (including strings)
814				#
815				sub _star2mpz {
816				my ($x) = @_;
817
818				$x =
819				ref($x) eq __PACKAGE__ ? $$x
820				: ref($x) ? _star2obj($x)
821				: _str2obj($x);
822
823				ref($x) eq 'Math::GMPz' and return $x;
824
825				(@_) = $x;
826				ref($x) eq 'Math::GMPq' and goto &_mpq2mpz;
827				goto &_any2mpz;
828				}
829
830				#
831				## Anything to MPFR or MPC, in this order (including strings)
832				#
833				sub _star2mpfr_mpc {
834				my ($x) = @_;
835
836				$x =
837				ref($x) eq __PACKAGE__ ? $$x
838				: ref($x) ? _star2obj($x)
839				: _str2obj($x);
840
841				if ( ref($x) eq 'Math::MPFR'
842				or ref($x) eq 'Math::MPC') {
843				return $x;
844				}
845
846				(@_) = $x;
847				ref($x) eq 'Math::GMPz' && goto &_mpz2mpfr;
848				ref($x) eq 'Math::GMPq' && goto &_mpq2mpfr;
849				goto &_any2mpfr; # this should not happen
850				}
851
852				# Anything to a {GMP*, MPFR or MPC} object
853				sub _star2obj {
854				my ($x) = @_;
855
856				ref($x) \|\| goto &_str2obj;
857
858				if (ref($x) eq __PACKAGE__) {
859				$$x;
860				}
861				elsif (
862				ref($x)
863				and ( ref($x) eq 'Math::GMPz'
864				or ref($x) eq 'Math::GMPq'
865				or ref($x) eq 'Math::MPFR'
866				or ref($x) eq 'Math::MPC')
867				) {
868				$x;
869				}
870				else {
871				(@_) = "$x";
872				goto &_str2obj;
873				}
874				}
875
876				sub new {
877				my ($class, $num, $base) = @_;
878
879				my $ref = ref($num);
880
881				# Special string values
882				if (!$ref and (!defined($base) or CORE::int($base) == 10)) {
883				return bless \_str2obj($num), $class;
884				}
885
886				# Special case
887				if (!defined($base) and $ref eq __PACKAGE__) {
888				return $num;
889				}
890
891				# Number with base
892				if (defined($base) and CORE::int($base) != 10) {
893
894				my $int_base = CORE::int($base);
895
896				if ($int_base < 2 or $int_base > 36) {
897				require Carp;
898				Carp::croak("base must be between 2 and 36, got $base");
899				}
900
901				$num = defined($num) ? "$num" : '0';
902
903				if (index($num, '/') != -1) {
904				my $r = Math::GMPq::Rmpq_init();
905				eval { Math::GMPq::Rmpq_set_str($r, $num, $int_base); 1 } // goto &nan;
906
907				if (Math::GMPq::Rmpq_get_str($r, 10) !~ m{^\s[-+]?[0-9]+\s/\s[-+]?[1-9]+[0-9]\s*\z}) {
908				goto &nan;
909				}
910
911				Math::GMPq::Rmpq_canonicalize($r);
912				return bless \$r, $class;
913				}
914				elsif (index($num, '.') != -1) {
915				my $r = Math::MPFR::Rmpfr_init2($PREC);
916				if (Math::MPFR::Rmpfr_set_str($r, $num, $int_base, $ROUND)) {
917				Math::MPFR::Rmpfr_set_nan($r);
918				}
919				return bless \$r, $class;
920				}
921				else {
922				return bless \(eval { Math::GMPz::Rmpz_init_set_str($num, $int_base) } // goto &nan), $class;
923				}
924				}
925
926				bless \_star2obj($num), $class;
927				}
928
929				sub new_si {
930				my ($class, $si) = @_;
931				bless \Math::GMPz::Rmpz_init_set_si($si), $class;
932				}
933
934				sub new_ui {
935				my ($class, $ui) = @_;
936				bless \Math::GMPz::Rmpz_init_set_ui($ui), $class;
937				}
938
939				sub new_z {
940				my ($class, $str, $base) = @_;
941				bless \Math::GMPz::Rmpz_init_set_str($str, $base // 10), $class;
942				}
943
944				sub new_q {
945				my ($class, $num, $den, $base) = @_;
946				my $r = Math::GMPq::Rmpq_init();
947
948				if (defined($den)) {
949				Math::GMPq::Rmpq_set_str($r, "$num/$den", $base // 10);
950				}
951				else {
952				Math::GMPq::Rmpq_set_str($r, "$num", $base // 10);
953				}
954
955				Math::GMPq::Rmpq_canonicalize($r);
956				bless \$r, $class;
957				}
958
959				sub new_f {
960				my ($class, $str, $base) = @_;
961				my $r = Math::MPFR::Rmpfr_init2($PREC);
962				Math::MPFR::Rmpfr_set_str($r, $str, $base // 10, $ROUND);
963				bless \$r, $class;
964				}
965
966				sub new_c {
967				my ($class, $real, $imag, $base) = @_;
968
969				my $c = Math::MPC::Rmpc_init2($PREC);
970
971				if (defined($imag)) {
972				my $re = Math::MPFR::Rmpfr_init2($PREC);
973				my $im = Math::MPFR::Rmpfr_init2($PREC);
974
975				Math::MPFR::Rmpfr_set_str($re, $real, $base // 10, $ROUND);
976				Math::MPFR::Rmpfr_set_str($im, $imag, $base // 10, $ROUND);
977
978				Math::MPC::Rmpc_set_fr_fr($c, $re, $im, $ROUND);
979				}
980				else {
981				Math::MPC::Rmpc_set_str($c, $real, $base // 10, $ROUND);
982				}
983
984				bless \$c, $class;
985				}
986
987				sub _nan {
988				state $nan = do {
989				my $r = Math::MPFR::Rmpfr_init2($PREC);
990				Math::MPFR::Rmpfr_set_nan($r);
991				$r;
992				};
993				}
994
995				sub nan {
996				state $nan = do {
997				my $r = Math::MPFR::Rmpfr_init2($PREC);
998				Math::MPFR::Rmpfr_set_nan($r);
999				bless \$r;
1000				};
1001				}
1002
1003				sub _inf {
1004				state $inf = do {
1005				my $r = Math::MPFR::Rmpfr_init2($PREC);
1006				Math::MPFR::Rmpfr_set_inf($r, 1);
1007				$r;
1008				};
1009				}
1010
1011				sub inf {
1012				state $inf = do {
1013				my $r = Math::MPFR::Rmpfr_init2($PREC);
1014				Math::MPFR::Rmpfr_set_inf($r, 1);
1015				bless \$r;
1016				};
1017				}
1018
1019				sub _ninf {
1020				state $ninf = do {
1021				my $r = Math::MPFR::Rmpfr_init2($PREC);
1022				Math::MPFR::Rmpfr_set_inf($r, -1);
1023				$r;
1024				};
1025				}
1026
1027				sub ninf {
1028				state $ninf = do {
1029				my $r = Math::MPFR::Rmpfr_init2($PREC);
1030				Math::MPFR::Rmpfr_set_inf($r, -1);
1031				bless \$r;
1032				};
1033				}
1034
1035				sub _zero {
1036				state $zero = Math::GMPz::Rmpz_init_set_ui(0);
1037				}
1038
1039				sub zero {
1040				state $zero = do {
1041				my $r = Math::GMPz::Rmpz_init_set_ui(0);
1042				bless \$r;
1043				};
1044				}
1045
1046				sub _one {
1047				state $one = Math::GMPz::Rmpz_init_set_ui(1);
1048				}
1049
1050				sub one {
1051				state $one = do {
1052				my $r = Math::GMPz::Rmpz_init_set_ui(1);
1053				bless \$r;
1054				};
1055				}
1056
1057				sub _mone {
1058				state $mone = Math::GMPz::Rmpz_init_set_si(-1);
1059				}
1060
1061				sub mone {
1062				state $mone = do {
1063				my $r = Math::GMPz::Rmpz_init_set_si(-1);
1064				bless \$r;
1065				};
1066				}
1067
1068				#
1069				## CONSTANTS
1070				#
1071
1072				sub pi {
1073				my $pi = Math::MPFR::Rmpfr_init2($PREC);
1074				Math::MPFR::Rmpfr_const_pi($pi, $ROUND);
1075				bless \$pi;
1076				}
1077
1078				sub tau {
1079				my $tau = Math::MPFR::Rmpfr_init2($PREC);
1080				Math::MPFR::Rmpfr_const_pi($tau, $ROUND);
1081				Math::MPFR::Rmpfr_mul_2ui($tau, $tau, 1, $ROUND);
1082				bless \$tau;
1083				}
1084
1085				sub ln2 {
1086				my $ln2 = Math::MPFR::Rmpfr_init2($PREC);
1087				Math::MPFR::Rmpfr_const_log2($ln2, $ROUND);
1088				bless \$ln2;
1089				}
1090
1091				sub euler {
1092				my $euler = Math::MPFR::Rmpfr_init2($PREC);
1093				Math::MPFR::Rmpfr_const_euler($euler, $ROUND);
1094				bless \$euler;
1095				}
1096
1097				sub catalan {
1098				my $catalan = Math::MPFR::Rmpfr_init2($PREC);
1099				Math::MPFR::Rmpfr_const_catalan($catalan, $ROUND);
1100				bless \$catalan;
1101				}
1102
1103				sub i {
1104				my $i = Math::MPC::Rmpc_init2($PREC);
1105				Math::MPC::Rmpc_set_ui_ui($i, 0, 1, $ROUND);
1106				bless \$i;
1107				}
1108
1109				sub e {
1110				state $one_f = (Math::MPFR::Rmpfr_init_set_ui_nobless(1, $ROUND))[0];
1111				my $e = Math::MPFR::Rmpfr_init2($PREC);
1112				Math::MPFR::Rmpfr_exp($e, $one_f, $ROUND);
1113				bless \$e;
1114				}
1115
1116				sub phi {
1117				state $five4_f = (Math::MPFR::Rmpfr_init_set_d_nobless(1.25, $ROUND))[0];
1118
1119				my $phi = Math::MPFR::Rmpfr_init2($PREC);
1120				Math::MPFR::Rmpfr_sqrt($phi, $five4_f, $ROUND);
1121				Math::MPFR::Rmpfr_add_d($phi, $phi, 0.5, $ROUND);
1122
1123				bless \$phi;
1124				}
1125
1126				#
1127				## OTHER
1128				#
1129
1130				sub stringify { # used in overloading
1131				require Math::AnyNum::stringify;
1132				(@_) = (${$_[0]});
1133				goto &__stringify__;
1134				}
1135
1136				sub numify { # used in overloading
1137				require Math::AnyNum::numify;
1138				(@_) = (${$_[0]});
1139				goto &__numify__;
1140				}
1141
1142				sub boolify { # used in overloading
1143				require Math::AnyNum::boolify;
1144				(@_) = (${$_[0]});
1145				goto &__boolify__;
1146				}
1147
1148				#
1149				## EQUALITY
1150				#
1151
1152				sub eq { # used in overloading
1153				require Math::AnyNum::eq;
1154				my ($x, $y) = @_;
1155
1156				if (ref($y) eq __PACKAGE__) {
1157				(@_) = ($$x, $$y);
1158				goto &__eq__;
1159				}
1160
1161				if (!ref($y)) {
1162				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1163				(@_) = ($$x, $y);
1164				}
1165				else {
1166				(@_) = ($$x, _str2obj($y));
1167				}
1168				goto &__eq__;
1169				}
1170
1171				(@_) = ($$x, _star2obj($y));
1172				goto &__eq__;
1173				}
1174
1175				#
1176				## INEQUALITY
1177				#
1178
1179				sub ne { # used in overloading
1180				require Math::AnyNum::ne;
1181				my ($x, $y) = @_;
1182
1183				if (ref($y) eq __PACKAGE__) {
1184				(@_) = ($$x, $$y);
1185				goto &__ne__;
1186				}
1187
1188				if (!ref($y)) {
1189				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1190				(@_) = ($$x, $y);
1191				}
1192				else {
1193				(@_) = ($$x, _str2obj($y));
1194				}
1195				goto &__ne__;
1196				}
1197
1198				(@_) = ($$x, _star2obj($y));
1199				goto &__ne__;
1200				}
1201
1202				#
1203				## COMPARISON
1204				#
1205
1206				sub cmp ($$) {
1207				require Math::AnyNum::cmp;
1208				my ($x, $y) = @_;
1209
1210				if (ref($y) eq __PACKAGE__) {
1211				(@_) = ($$x, $$y);
1212				goto &__cmp__;
1213				}
1214
1215				if (!ref($y)) {
1216				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1217				(@_) = ($$x, $y);
1218				}
1219				else {
1220				(@_) = ($$x, _str2obj($y));
1221				}
1222				goto &__cmp__;
1223				}
1224
1225				(@_) = ($$x, _star2obj($y));
1226				goto &__cmp__;
1227				}
1228
1229				sub acmp ($$) {
1230				require Math::AnyNum::abs;
1231				require Math::AnyNum::cmp;
1232				my ($x, $y) = @_;
1233
1234				if (!ref($y) and CORE::int($y) eq $y and $y >= 0 and $y <= ULONG_MAX) {
1235				## `y` is a native unsigned integer
1236				}
1237				else {
1238				$y = __abs__(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y));
1239				}
1240
1241				__cmp__(__abs__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), $y);
1242				}
1243
1244				#
1245				## GREATER THAN
1246				#
1247
1248				sub gt { # used in overloading
1249				require Math::AnyNum::cmp;
1250				my ($x, $y) = @_;
1251
1252				if (ref($y) eq __PACKAGE__) {
1253				return ((__cmp__($$x, $$y) // return undef) > 0);
1254				}
1255
1256				if (!ref($y)) {
1257				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1258				return ((__cmp__($$x, $y) // return undef) > 0);
1259				}
1260				return ((__cmp__($$x, _str2obj($y)) // return undef) > 0);
1261				}
1262
1263				(__cmp__($$x, _star2obj($y)) // return undef) > 0;
1264				}
1265
1266				#
1267				## EQUAL OR GREATER THAN
1268				#
1269
1270				sub ge { # used in overloading
1271				require Math::AnyNum::cmp;
1272				my ($x, $y) = @_;
1273
1274				if (ref($y) eq __PACKAGE__) {
1275				return ((__cmp__($$x, $$y) // return undef) >= 0);
1276				}
1277
1278				if (!ref($y)) {
1279				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1280				return ((__cmp__($$x, $y) // return undef) >= 0);
1281				}
1282				return ((__cmp__($$x, _str2obj($y)) // return undef) >= 0);
1283				}
1284
1285				(__cmp__($$x, _star2obj($y)) // return undef) >= 0;
1286				}
1287
1288				#
1289				## LESS THAN
1290				#
1291
1292				sub lt { # used in overloading
1293				require Math::AnyNum::cmp;
1294				my ($x, $y) = @_;
1295
1296				if (ref($y) eq __PACKAGE__) {
1297				return ((__cmp__($$x, $$y) // return undef) < 0);
1298				}
1299
1300				if (!ref($y)) {
1301				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1302				return ((__cmp__($$x, $y) // return undef) < 0);
1303				}
1304				return ((__cmp__($$x, _str2obj($y)) // return undef) < 0);
1305				}
1306
1307				(__cmp__($$x, _star2obj($y)) // return undef) < 0;
1308				}
1309
1310				#
1311				## EQUAL OR LESS THAN
1312				#
1313
1314				sub le { # used in overloading
1315				require Math::AnyNum::cmp;
1316				my ($x, $y) = @_;
1317
1318				if (ref($y) eq __PACKAGE__) {
1319				return ((__cmp__($$x, $$y) // return undef) <= 0);
1320				}
1321
1322				if (!ref($y)) {
1323				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1324				return ((__cmp__($$x, $y) // return undef) <= 0);
1325				}
1326				return ((__cmp__($$x, _str2obj($y)) // return undef) <= 0);
1327				}
1328
1329				(__cmp__($$x, _star2obj($y)) // return undef) <= 0;
1330				}
1331
1332				#
1333				## COPY
1334				#
1335
1336				sub _copy {
1337				my ($x) = @_;
1338				my $ref = ref($x);
1339
1340				if ($ref eq 'Math::GMPz') {
1341				Math::GMPz::Rmpz_init_set($x);
1342				}
1343				elsif ($ref eq 'Math::MPFR') {
1344				my $r = Math::MPFR::Rmpfr_init2($PREC);
1345				Math::MPFR::Rmpfr_set($r, $x, $ROUND);
1346				$r;
1347				}
1348				elsif ($ref eq 'Math::GMPq') {
1349				my $r = Math::GMPq::Rmpq_init();
1350				Math::GMPq::Rmpq_set($r, $x);
1351				$r;
1352				}
1353				elsif ($ref eq 'Math::MPC') {
1354				my $r = Math::MPC::Rmpc_init2($PREC);
1355				Math::MPC::Rmpc_set($r, $x, $ROUND);
1356				$r;
1357				}
1358				else {
1359				_str2obj("$x"); # this should not happen
1360				}
1361				}
1362
1363				sub copy ($) {
1364				my ($x) = @_;
1365				bless \_copy($$x);
1366				}
1367
1368				#
1369				## CONVERSION TO INTEGER
1370				#
1371
1372				sub int { # used in overloading
1373				my ($x) = @_;
1374
1375				bless \(
1376				(
1377				ref($x) eq __PACKAGE__
1378				? ref($$x) eq 'Math::GMPz'
1379				? (return $x)
1380				: _any2mpz($$x)
1381				: _star2mpz($x)
1382				) // goto &nan
1383				);
1384				}
1385
1386				#
1387				## CONVERSION TO RATIONAL
1388				#
1389
1390				sub rat ($) {
1391				my ($x) = @_;
1392				if (ref($x) eq __PACKAGE__) {
1393				ref($$x) eq 'Math::GMPq' && return $x;
1394				bless \(_any2mpq($$x) // (goto &nan));
1395				}
1396				else {
1397
1398				# Parse a decimal number as an exact fraction
1399				if ("$x" =~ /^([+-]?+(?=\.?[0-9])[0-9_]*+(?:\.[0-9_]++)?(?:[Ee](?:[+-]?+[0-9_]+))?)\z/) {
1400				my $frac = _str2frac(lc($1));
1401				my $q = Math::GMPq::Rmpq_init();
1402				Math::GMPq::Rmpq_set_str($q, $frac, 10);
1403				Math::GMPq::Rmpq_canonicalize($q) if (index($frac, '/') != -1);
1404				return bless \$q;
1405				}
1406
1407				my $r = __PACKAGE__->new($x);
1408				ref($$r) eq 'Math::GMPq' && return $r;
1409				bless(\_any2mpq($$r) // goto &nan);
1410				}
1411				}
1412
1413				#
1414				## CONVERSION TO FLOATING-POINT
1415				#
1416
1417				sub float ($) {
1418				my ($x) = @_;
1419
1420				bless \(
1421				ref($x) eq __PACKAGE__
1422				? ref($$x) eq 'Math::MPFR'
1423				? (return $x)
1424				: _any2mpfr($$x)
1425				: _star2mpfr($x)
1426				);
1427				}
1428
1429				#
1430				## CONVERSION TO COMPLEX
1431				#
1432
1433				sub complex ($) {
1434				my ($x) = @_;
1435				bless \(
1436				ref($x) eq __PACKAGE__
1437				? ref($$x) eq 'Math::MPC'
1438				? (return $x)
1439				: _any2mpc($$x)
1440				: _any2mpc(_star2obj($x))
1441				);
1442				}
1443
1444				#
1445				## NEGATION
1446				#
1447
1448				sub neg { # used in overloading
1449				require Math::AnyNum::neg;
1450				my ($x) = @_;
1451				bless \__neg__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
1452				}
1453
1454				#
1455				## ABSOLUTE VALUE
1456				#
1457
1458				sub abs { # used in overloading
1459				require Math::AnyNum::abs;
1460				my ($x) = @_;
1461				bless \__abs__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
1462				}
1463
1464				#
1465				## MULTIPLICATIVE INVERSE
1466				#
1467
1468				sub inv ($) {
1469				require Math::AnyNum::inv;
1470				my ($x) = @_;
1471				bless \__inv__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
1472				}
1473
1474				#
1475				## INCREMENTATION BY ONE
1476				#
1477
1478				sub inc ($) {
1479				require Math::AnyNum::inc;
1480				my ($x) = @_;
1481				bless \__inc__($$x);
1482				}
1483
1484				#
1485				## DECREMENTATION BY ONE
1486				#
1487
1488				sub dec ($) {
1489				require Math::AnyNum::dec;
1490				my ($x) = @_;
1491				bless \__dec__($$x);
1492				}
1493
1494				sub conj ($) {
1495				my ($x) = @_;
1496
1497				if (ref($x) ne __PACKAGE__) {
1498				$x = __PACKAGE__->new($x);
1499				}
1500
1501				if (ref($$x) eq 'Math::MPC') {
1502				my $r = Math::MPC::Rmpc_init2($PREC);
1503				Math::MPC::Rmpc_conj($r, $$x, $ROUND);
1504				bless \$r;
1505				}
1506				else {
1507				$x;
1508				}
1509				}
1510
1511				sub real ($) {
1512				my ($x) = @_;
1513
1514				if (ref($x) ne __PACKAGE__) {
1515				$x = __PACKAGE__->new($x);
1516				}
1517
1518				if (ref($$x) eq 'Math::MPC') {
1519				my $r = Math::MPFR::Rmpfr_init2($PREC);
1520				Math::MPC::RMPC_RE($r, $$x);
1521				bless \$r;
1522				}
1523				else {
1524				$x;
1525				}
1526				}
1527
1528				sub imag ($) {
1529				my ($x) = @_;
1530
1531				if (ref($x) ne __PACKAGE__) {
1532				$x = __PACKAGE__->new($x);
1533				}
1534
1535				if (ref($$x) eq 'Math::MPC') {
1536				my $r = Math::MPFR::Rmpfr_init2($PREC);
1537				Math::MPC::RMPC_IM($r, $$x);
1538				bless \$r;
1539				}
1540				else {
1541				goto &zero;
1542				}
1543				}
1544
1545				sub reals ($) {
1546				my ($x) = @_;
1547
1548				if (ref($x) ne __PACKAGE__) {
1549				$x = __PACKAGE__->new($x);
1550				}
1551
1552				($x->real, $x->imag);
1553				}
1554
1555				#
1556				## ADDITION
1557				#
1558
1559				sub add { # used in overloading
1560				require Math::AnyNum::add;
1561				my ($x, $y) = @_;
1562
1563				if (ref($y) eq __PACKAGE__) {
1564				return bless \__add__($$x, $$y);
1565				}
1566
1567				$x = $$x;
1568
1569				if (!ref($y)) {
1570				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1571				if (ref($x) eq 'Math::GMPq') {
1572				my $r = Math::GMPq::Rmpq_init();
1573				$y < 0
1574				? Math::GMPq::Rmpq_set_si($r, $y, 1)
1575				: Math::GMPq::Rmpq_set_ui($r, $y, 1);
1576				Math::GMPq::Rmpq_add($r, $r, $x);
1577				return bless \$r;
1578				}
1579
1580				return bless \__add__($x, $y);
1581				}
1582
1583				return bless \__add__($x, _str2obj($y));
1584				}
1585
1586				bless \__add__($x, _star2obj($y));
1587				}
1588
1589				#
1590				## SUBTRACTION
1591				#
1592
1593				sub sub { # used in overloading
1594				require Math::AnyNum::sub;
1595				my ($x, $y) = @_;
1596
1597				$x =
1598				ref($x) eq __PACKAGE__ ? $$x
1599				: ref($x) ? _star2obj($x)
1600				: _str2obj($x);
1601
1602				if (ref($y) eq __PACKAGE__) {
1603				return bless \__sub__($x, $$y);
1604				}
1605
1606				if (!ref($y)) {
1607				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1608				if (ref($x) eq 'Math::GMPq') {
1609				my $r = Math::GMPq::Rmpq_init();
1610				$y < 0
1611				? Math::GMPq::Rmpq_set_si($r, $y, 1)
1612				: Math::GMPq::Rmpq_set_ui($r, $y, 1);
1613				Math::GMPq::Rmpq_sub($r, $x, $r);
1614				return bless \$r;
1615				}
1616
1617				return bless \__sub__($x, $y);
1618				}
1619
1620				return bless \__sub__($x, _str2obj($y));
1621				}
1622
1623				bless \__sub__($x, _star2obj($y));
1624				}
1625
1626				#
1627				## MULTIPLICATION
1628				#
1629
1630				sub mul { # used in overloading
1631				require Math::AnyNum::mul;
1632				my ($x, $y) = @_;
1633
1634				if (ref($y) eq __PACKAGE__) {
1635				return bless \__mul__($$x, $$y);
1636				}
1637
1638				$x = $$x;
1639
1640				if (!ref($y)) {
1641				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1642				if (ref($x) eq 'Math::GMPq') {
1643				my $r = Math::GMPq::Rmpq_init();
1644				$y < 0
1645				? Math::GMPq::Rmpq_set_si($r, $y, 1)
1646				: Math::GMPq::Rmpq_set_ui($r, $y, 1);
1647				Math::GMPq::Rmpq_mul($r, $r, $x);
1648				return bless \$r;
1649				}
1650
1651				return bless \__mul__($x, $y);
1652				}
1653
1654				return bless \__mul__($x, _str2obj($y));
1655				}
1656
1657				bless \__mul__($x, _star2obj($y));
1658				}
1659
1660				#
1661				## DIVISION
1662				#
1663
1664				sub div { # used in overloading
1665				require Math::AnyNum::div;
1666				my ($x, $y) = @_;
1667
1668				$x =
1669				ref($x) eq __PACKAGE__ ? $$x
1670				: ref($x) ? _star2obj($x)
1671				: _str2obj($x);
1672
1673				if (ref($y) eq __PACKAGE__) {
1674				return bless \__div__($x, $$y);
1675				}
1676
1677				if (!ref($y)) {
1678				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN and CORE::int($y)) {
1679				if (ref($x) eq 'Math::GMPq') {
1680				my $r = Math::GMPq::Rmpq_init();
1681				$y < 0
1682				? Math::GMPq::Rmpq_set_si($r, -1, -$y)
1683				: Math::GMPq::Rmpq_set_ui($r, 1, $y);
1684				Math::GMPq::Rmpq_mul($r, $r, $x);
1685				return bless \$r;
1686				}
1687
1688				return bless \__div__($x, $y);
1689				}
1690
1691				return bless \__div__($x, _str2obj($y));
1692				}
1693
1694				bless \__div__($x, _star2obj($y));
1695				}
1696
1697				#
1698				## INTEGER ADDITION
1699				#
1700
1701				sub iadd ($$) {
1702				my ($x, $y) = @_;
1703
1704				if (!ref($x) and ref($y)) {
1705				($x, $y) = ($y, $x);
1706				}
1707
1708				$x = _star2mpz($x) // goto &nan;
1709
1710				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1711				my $r = Math::GMPz::Rmpz_init();
1712				$y < 0
1713				? Math::GMPz::Rmpz_sub_ui($r, $x, -$y)
1714				: Math::GMPz::Rmpz_add_ui($r, $x, $y);
1715				return bless \$r;
1716				}
1717
1718				$y = _star2mpz($y) // goto &nan;
1719
1720				my $r = Math::GMPz::Rmpz_init();
1721				Math::GMPz::Rmpz_add($r, $x, $y);
1722				bless \$r;
1723				}
1724
1725				#
1726				## INTEGER SUBTRACTION
1727				#
1728
1729				sub isub ($$) {
1730				my ($x, $y) = @_;
1731
1732				$x = _star2mpz($x) // goto &nan;
1733
1734				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1735				my $r = Math::GMPz::Rmpz_init();
1736				$y < 0
1737				? Math::GMPz::Rmpz_add_ui($r, $x, -$y)
1738				: Math::GMPz::Rmpz_sub_ui($r, $x, $y);
1739				return bless \$r;
1740				}
1741
1742				$y = _star2mpz($y) // goto &nan;
1743
1744				my $r = Math::GMPz::Rmpz_init();
1745				Math::GMPz::Rmpz_sub($r, $x, $y);
1746				bless \$r;
1747				}
1748
1749				#
1750				## INTEGER MULTIPLICATION
1751				#
1752
1753				sub imul ($$) {
1754				my ($x, $y) = @_;
1755
1756				if (!ref($x) and ref($y)) {
1757				($x, $y) = ($y, $x);
1758				}
1759
1760				$x = _star2mpz($x) // goto &nan;
1761
1762				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1763				my $r = Math::GMPz::Rmpz_init();
1764				Math::GMPz::Rmpz_mul_ui($r, $x, CORE::abs($y));
1765				Math::GMPz::Rmpz_neg($r, $r) if $y < 0;
1766				return bless \$r;
1767				}
1768
1769				$y = _star2mpz($y) // goto &nan;
1770
1771				my $r = Math::GMPz::Rmpz_init();
1772				Math::GMPz::Rmpz_mul($r, $x, $y);
1773				bless \$r;
1774				}
1775
1776				#
1777				## INTEGER DIVISION
1778				#
1779
1780				sub idiv ($$) {
1781				my ($x, $y) = @_;
1782
1783				$x = _star2mpz($x) // goto &nan;
1784
1785				if (!ref($y) and CORE::int($y) eq $y and CORE::int($y) and CORE::abs($y) <= ULONG_MAX) {
1786				my $r = Math::GMPz::Rmpz_init();
1787				Math::GMPz::Rmpz_tdiv_q_ui($r, $x, CORE::abs($y));
1788				Math::GMPz::Rmpz_neg($r, $r) if $y < 0;
1789				return bless \$r;
1790				}
1791
1792				$y = _star2mpz($y) // goto &nan;
1793
1794				# Detect division by zero
1795				Math::GMPz::Rmpz_sgn($y) \|\| do {
1796				my $sign = Math::GMPz::Rmpz_sgn($x);
1797
1798				if ($sign == 0) { # 0/0
1799				goto &nan;
1800				}
1801				elsif ($sign > 0) { # x/0 where: x > 0
1802				goto &inf;
1803				}
1804				else { # x/0 where: x < 0
1805				goto &ninf;
1806				}
1807				};
1808
1809				my $r = Math::GMPz::Rmpz_init();
1810				Math::GMPz::Rmpz_tdiv_q($r, $x, $y);
1811				bless \$r;
1812				}
1813
1814				#
1815				## POWER
1816				#
1817
1818				sub pow ($$) {
1819				require Math::AnyNum::pow;
1820				my ($x, $y) = @_;
1821
1822				$x =
1823				ref($x) eq __PACKAGE__ ? $$x
1824				: ref($x) ? _star2obj($x)
1825				: _str2obj($x);
1826
1827				if (ref($y) eq __PACKAGE__) {
1828				return bless \__pow__($x, $$y);
1829				}
1830
1831				if (!ref($y)) {
1832				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1833				return bless \__pow__($x, $y);
1834				}
1835
1836				return bless \__pow__($x, _str2obj($y));
1837				}
1838
1839				bless \__pow__($x, _star2obj($y));
1840				}
1841
1842				#
1843				## INTEGER POWER
1844				#
1845
1846				sub ipow ($$) {
1847				my ($x, $y) = @_;
1848
1849				# Both `x` and `y` are strings
1850				if ( !ref($x)
1851				and !ref($y)
1852				and CORE::int($x) eq $x
1853				and $x >= 0
1854				and $x <= ULONG_MAX
1855				and CORE::int($y) eq $y
1856				and $y >= 0
1857				and $y <= ULONG_MAX) {
1858
1859				my $r = Math::GMPz::Rmpz_init();
1860				Math::GMPz::Rmpz_ui_pow_ui($r, $x, $y);
1861				return bless \$r;
1862				}
1863
1864				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
1865
1866				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1867				## `y` is already a native integer
1868				}
1869				else {
1870				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // (goto &nan);
1871				}
1872
1873				my $r = Math::GMPz::Rmpz_init();
1874				Math::GMPz::Rmpz_pow_ui($r, $x, CORE::abs($y));
1875
1876				if ($y < 0) {
1877				Math::GMPz::Rmpz_sgn($r) \|\| goto &inf; # 0^(-y) = Inf
1878				state $ONE_Z = Math::GMPz::Rmpz_init_set_ui_nobless(1);
1879				Math::GMPz::Rmpz_tdiv_q($r, $ONE_Z, $r);
1880				}
1881
1882				bless \$r;
1883				}
1884
1885				#
1886				## IPOW2
1887				#
1888
1889				sub ipow2 ($) {
1890				my ($n) = @_;
1891
1892				if (ref($n) eq __PACKAGE__) {
1893				$n = _any2si($$n) // goto &nan;
1894				}
1895				elsif ( !ref($n)
1896				and CORE::int($n) eq $n
1897				and $n >= LONG_MIN
1898				and $n <= ULONG_MAX) {
1899				## `n` is already a native integer
1900				}
1901				else {
1902				$n = _any2si(_star2obj($n)) // goto &nan;
1903				}
1904
1905				goto &zero if $n < 0;
1906				state $one = Math::GMPz::Rmpz_init_set_ui_nobless(1);
1907
1908				my $r = Math::GMPz::Rmpz_init();
1909				Math::GMPz::Rmpz_mul_2exp($r, $one, $n);
1910				bless \$r;
1911				}
1912
1913				#
1914				## IPOW10
1915				#
1916
1917				sub ipow10 ($) {
1918				my ($n) = @_;
1919
1920				if (ref($n) eq __PACKAGE__) {
1921				$n = _any2si($$n) // goto &nan;
1922				}
1923				elsif ( !ref($n)
1924				and CORE::int($n) eq $n
1925				and $n >= LONG_MIN
1926				and $n <= ULONG_MAX) {
1927				## $n is a native integer
1928				}
1929				else {
1930				$n = _any2si(_star2obj($n)) // goto &nan;
1931				}
1932
1933				goto &zero if $n < 0;
1934
1935				my $r = Math::GMPz::Rmpz_init();
1936				Math::GMPz::Rmpz_ui_pow_ui($r, 10, $n);
1937				bless \$r;
1938				}
1939
1940				#
1941				## ROOT
1942				#
1943
1944				sub root ($$) {
1945				require Math::AnyNum::pow;
1946				require Math::AnyNum::inv;
1947				my ($x, $y) = @_;
1948
1949				$x =
1950				ref($x) eq __PACKAGE__ ? $$x
1951				: ref($x) ? _star2obj($x)
1952				: _str2obj($x);
1953
1954				$y =
1955				ref($y) eq __PACKAGE__ ? $$y
1956				: ref($y) ? _star2obj($y)
1957				: _str2obj($y);
1958
1959				bless \__pow__($x, __inv__($y));
1960				}
1961
1962				#
1963				## Polygonal root
1964				#
1965
1966				sub polygonal_root ($$) {
1967				require Math::AnyNum::polygonal_root;
1968				bless \__polygonal_root__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
1969				}
1970
1971				#
1972				## Second polygonal root
1973				#
1974
1975				sub polygonal_root2 ($$) {
1976				require Math::AnyNum::polygonal_root;
1977				bless \__polygonal_root__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]), 1);
1978				}
1979
1980				#
1981				## isqrt
1982				#
1983
1984				sub isqrt ($) {
1985				my ($x) = @_;
1986
1987				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
1988
1989				Math::GMPz::Rmpz_sgn($x) < 0 and goto &nan;
1990				my $r = Math::GMPz::Rmpz_init();
1991				Math::GMPz::Rmpz_sqrt($r, $x);
1992				bless \$r;
1993				}
1994
1995				#
1996				## icbrt
1997				#
1998
1999				sub icbrt ($) {
2000				my ($x) = @_;
2001				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
2002				my $r = Math::GMPz::Rmpz_init();
2003				Math::GMPz::Rmpz_root($r, $x, 3);
2004				bless \$r;
2005				}
2006
2007				#
2008				## IROOT
2009				#
2010
2011				sub iroot ($$) {
2012				my ($x, $y) = @_;
2013
2014				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
2015
2016				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
2017				## `y`is native integer
2018				}
2019				elsif (ref($y) eq __PACKAGE__) {
2020				$y = _any2si($$y) // goto &nan;
2021				}
2022				else {
2023				$y = _any2si(_star2obj($y)) // goto &nan;
2024				}
2025
2026				if ($y == 0) {
2027				Math::GMPz::Rmpz_sgn($x) \|\| goto &zero; # 0^Inf = 0
2028
2029				# 1^Inf = 1 ; (-1)^Inf = 1
2030				if (Math::GMPz::Rmpz_cmpabs_ui($x, 1) == 0) {
2031				goto &one;
2032				}
2033
2034				goto &inf;
2035				}
2036
2037				if ($y < 0) {
2038				my $sign = Math::GMPz::Rmpz_sgn($x)
2039				\|\| goto &inf; # 1 / 0^k = Inf
2040
2041				if ($sign < 0) {
2042				goto &nan;
2043				}
2044
2045				if (Math::GMPz::Rmpz_cmp_ui($x, 1) == 0) { # 1 / 1^k = 1
2046				goto &one;
2047				}
2048
2049				goto &zero;
2050				}
2051
2052				if ($y % 2 == 0 and Math::GMPz::Rmpz_sgn($x) < 0) {
2053				goto &nan;
2054				}
2055
2056				my $r = Math::GMPz::Rmpz_init();
2057				$y == 2
2058				? Math::GMPz::Rmpz_sqrt($r, $x)
2059				: Math::GMPz::Rmpz_root($r, $x, $y);
2060				bless \$r;
2061				}
2062
2063				#
2064				## ISQRTREM
2065				#
2066
2067				sub isqrtrem ($) {
2068				my ($x) = @_;
2069
2070				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // return (nan(), nan());
2071
2072				Math::GMPz::Rmpz_sgn($x) < 0
2073				and return (nan(), nan());
2074
2075				my $r = Math::GMPz::Rmpz_init();
2076				my $s = Math::GMPz::Rmpz_init();
2077
2078				Math::GMPz::Rmpz_sqrtrem($r, $s, $x);
2079
2080				((bless \$r), (bless \$s));
2081				}
2082
2083				#
2084				## IROOTREM
2085				#
2086
2087				sub irootrem ($$) {
2088				my ($x, $y) = @_;
2089
2090				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // return (nan(), nan());
2091
2092				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
2093				## `y` is a native integer
2094				}
2095				else {
2096				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // (return (nan(), nan()));
2097				}
2098
2099				if ($y == 0) {
2100
2101				# 0^Inf = 0
2102				if (Math::GMPz::Rmpz_sgn($x) == 0) {
2103				return (zero(), mone());
2104				}
2105
2106				my $r = Math::GMPz::Rmpz_init();
2107				Math::GMPz::Rmpz_sub_ui($r, $x, 1);
2108
2109				# 1^Inf = 1 ; (-1)^Inf = 1
2110				if (Math::GMPz::Rmpz_cmpabs_ui($x, 1) == 0) {
2111				return (one(), (bless \$r));
2112				}
2113
2114				return (inf(), (bless \$r));
2115				}
2116
2117				if ($y < 0) {
2118				my $sgn = Math::GMPz::Rmpz_sgn($x);
2119
2120				# 1 / 0^k = Inf
2121				if ($sgn == 0) {
2122				return (inf(), zero());
2123				}
2124
2125				# 1 / 1^k = 1
2126				if (Math::GMPz::Rmpz_cmp_ui($x, 1) == 0) {
2127				return (one(), zero());
2128				}
2129
2130				# x is negative
2131				if ($sgn < 0) {
2132				return (nan(), nan());
2133				}
2134
2135				return (zero(), ninf());
2136				}
2137
2138				if ($y % 2 == 0 and Math::GMPz::Rmpz_sgn($x) < 0) {
2139				return (nan(), nan());
2140				}
2141
2142				my $r = Math::GMPz::Rmpz_init();
2143				my $s = Math::GMPz::Rmpz_init();
2144
2145				$y == 2
2146				? Math::GMPz::Rmpz_sqrtrem($r, $s, $x)
2147				: Math::GMPz::Rmpz_rootrem($r, $s, $x, $y);
2148
2149				((bless \$r), (bless \$s));
2150				}
2151
2152				#
2153				## MOD
2154				#
2155
2156				sub mod ($$) {
2157				require Math::AnyNum::mod;
2158				my ($x, $y) = @_;
2159
2160				$x =
2161				ref($x) eq __PACKAGE__ ? $$x
2162				: ref($x) ? _star2obj($x)
2163				: _str2obj($x);
2164
2165				if (ref($y) eq __PACKAGE__) {
2166				return bless \__mod__($x, $$y);
2167				}
2168
2169				if (!ref($y)) {
2170
2171				if ( ref($x) ne 'Math::GMPq'
2172				and CORE::int($y) eq $y
2173				and $y > 0
2174				and $y <= ULONG_MAX) {
2175				return bless \__mod__($x, $y);
2176				}
2177
2178				return bless \__mod__($x, _str2obj($y));
2179				}
2180
2181				bless \__mod__($x, _star2obj($y));
2182				}
2183
2184				#
2185				## IMOD
2186				#
2187
2188				sub imod ($$) {
2189				my ($x, $y) = @_;
2190
2191				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
2192
2193				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
2194
2195				CORE::int($y) \|\| goto &nan;
2196
2197				my $neg_y = $y < 0;
2198				$y = -$y if $neg_y;
2199
2200				my $r = Math::GMPz::Rmpz_init();
2201				Math::GMPz::Rmpz_mod_ui($r, $x, $y);
2202
2203				if (!Math::GMPz::Rmpz_sgn($r)) {
2204				## OK
2205				}
2206				elsif ($neg_y) {
2207				Math::GMPz::Rmpz_sub_ui($r, $r, $y);
2208				}
2209
2210				return bless \$r;
2211				}
2212
2213				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
2214
2215				my $sign_y = Math::GMPz::Rmpz_sgn($y) \|\| goto &nan;
2216
2217				my $r = Math::GMPz::Rmpz_init();
2218				Math::GMPz::Rmpz_mod($r, $x, $y);
2219
2220				if (!Math::GMPz::Rmpz_sgn($r)) {
2221				## OK
2222				}
2223				elsif ($sign_y < 0) {
2224				Math::GMPz::Rmpz_add($r, $r, $y);
2225				}
2226
2227				bless \$r;
2228				}
2229
2230				#
2231				## DIVMOD
2232				#
2233
2234				sub divmod ($$) {
2235				my ($x, $y) = @_;
2236
2237				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // return (nan(), nan());
2238				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // return (nan(), nan());
2239
2240				Math::GMPz::Rmpz_sgn($y)
2241				\|\| return (nan(), nan());
2242
2243				my $r = Math::GMPz::Rmpz_init();
2244				my $s = Math::GMPz::Rmpz_init();
2245
2246				Math::GMPz::Rmpz_divmod($r, $s, $x, $y);
2247
2248				((bless \$r), (bless \$s));
2249				}
2250
2251				#
2252				## is_div
2253				#
2254
2255				sub is_div ($$) {
2256				require Math::AnyNum::eq;
2257				(@_) = (${mod($_[0], $_[1])}, 0);
2258				goto &__eq__;
2259				}
2260
2261				#
2262				## SPECIAL
2263				#
2264
2265				sub ln { # used in overloading
2266				require Math::AnyNum::log;
2267				bless \__log__(_star2mpfr_mpc($_[0]));
2268				}
2269
2270				sub log2 ($) {
2271				require Math::AnyNum::log;
2272				bless \__log2__(_star2mpfr_mpc($_[0]));
2273				}
2274
2275				sub log10 ($) {
2276				require Math::AnyNum::log;
2277				bless \__log10__(_star2mpfr_mpc($_[0]));
2278				}
2279
2280				sub length ($) {
2281				my ($z) = _star2mpz($_[0]) // return -1;
2282				CORE::length(Math::GMPz::Rmpz_get_str($z, 10)) - (Math::GMPz::Rmpz_sgn($z) < 0 ? 1 : 0);
2283				}
2284
2285				sub log (_;$) {
2286				require Math::AnyNum::log;
2287				my ($x, $y) = @_;
2288
2289				if (!defined($y)) {
2290				return bless \__log__(_star2mpfr_mpc($x));
2291				}
2292
2293				require Math::AnyNum::div;
2294				bless \__div__(__log__(_star2mpfr_mpc($x)), __log__(_star2mpfr_mpc($y)));
2295				}
2296
2297				#
2298				## ILOG
2299				#
2300
2301				sub ilog2 ($) {
2302				require Math::AnyNum::ilog;
2303				state $two = Math::GMPz::Rmpz_init_set_ui(2);
2304				bless \__ilog__((_star2mpz($_[0]) // goto &nan), $two);
2305				}
2306
2307				sub ilog10 ($) {
2308				require Math::AnyNum::ilog;
2309				state $ten = Math::GMPz::Rmpz_init_set_ui(10);
2310				bless \__ilog__((_star2mpz($_[0]) // goto &nan), $ten);
2311				}
2312
2313				sub ilog ($;$) {
2314				my ($x, $y) = @_;
2315
2316				if (!defined($y)) {
2317				require Math::AnyNum::log;
2318				return bless \(_any2mpz(__log__(_star2mpfr_mpc($x))) // goto &nan);
2319				}
2320
2321				require Math::AnyNum::ilog;
2322				bless \__ilog__((_star2mpz($x) // goto &nan), (_star2mpz($y) // goto &nan));
2323				}
2324
2325				#
2326				## SQRT
2327				#
2328
2329				sub sqrt { # used in overloading
2330				require Math::AnyNum::sqrt;
2331				bless \__sqrt__(_star2mpfr_mpc($_[0]));
2332				}
2333
2334				sub cbrt ($) {
2335				require Math::AnyNum::cbrt;
2336				bless \__cbrt__(_star2mpfr_mpc($_[0]));
2337				}
2338
2339				sub sqr ($) {
2340				require Math::AnyNum::mul;
2341				my ($x) = @_;
2342
2343				$x =
2344				ref($x) eq __PACKAGE__
2345				? $$x
2346				: _star2obj($x);
2347
2348				bless \__mul__($x, $x);
2349				}
2350
2351				sub norm ($) {
2352				require Math::AnyNum::norm;
2353				my ($x) = @_;
2354
2355				$x =
2356				ref($x) eq __PACKAGE__
2357				? $$x
2358				: _star2obj($x);
2359
2360				bless \__norm__($x);
2361				}
2362
2363				sub exp { # used in overloading
2364				require Math::AnyNum::exp;
2365				bless \__exp__(_star2mpfr_mpc($_[0]));
2366				}
2367
2368				sub exp2 ($) {
2369				require Math::AnyNum::pow;
2370				my ($x) = @_;
2371
2372				state $base = Math::GMPz::Rmpz_init_set_ui(2);
2373
2374				if (ref($x) eq __PACKAGE__) {
2375				bless \__pow__($base, $$x);
2376				}
2377				elsif (!ref($x) and CORE::int($x) eq $x and $x >= LONG_MIN and $x <= ULONG_MAX) {
2378				bless \__pow__($base, $x);
2379				}
2380				else {
2381				bless \__pow__($base, _star2obj($x));
2382				}
2383				}
2384
2385				sub exp10 ($) {
2386				require Math::AnyNum::pow;
2387				my ($x) = @_;
2388
2389				state $base = Math::GMPz::Rmpz_init_set_ui(10);
2390
2391				if (ref($x) eq __PACKAGE__) {
2392				bless \__pow__($base, $$x);
2393				}
2394				elsif (!ref($x) and CORE::int($x) eq $x and $x >= LONG_MIN and $x <= ULONG_MAX) {
2395				bless \__pow__($base, $x);
2396				}
2397				else {
2398				bless \__pow__($base, _star2obj($x));
2399				}
2400				}
2401
2402				sub floor ($) {
2403				require Math::AnyNum::floor;
2404				my ($x) = @_;
2405
2406				if (ref($x) ne __PACKAGE__) {
2407				$x = __PACKAGE__->new($x);
2408				}
2409
2410				ref($$x) eq 'Math::GMPz' and return $x; # already an integer
2411				bless \__floor__($$x);
2412				}
2413
2414				sub ceil ($) {
2415				require Math::AnyNum::ceil;
2416				my ($x) = @_;
2417
2418				if (ref($x) ne __PACKAGE__) {
2419				$x = __PACKAGE__->new($x);
2420				}
2421
2422				ref($$x) eq 'Math::GMPz' and return $x; # already an integer
2423				bless \__ceil__($$x);
2424				}
2425
2426				#
2427				## sin / sinh / asin / asinh
2428				#
2429
2430				sub sin { # used in overloading
2431				require Math::AnyNum::sin;
2432				bless \__sin__(_star2mpfr_mpc($_[0]));
2433				}
2434
2435				sub sinh ($) {
2436				require Math::AnyNum::sinh;
2437				bless \__sinh__(_star2mpfr_mpc($_[0]));
2438				}
2439
2440				sub asin ($) {
2441				require Math::AnyNum::asin;
2442				bless \__asin__(_star2mpfr_mpc($_[0]));
2443				}
2444
2445				sub asinh ($) {
2446				require Math::AnyNum::asinh;
2447				bless \__asinh__(_star2mpfr_mpc($_[0]));
2448				}
2449
2450				#
2451				## cos / cosh / acos / acosh
2452				#
2453
2454				sub cos { # used in overloading
2455				require Math::AnyNum::cos;
2456				bless \__cos__(_star2mpfr_mpc($_[0]));
2457				}
2458
2459				sub cosh ($) {
2460				require Math::AnyNum::cosh;
2461				bless \__cosh__(_star2mpfr_mpc($_[0]));
2462				}
2463
2464				sub acos ($) {
2465				require Math::AnyNum::acos;
2466				bless \__acos__(_star2mpfr_mpc($_[0]));
2467				}
2468
2469				sub acosh ($) {
2470				require Math::AnyNum::acosh;
2471				bless \__acosh__(_star2mpfr_mpc($_[0]));
2472				}
2473
2474				#
2475				## tan / tanh / atan / atanh
2476				#
2477
2478				sub tan ($) {
2479				require Math::AnyNum::tan;
2480				bless \__tan__(_star2mpfr_mpc($_[0]));
2481				}
2482
2483				sub tanh ($) {
2484				require Math::AnyNum::tanh;
2485				bless \__tanh__(_star2mpfr_mpc($_[0]));
2486				}
2487
2488				sub atan ($) {
2489				require Math::AnyNum::atan;
2490				bless \__atan__(_star2mpfr_mpc($_[0]));
2491				}
2492
2493				sub atanh ($) {
2494				require Math::AnyNum::atanh;
2495				bless \__atanh__(_star2mpfr_mpc($_[0]));
2496				}
2497
2498				sub atan2 ($$) {
2499				require Math::AnyNum::atan2;
2500				bless \__atan2__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
2501				}
2502
2503				#
2504				## sec / sech / asec / asech
2505				#
2506
2507				sub sec ($) {
2508				require Math::AnyNum::sec;
2509				bless \__sec__(_star2mpfr_mpc($_[0]));
2510				}
2511
2512				sub sech ($) {
2513				require Math::AnyNum::sech;
2514				bless \__sech__(_star2mpfr_mpc($_[0]));
2515				}
2516
2517				sub asec ($) {
2518				require Math::AnyNum::asec;
2519				bless \__asec__(_star2mpfr_mpc($_[0]));
2520				}
2521
2522				sub asech ($) {
2523				require Math::AnyNum::asech;
2524				bless \__asech__(_star2mpfr_mpc($_[0]));
2525				}
2526
2527				#
2528				## csc / csch / acsc / acsch
2529				#
2530
2531				sub csc ($) {
2532				require Math::AnyNum::csc;
2533				bless \__csc__(_star2mpfr_mpc($_[0]));
2534				}
2535
2536				sub csch ($) {
2537				require Math::AnyNum::csch;
2538				bless \__csch__(_star2mpfr_mpc($_[0]));
2539				}
2540
2541				sub acsc ($) {
2542				require Math::AnyNum::acsc;
2543				bless \__acsc__(_star2mpfr_mpc($_[0]));
2544				}
2545
2546				sub acsch ($) {
2547				require Math::AnyNum::acsch;
2548				bless \__acsch__(_star2mpfr_mpc($_[0]));
2549				}
2550
2551				#
2552				## cot / coth / acot / acoth
2553				#
2554
2555				sub cot ($) {
2556				require Math::AnyNum::cot;
2557				bless \__cot__(_star2mpfr_mpc($_[0]));
2558				}
2559
2560				sub coth ($) {
2561				require Math::AnyNum::coth;
2562				bless \__coth__(_star2mpfr_mpc($_[0]));
2563				}
2564
2565				sub acot ($) {
2566				require Math::AnyNum::acot;
2567				bless \__acot__(_star2mpfr_mpc($_[0]));
2568				}
2569
2570				sub acoth ($) {
2571				require Math::AnyNum::acoth;
2572				bless \__acoth__(_star2mpfr_mpc($_[0]));
2573				}
2574
2575				sub deg2rad ($) {
2576				require Math::AnyNum::mul;
2577				my ($x) = @_;
2578				my $f = Math::MPFR::Rmpfr_init2($PREC);
2579				Math::MPFR::Rmpfr_const_pi($f, $ROUND);
2580				Math::MPFR::Rmpfr_div_ui($f, $f, 180, $ROUND);
2581				bless \__mul__(_star2mpfr_mpc($x), $f);
2582				}
2583
2584				sub rad2deg ($) {
2585				require Math::AnyNum::mul;
2586				my ($x) = @_;
2587				my $f = Math::MPFR::Rmpfr_init2($PREC);
2588				Math::MPFR::Rmpfr_const_pi($f, $ROUND);
2589				Math::MPFR::Rmpfr_ui_div($f, 180, $f, $ROUND);
2590				bless \__mul__(_star2mpfr_mpc($x), $f);
2591				}
2592
2593				#
2594				## gamma
2595				#
2596
2597				sub gamma ($) {
2598				my $r = Math::MPFR::Rmpfr_init2($PREC);
2599				Math::MPFR::Rmpfr_gamma($r, _star2mpfr($_[0]), $ROUND);
2600				bless \$r;
2601				}
2602
2603				#
2604				## lgamma
2605				#
2606
2607				sub lgamma ($) {
2608				my $r = Math::MPFR::Rmpfr_init2($PREC);
2609				Math::MPFR::Rmpfr_lgamma($r, _star2mpfr($_[0]), $ROUND);
2610				bless \$r;
2611				}
2612
2613				#
2614				## lngamma
2615				#
2616
2617				sub lngamma ($) {
2618				my $r = Math::MPFR::Rmpfr_init2($PREC);
2619				Math::MPFR::Rmpfr_lngamma($r, _star2mpfr($_[0]), $ROUND);
2620				bless \$r;
2621				}
2622
2623				#
2624				## digamma
2625				#
2626
2627				sub digamma ($) {
2628				my $r = Math::MPFR::Rmpfr_init2($PREC);
2629				Math::MPFR::Rmpfr_digamma($r, _star2mpfr($_[0]), $ROUND);
2630				bless \$r;
2631				}
2632
2633				#
2634				## zeta
2635				#
2636
2637				sub zeta ($) {
2638				my ($x) = @_;
2639
2640				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2641				## $x is an unsigned integer
2642				}
2643				else {
2644				$x = _star2mpfr($x);
2645
2646				# If $x fits inside an unsigned integer, then unpack it.
2647				if ( Math::MPFR::Rmpfr_integer_p($x)
2648				and Math::MPFR::Rmpfr_fits_ulong_p($x, $ROUND)) {
2649				$x = Math::MPFR::Rmpfr_get_ui($x, $ROUND);
2650				}
2651				}
2652
2653				my $r = Math::MPFR::Rmpfr_init2($PREC);
2654
2655				ref($x)
2656				? Math::MPFR::Rmpfr_zeta($r, $x, $ROUND)
2657				: Math::MPFR::Rmpfr_zeta_ui($r, $x, $ROUND);
2658
2659				bless \$r;
2660				}
2661
2662				#
2663				## eta
2664				#
2665
2666				sub eta ($) {
2667				require Math::AnyNum::eta;
2668				bless \__eta__(_star2mpfr($_[0]));
2669				}
2670
2671				#
2672				## beta
2673				#
2674				sub beta ($$) {
2675				require Math::AnyNum::beta;
2676				bless \__beta__(_star2mpfr($_[0]), _star2mpfr($_[1]));
2677				}
2678
2679				#
2680				## Airy function (Ai)
2681				#
2682
2683				sub Ai ($) {
2684				my $r = Math::MPFR::Rmpfr_init2($PREC);
2685				Math::MPFR::Rmpfr_ai($r, _star2mpfr($_[0]), $ROUND);
2686				bless \$r;
2687				}
2688
2689				#
2690				## Exponential integral (Ei)
2691				#
2692
2693				sub Ei ($) {
2694				my $r = Math::MPFR::Rmpfr_init2($PREC);
2695				Math::MPFR::Rmpfr_eint($r, _star2mpfr($_[0]), $ROUND);
2696				bless \$r;
2697				}
2698
2699				#
2700				## Logarithmic integral (Li)
2701				#
2702				sub Li ($) {
2703				my $r = Math::MPFR::Rmpfr_init2($PREC);
2704				Math::MPFR::Rmpfr_log($r, _star2mpfr($_[0]), $ROUND);
2705				Math::MPFR::Rmpfr_eint($r, $r, $ROUND);
2706				bless \$r;
2707				}
2708
2709				#
2710				## Dilogarithm function (Li_2)
2711				#
2712				sub Li2 ($) {
2713				my $r = Math::MPFR::Rmpfr_init2($PREC);
2714				Math::MPFR::Rmpfr_li2($r, _star2mpfr($_[0]), $ROUND);
2715				bless \$r;
2716				}
2717
2718				#
2719				## Error function
2720				#
2721				sub erf ($) {
2722				my $r = Math::MPFR::Rmpfr_init2($PREC);
2723				Math::MPFR::Rmpfr_erf($r, _star2mpfr($_[0]), $ROUND);
2724				bless \$r;
2725				}
2726
2727				#
2728				## Complementary error function
2729				#
2730				sub erfc ($) {
2731				my $r = Math::MPFR::Rmpfr_init2($PREC);
2732				Math::MPFR::Rmpfr_erfc($r, _star2mpfr($_[0]), $ROUND);
2733				bless \$r;
2734				}
2735
2736				#
2737				## Lambert W
2738				#
2739
2740				sub LambertW ($) {
2741				require Math::AnyNum::LambertW;
2742				bless \__LambertW__(_star2mpfr_mpc($_[0]));
2743				}
2744
2745				#
2746				## lgrt -- logarithmic root
2747				#
2748
2749				sub lgrt ($) {
2750				require Math::AnyNum::lgrt;
2751				bless \__lgrt__(_star2mpfr_mpc($_[0]));
2752				}
2753
2754				#
2755				## agm
2756				#
2757				sub agm ($$) {
2758				require Math::AnyNum::agm;
2759				bless \__agm__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
2760				}
2761
2762				#
2763				## hypot
2764				#
2765
2766				sub hypot ($$) {
2767				require Math::AnyNum::hypot;
2768				bless \__hypot__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
2769				}
2770
2771				#
2772				## BesselJ
2773				#
2774
2775				sub BesselJ ($$) {
2776				require Math::AnyNum::BesselJ;
2777				my ($x, $y) = @_;
2778
2779				$x = ref($x) eq __PACKAGE__ ? _any2mpfr($$x) : _star2mpfr($x);
2780
2781				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
2782				return bless \__BesselJ__($x, $y);
2783				}
2784
2785				bless \__BesselJ__($x, _star2mpz($y) // (goto &nan));
2786				}
2787
2788				#
2789				## BesselY
2790				#
2791
2792				sub BesselY ($$) {
2793				require Math::AnyNum::BesselY;
2794				my ($x, $y) = @_;
2795
2796				$x = ref($x) eq __PACKAGE__ ? _any2mpfr($$x) : _star2mpfr($x);
2797
2798				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
2799				return bless \__BesselY__($x, $y);
2800				}
2801
2802				bless \__BesselY__($x, _star2mpz($y) // (goto &nan));
2803				}
2804
2805				#
2806				## ROUND
2807				#
2808
2809				sub round ($;$) {
2810				require Math::AnyNum::round;
2811				my ($x, $y) = @_;
2812
2813				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
2814
2815				if (!defined($y)) {
2816				return bless \__round__($x, 0);
2817				}
2818
2819				if (!ref($y)) {
2820				if (CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
2821				## `y` is a native integer
2822				}
2823				else {
2824				$y = _any2si(_str2obj($y)) // (goto &nan);
2825				}
2826				}
2827				elsif (ref($y) eq __PACKAGE__) {
2828				$y = _any2si($$y) // (goto &nan);
2829				}
2830				else {
2831				$y = _any2si(_star2obj($y)) // (goto &nan);
2832				}
2833
2834				bless \__round__($x, $y);
2835				}
2836
2837				#
2838				## RAND / IRAND
2839				#
2840
2841				{
2842				my $srand = srand();
2843
2844				{
2845				state $state = Math::MPFR::Rmpfr_randinit_mt_nobless();
2846				Math::MPFR::Rmpfr_randseed_ui($state, $srand);
2847
2848				sub rand (;$;$) {
2849				require Math::AnyNum::mul;
2850				my ($x, $y) = @_;
2851
2852				if (@_ == 0) {
2853				$x = one();
2854				}
2855
2856				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
2857
2858				if (!defined($y)) {
2859				my $rand = Math::MPFR::Rmpfr_init2($PREC);
2860				Math::MPFR::Rmpfr_urandom($rand, $state, $ROUND);
2861				return bless \__mul__($rand, $x);
2862				}
2863
2864				require Math::AnyNum::sub;
2865				require Math::AnyNum::add;
2866
2867				$y = ref($y) eq __PACKAGE__ ? $$y : _star2obj($y);
2868
2869				my $rand = Math::MPFR::Rmpfr_init2($PREC);
2870				Math::MPFR::Rmpfr_urandom($rand, $state, $ROUND);
2871				$rand = __mul__($rand, __sub__($y, $x));
2872				bless \__add__($rand, $x);
2873				}
2874
2875				sub seed ($) {
2876				my $z = _star2mpz($_[0]) // do {
2877				require Carp;
2878				Carp::croak("seed(): invalid seed value <<$_[0]>> (expected an integer)");
2879				};
2880				Math::MPFR::Rmpfr_randseed($state, $z);
2881				bless \$z;
2882				}
2883				}
2884
2885				{
2886				state $state = Math::GMPz::zgmp_randinit_mt_nobless();
2887				Math::GMPz::zgmp_randseed_ui($state, $srand);
2888
2889				sub irand ($;$) {
2890				require Math::AnyNum::irand;
2891				my ($x, $y) = @_;
2892
2893				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
2894
2895				if (!defined($y)) {
2896				return bless \__irand__($x, undef, $state);
2897				}
2898
2899				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
2900				bless \__irand__($x, $y, $state);
2901				}
2902
2903				sub iseed ($) {
2904				my $z = _star2mpz($_[0]) // do {
2905				require Carp;
2906				Carp::croak("iseed(): invalid seed value <<$_[0]>> (expected an integer)");
2907				};
2908				Math::GMPz::zgmp_randseed($state, $z);
2909				bless \$z;
2910				}
2911				}
2912				}
2913
2914				#
2915				## Fibonacci
2916				#
2917				sub fibonacci ($) {
2918				my ($x) = @_;
2919
2920				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2921				## `x` is a native unsigned integer
2922				}
2923				elsif (ref($x) eq __PACKAGE__) {
2924				$x = _any2ui($$x) // goto &nan;
2925				}
2926				else {
2927				$x = _any2ui(_star2obj($x)) // goto &nan;
2928				}
2929
2930				my $r = Math::GMPz::Rmpz_init();
2931				Math::GMPz::Rmpz_fib_ui($r, $x);
2932				bless \$r;
2933				}
2934
2935				#
2936				## Lucas
2937				#
2938				sub lucas ($) {
2939				my ($x) = @_;
2940
2941				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2942				## `x` is a native unsigned integer
2943				}
2944				elsif (ref($x) eq __PACKAGE__) {
2945				$x = _any2ui($$x) // goto &nan;
2946				}
2947				else {
2948				$x = _any2ui(_star2obj($x)) // goto &nan;
2949				}
2950
2951				my $r = Math::GMPz::Rmpz_init();
2952				Math::GMPz::Rmpz_lucnum_ui($r, $x);
2953				bless \$r;
2954				}
2955
2956				#
2957				## Primorial
2958				#
2959				sub primorial ($) {
2960				my ($x) = @_;
2961
2962				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2963				## `x` is a native unsigned integer
2964				}
2965				elsif (ref($x) eq __PACKAGE__) {
2966				$x = _any2ui($$x) // goto &nan;
2967				}
2968				else {
2969				$x = _any2ui(_star2obj($x)) // goto &nan;
2970				}
2971
2972				my $r = Math::GMPz::Rmpz_init();
2973				Math::GMPz::Rmpz_primorial_ui($r, $x);
2974				bless \$r;
2975				}
2976
2977				#
2978				## bernfrac
2979				#
2980
2981				sub bernfrac ($) {
2982				require Math::AnyNum::bernfrac;
2983				my ($x) = @_;
2984
2985				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2986				## `x` is a native unsigned integer
2987				}
2988				elsif (ref($x) eq __PACKAGE__) {
2989				$x = _any2ui($$x) // goto &nan;
2990				}
2991				else {
2992				$x = _any2ui(_star2obj($x)) // goto &nan;
2993				}
2994
2995				bless \__bernfrac__($x);
2996				}
2997
2998				#
2999				## harmfrac
3000				#
3001
3002				sub harmfrac ($) {
3003				require Math::AnyNum::harmfrac;
3004				my ($x) = @_;
3005
3006				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3007				## `x` is a native unsigned integer
3008				}
3009				elsif (ref($x) eq __PACKAGE__) {
3010				$x = _any2ui($$x) // goto &nan;
3011				}
3012				else {
3013				$x = _any2ui(_star2obj($x)) // goto &nan;
3014				}
3015
3016				bless \__harmfrac__($x);
3017				}
3018
3019				#
3020				## bernreal
3021				#
3022
3023				sub bernreal ($) {
3024				require Math::AnyNum::bernreal;
3025				my ($x) = @_;
3026
3027				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3028				## `x` is a native unsigned integer
3029				}
3030				elsif (ref($x) eq __PACKAGE__) {
3031				$x = _any2ui($$x) // goto &nan;
3032				}
3033				else {
3034				$x = _any2ui(_star2obj($x)) // goto &nan;
3035				}
3036
3037				bless \__bernreal__($x);
3038				}
3039
3040				#
3041				## harmreal
3042				#
3043
3044				sub harmreal ($) {
3045				require Math::AnyNum::harmreal;
3046				bless \__harmreal__(_star2mpfr($_[0]) // (goto &nan));
3047				}
3048
3049				#
3050				## Factorial
3051				#
3052				sub factorial ($) {
3053				my ($x) = @_;
3054
3055				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3056				## `x` is a native unsigned integer
3057				}
3058				elsif (ref($x) eq __PACKAGE__) {
3059				$x = _any2ui($$x) // goto &nan;
3060				}
3061				else {
3062				$x = _any2ui(_star2obj($x)) // goto &nan;
3063				}
3064
3065				my $r = Math::GMPz::Rmpz_init();
3066				Math::GMPz::Rmpz_fac_ui($r, $x);
3067				bless \$r;
3068				}
3069
3070				#
3071				## Double-factorial
3072				#
3073
3074				sub dfactorial ($) {
3075				my ($x) = @_;
3076
3077				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3078				## `x` is a native unsigned integer
3079				}
3080				elsif (ref($x) eq __PACKAGE__) {
3081				$x = _any2ui($$x) // goto &nan;
3082				}
3083				else {
3084				$x = _any2ui(_star2obj($x)) // goto &nan;
3085				}
3086
3087				my $r = Math::GMPz::Rmpz_init();
3088				Math::GMPz::Rmpz_2fac_ui($r, $x);
3089				bless \$r;
3090				}
3091
3092				#
3093				## M-factorial
3094				#
3095
3096				sub mfactorial ($$) {
3097				my ($x, $y) = @_;
3098
3099				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3100				## `x` is an unsigned native integer
3101				}
3102				elsif (ref($x) eq __PACKAGE__) {
3103				$x = _any2ui($$x) // goto &nan;
3104				}
3105				else {
3106				$x = _any2ui(_star2obj($x)) // goto &nan;
3107				}
3108
3109				if (!ref($y) and CORE::int($y) eq $y and $y >= 0 and $y <= ULONG_MAX) {
3110				## `y` is an unsigned native integer
3111				}
3112				elsif (ref($y) eq __PACKAGE__) {
3113				$y = _any2ui($$y) // goto &nan;
3114				}
3115				else {
3116				$y = _any2ui(_star2obj($y)) // goto &nan;
3117				}
3118
3119				my $r = Math::GMPz::Rmpz_init();
3120				Math::GMPz::Rmpz_mfac_uiui($r, $x, $y);
3121				bless \$r;
3122				}
3123
3124				#
3125				## falling_factorial(x, +y) = binomial(x, y) * y!
3126				## falling_factorial(x, -y) = 1/falling_factorial(x + y, y)
3127				#
3128				sub falling_factorial ($$) {
3129				my ($x, $y) = @_;
3130
3131				$x = _star2mpz($x) // goto &nan;
3132
3133				if (ref($y) eq __PACKAGE__) {
3134				$y = _any2si($$y) // goto &nan;
3135				}
3136				elsif (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3137				## `y` is a native integer
3138				}
3139				else {
3140				$y = _any2si(_star2obj($y)) // goto &nan;
3141				}
3142
3143				my $r = Math::GMPz::Rmpz_init_set($x);
3144
3145				if ($y < 0) {
3146				Math::GMPz::Rmpz_add_ui($r, $r, CORE::abs($y));
3147				}
3148
3149				Math::GMPz::Rmpz_fits_ulong_p($r)
3150				? Math::GMPz::Rmpz_bin_uiui($r, Math::GMPz::Rmpz_get_ui($r), CORE::abs($y))
3151				: Math::GMPz::Rmpz_bin_ui($r, $r, CORE::abs($y));
3152
3153				Math::GMPz::Rmpz_sgn($r) \|\| do {
3154				$y < 0
3155				? (goto &nan)
3156				: (goto &zero);
3157				};
3158
3159				state $t = Math::GMPz::Rmpz_init_nobless();
3160				Math::GMPz::Rmpz_fac_ui($t, CORE::abs($y));
3161				Math::GMPz::Rmpz_mul($r, $r, $t);
3162
3163				if ($y < 0) {
3164				my $q = Math::GMPq::Rmpq_init();
3165				Math::GMPq::Rmpq_set_z($q, $r);
3166				Math::GMPq::Rmpq_inv($q, $q);
3167				return bless \$q;
3168				}
3169
3170				bless \$r;
3171				}
3172
3173				#
3174				## rising_factorial(x, +y) = binomial(x + y - 1, y) * y!
3175				## rising_factorial(x, -y) = 1/rising_factorial(x - y, y)
3176				#
3177				sub rising_factorial ($$) {
3178				my ($x, $y) = @_;
3179
3180				$x = _star2mpz($x) // goto &nan;
3181
3182				if (ref($y) eq __PACKAGE__) {
3183				$y = _any2si($$y) // goto &nan;
3184				}
3185				elsif (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3186				## `y` is a native integer
3187				}
3188				else {
3189				$y = _any2si(_star2obj($y)) // goto &nan;
3190				}
3191
3192				my $r = Math::GMPz::Rmpz_init_set($x);
3193				Math::GMPz::Rmpz_add_ui($r, $r, CORE::abs($y));
3194				Math::GMPz::Rmpz_sub_ui($r, $r, 1);
3195
3196				if ($y < 0) {
3197				Math::GMPz::Rmpz_sub_ui($r, $r, CORE::abs($y));
3198				}
3199
3200				Math::GMPz::Rmpz_fits_ulong_p($r)
3201				? Math::GMPz::Rmpz_bin_uiui($r, Math::GMPz::Rmpz_get_ui($r), CORE::abs($y))
3202				: Math::GMPz::Rmpz_bin_ui($r, $r, CORE::abs($y));
3203
3204				Math::GMPz::Rmpz_sgn($r) \|\| do {
3205				$y < 0
3206				? (goto &nan)
3207				: (goto &zero);
3208				};
3209
3210				state $t = Math::GMPz::Rmpz_init_nobless();
3211				Math::GMPz::Rmpz_fac_ui($t, CORE::abs($y));
3212				Math::GMPz::Rmpz_mul($r, $r, $t);
3213
3214				if ($y < 0) {
3215				my $q = Math::GMPq::Rmpq_init();
3216				Math::GMPq::Rmpq_set_z($q, $r);
3217				Math::GMPq::Rmpq_inv($q, $q);
3218				return bless \$q;
3219				}
3220
3221				bless \$r;
3222				}
3223
3224				#
3225				## Greatest common multiple
3226				#
3227
3228				sub gcd ($$) {
3229				my ($x, $y) = @_;
3230
3231				if (ref($y) and !ref($x)) {
3232				($x, $y) = ($y, $x);
3233				}
3234
3235				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
3236
3237				my $r = Math::GMPz::Rmpz_init();
3238
3239				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
3240				Math::GMPz::Rmpz_gcd_ui($r, $x, CORE::abs($y));
3241				}
3242				else {
3243				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3244				Math::GMPz::Rmpz_gcd($r, $x, $y);
3245				}
3246
3247				bless \$r;
3248				}
3249
3250				#
3251				## Least common multiple
3252				#
3253
3254				sub lcm ($$) {
3255				my ($x, $y) = @_;
3256
3257				if (ref($y) and !ref($x)) {
3258				($x, $y) = ($y, $x);
3259				}
3260
3261				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
3262
3263				my $r = Math::GMPz::Rmpz_init();
3264
3265				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
3266				Math::GMPz::Rmpz_lcm_ui($r, $x, CORE::abs($y));
3267				}
3268				else {
3269				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3270				Math::GMPz::Rmpz_lcm($r, $x, $y);
3271				}
3272
3273				bless \$r;
3274				}
3275
3276				#
3277				## Next prime after `x`.
3278				#
3279
3280				sub next_prime ($) {
3281				my ($x) = @_;
3282				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3283				my $r = Math::GMPz::Rmpz_init();
3284				Math::GMPz::Rmpz_nextprime($r, $x);
3285				bless \$r;
3286				}
3287
3288				#
3289				## Is prime?
3290				#
3291
3292				sub is_prime ($;$) {
3293				require Math::AnyNum::is_int;
3294				my ($x, $y) = @_;
3295
3296				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3297
3298				__is_int__($x) \|\| return 0;
3299
3300				$y = defined($y) ? (CORE::abs(CORE::int($y)) \|\| 20) : 20;
3301				Math::GMPz::Rmpz_probab_prime_p(_any2mpz($x) // (return 0), $y);
3302				}
3303
3304				#
3305				## Is `x` coprime to `y`?
3306				#
3307
3308				sub is_coprime ($$) {
3309				require Math::AnyNum::is_int;
3310				my ($x, $y) = @_;
3311
3312				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3313
3314				__is_int__($x) \|\| return 0;
3315
3316				$x = _any2mpz($x) // return 0;
3317
3318				if (!ref($y) and CORE::int($y) eq $y and $y >= 0 and $y <= ULONG_MAX) {
3319				## `y` is a native integer
3320				}
3321				else {
3322				$y = ref($y) eq __PACKAGE__ ? $$y : _star2obj($y);
3323				__is_int__($y) \|\| return 0;
3324				$y = _any2mpz($y) // return 0;
3325				}
3326
3327				state $t = Math::GMPz::Rmpz_init_nobless();
3328
3329				ref($y)
3330				? Math::GMPz::Rmpz_gcd($t, $x, $y)
3331				: Math::GMPz::Rmpz_gcd_ui($t, $x, $y);
3332
3333				Math::GMPz::Rmpz_cmp_ui($t, 1) == 0;
3334				}
3335
3336				#
3337				## Is integer?
3338				#
3339
3340				sub is_int ($) {
3341				require Math::AnyNum::is_int;
3342				my ($x) = @_;
3343				__is_int__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
3344				}
3345
3346				#
3347				## Is rational?
3348				#
3349
3350				sub is_rat ($) {
3351				my ($x) = @_;
3352				my $ref = ref(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
3353				$ref eq 'Math::GMPz' or $ref eq 'Math::GMPq';
3354				}
3355
3356				#
3357				## Numerator of a number
3358				#
3359
3360				sub numerator ($) {
3361				my ($x) = @_;
3362
3363				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3364				{
3365				my $ref = ref($r);
3366				ref($r) eq 'Math::GMPz' && return $x; # is an integer
3367
3368				if (ref($r) eq 'Math::GMPq') {
3369				my $z = Math::GMPz::Rmpz_init();
3370				Math::GMPq::Rmpq_get_num($z, $r);
3371				return bless \$z;
3372				}
3373
3374				$r = _any2mpq($r) // goto &nan;
3375				redo;
3376				}
3377				}
3378
3379				#
3380				## Denominator of a number
3381				#
3382
3383				sub denominator ($) {
3384				my ($x) = @_;
3385
3386				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3387				{
3388				my $ref = ref($r);
3389				ref($r) eq 'Math::GMPz' && goto &one; # is an integer
3390
3391				if (ref($r) eq 'Math::GMPq') {
3392				my $z = Math::GMPz::Rmpz_init();
3393				Math::GMPq::Rmpq_get_den($z, $r);
3394				return bless \$z;
3395				}
3396				$r = _any2mpq($r) // goto &nan;
3397				redo;
3398				}
3399				}
3400
3401				#
3402				## (numerator, denominator)
3403				#
3404
3405				sub nude ($) {
3406				my ($x) = @_;
3407
3408				if (ref($x) ne __PACKAGE__) {
3409				$x = __PACKAGE__->new($x);
3410				}
3411
3412				($x->numerator, $x->denominator);
3413				}
3414
3415				#
3416				## Sign of a number
3417				#
3418
3419				sub sgn ($) {
3420				require Math::AnyNum::sgn;
3421				my ($x) = @_;
3422				my $r = __sgn__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
3423				ref($r) ? (bless \$r) : $r;
3424				}
3425
3426				#
3427				## Is a real number?
3428				#
3429
3430				sub is_real ($) {
3431				my ($x) = @_;
3432
3433				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3434				{
3435				my $ref = ref($r);
3436
3437				$ref eq 'Math::GMPz' && return 1;
3438				$ref eq 'Math::GMPq' && return 1;
3439				$ref eq 'Math::MPFR' && return Math::MPFR::Rmpfr_number_p($r);
3440
3441				$r = _any2mpfr($r);
3442				redo;
3443				}
3444				}
3445
3446				#
3447				## Is an imaginary number?
3448				#
3449
3450				sub is_imag ($) {
3451				my ($x) = @_;
3452
3453				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3454				ref($r) eq 'Math::MPC' or return 0;
3455
3456				my $f = Math::MPFR::Rmpfr_init2($PREC);
3457				Math::MPC::RMPC_RE($f, $r);
3458				Math::MPFR::Rmpfr_zero_p($f) \|\| return 0; # is complex
3459				Math::MPC::RMPC_IM($f, $r);
3460				!Math::MPFR::Rmpfr_zero_p($f);
3461				}
3462
3463				#
3464				## Is a complex number?
3465				#
3466
3467				sub is_complex ($) {
3468				my ($x) = @_;
3469
3470				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3471				ref($r) eq 'Math::MPC' or return 0;
3472
3473				my $f = Math::MPFR::Rmpfr_init2($PREC);
3474				Math::MPC::RMPC_IM($f, $r);
3475				Math::MPFR::Rmpfr_zero_p($f) && return 0; # is real
3476				Math::MPC::RMPC_RE($f, $r);
3477				!Math::MPFR::Rmpfr_zero_p($f);
3478				}
3479
3480				#
3481				## Is positive infinity?
3482				#
3483
3484				sub is_inf ($) {
3485				my ($x) = @_;
3486
3487				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3488				{
3489				my $ref = ref($r);
3490
3491				$ref eq 'Math::GMPz' && return 0;
3492				$ref eq 'Math::GMPq' && return 0;
3493				$ref eq 'Math::MPFR' && return (Math::MPFR::Rmpfr_inf_p($r) and Math::MPFR::Rmpfr_sgn($r) > 0);
3494
3495				$r = _any2mpfr($r);
3496				redo;
3497				}
3498				}
3499
3500				#
3501				## Is negative infinity?
3502				#
3503
3504				sub is_ninf ($) {
3505				my ($x) = @_;
3506
3507				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3508				{
3509				my $ref = ref($r);
3510
3511				$ref eq 'Math::GMPz' && return 0;
3512				$ref eq 'Math::GMPq' && return 0;
3513				$ref eq 'Math::MPFR' && return (Math::MPFR::Rmpfr_inf_p($r) and Math::MPFR::Rmpfr_sgn($r) < 0);
3514
3515				$r = _any2mpfr($r);
3516				redo;
3517				}
3518				}
3519
3520				#
3521				## Is Not-A-Number?
3522				#
3523
3524				sub is_nan ($) {
3525				my ($x) = @_;
3526
3527				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3528
3529				ref($x) eq 'Math::GMPz' && return 0;
3530				ref($x) eq 'Math::GMPq' && return 0;
3531				ref($x) eq 'Math::MPFR' && return Math::MPFR::Rmpfr_nan_p($x);
3532
3533				my $t = Math::MPFR::Rmpfr_init2($PREC);
3534
3535				Math::MPC::RMPC_RE($t, $x);
3536				Math::MPFR::Rmpfr_nan_p($t) && return 1;
3537
3538				Math::MPC::RMPC_IM($t, $x);
3539				Math::MPFR::Rmpfr_nan_p($t) && return 1;
3540
3541				return 0;
3542				}
3543
3544				#
3545				## Is an even integer?
3546				#
3547
3548				sub is_even ($) {
3549				require Math::AnyNum::is_int;
3550				my ($x) = @_;
3551
3552				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3553
3554				__is_int__($x)
3555				&& Math::GMPz::Rmpz_even_p(_any2mpz($x) // (return 0));
3556				}
3557
3558				#
3559				## Is an odd integer?
3560				#
3561
3562				sub is_odd ($) {
3563				require Math::AnyNum::is_int;
3564				my ($x) = @_;
3565
3566				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3567
3568				__is_int__($x)
3569				&& Math::GMPz::Rmpz_odd_p(_any2mpz($x) // (return 0));
3570				}
3571
3572				#
3573				## Is zero?
3574				#
3575
3576				sub is_zero ($) {
3577				require Math::AnyNum::eq;
3578				my ($x) = @_;
3579				(@_) = ((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 0);
3580				goto &__eq__;
3581				}
3582
3583				#
3584				## Is one?
3585				#
3586
3587				sub is_one ($) {
3588				require Math::AnyNum::eq;
3589				my ($x) = @_;
3590				(@_) = ((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 1);
3591				goto &__eq__;
3592				}
3593
3594				#
3595				## Is minus one?
3596				#
3597
3598				sub is_mone ($) {
3599				require Math::AnyNum::eq;
3600				my ($x) = @_;
3601				(@_) = ((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), -1);
3602				goto &__eq__;
3603				}
3604
3605				#
3606				## Is positive?
3607				#
3608
3609				sub is_pos ($) {
3610				require Math::AnyNum::cmp;
3611				my ($x) = @_;
3612				(__cmp__((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 0) // return undef) > 0;
3613				}
3614
3615				#
3616				## Is negative?
3617				#
3618
3619				sub is_neg ($) {
3620				require Math::AnyNum::cmp;
3621				my ($x) = @_;
3622				(__cmp__((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 0) // return undef) < 0;
3623				}
3624
3625				#
3626				## Is square?
3627				#
3628
3629				sub is_square ($) {
3630				require Math::AnyNum::is_int;
3631				my ($x, $y) = @_;
3632
3633				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3634
3635				__is_int__($x)
3636				&& Math::GMPz::Rmpz_perfect_square_p(_any2mpz($x) // (return 0));
3637				}
3638
3639				#
3640				## Is a polygonal number?
3641				#
3642
3643				sub is_polygonal ($$) {
3644				require Math::AnyNum::is_int;
3645				require Math::AnyNum::is_polygonal;
3646				my ($n, $k) = @_;
3647
3648				$n = ref($n) eq __PACKAGE__ ? $$n : _star2obj($n);
3649
3650				$n = (__is_int__($n) ? _any2mpz($n) : return 0) // return 0;
3651				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // return 0;
3652
3653				__is_polygonal__($n, $k);
3654				}
3655
3656				#
3657				## Is a second polygonal number?
3658				#
3659
3660				sub is_polygonal2 ($$) {
3661				require Math::AnyNum::is_int;
3662				require Math::AnyNum::is_polygonal;
3663				my ($n, $k) = @_;
3664
3665				$n = ref($n) eq __PACKAGE__ ? $$n : _star2obj($n);
3666
3667				$n = (__is_int__($n) ? _any2mpz($n) : return 0) // return 0;
3668				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // return 0;
3669
3670				__is_polygonal__($n, $k, 1);
3671				}
3672
3673				#
3674				## Integer polygonal root
3675				#
3676
3677				sub ipolygonal_root ($$) {
3678				require Math::AnyNum::ipolygonal_root;
3679				my ($n, $k) = @_;
3680
3681				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3682				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // goto &nan;
3683
3684				bless \__ipolygonal_root__($n, $k);
3685				}
3686
3687				#
3688				## Second integer polygonal root
3689				#
3690
3691				sub ipolygonal_root2 ($$) {
3692				require Math::AnyNum::ipolygonal_root;
3693				my ($n, $k) = @_;
3694
3695				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3696				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // goto &nan;
3697
3698				bless \__ipolygonal_root__($n, $k, 1);
3699				}
3700
3701				#
3702				## n-th k-gonal number
3703				#
3704
3705				sub polygonal ($$) {
3706				my ($n, $k) = @_;
3707
3708				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3709
3710				if (!ref($k) and CORE::int($k) eq $k and $k >= 0 and $k <= ULONG_MAX) {
3711				## `k` is a native unsigned integer
3712				}
3713				else {
3714				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // goto &nan;
3715				}
3716
3717				#
3718				## polygonal(n, k) = n * (kn - k - 2n + 4) / 2
3719				#
3720
3721				my $r = Math::GMPz::Rmpz_init();
3722
3723				if (!ref($k)) { # `k` is a native unsigned integer
3724				Math::GMPz::Rmpz_mul_ui($r, $n, $k); # r = n*k
3725				Math::GMPz::Rmpz_sub_ui($r, $r, $k); # r = r-k
3726				}
3727				else {
3728				Math::GMPz::Rmpz_mul($r, $n, $k); # r = n*k
3729				Math::GMPz::Rmpz_sub($r, $r, $k); # r = r-k
3730				}
3731
3732				Math::GMPz::Rmpz_submul_ui($r, $n, 2); # r = r-2*n
3733				Math::GMPz::Rmpz_add_ui($r, $r, 4); # r = r+4
3734				Math::GMPz::Rmpz_mul($r, $r, $n); # r = r*n
3735				Math::GMPz::Rmpz_div_2exp($r, $r, 1); # r = r/2
3736
3737				bless \$r;
3738				}
3739
3740				#
3741				## is_power
3742				#
3743
3744				sub is_power ($;$) {
3745				require Math::AnyNum::is_power;
3746				require Math::AnyNum::is_int;
3747				my ($x, $y) = @_;
3748
3749				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3750
3751				__is_int__($x) \|\| return 0;
3752				$x = _any2mpz($x) // goto &nan;
3753
3754				if (!defined($y)) {
3755				return Math::GMPz::Rmpz_perfect_power_p($x);
3756				}
3757
3758				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
3759				## `y` is a native integer
3760				}
3761				else {
3762				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // return 0;
3763				}
3764
3765				__is_power__($x, $y);
3766				}
3767
3768				#
3769				## kronecker
3770				#
3771
3772				sub kronecker ($$) {
3773				my ($x, $y) = @_;
3774
3775				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3776				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3777
3778				Math::GMPz::Rmpz_kronecker($x, $y);
3779				}
3780
3781				#
3782				## valuation
3783				#
3784
3785				sub valuation ($$) {
3786				require Math::AnyNum::valuation;
3787				my ($x, $y) = @_;
3788
3789				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3790				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3791
3792				(__valuation__($x, $y))[0];
3793				}
3794
3795				#
3796				## remdiv
3797				#
3798
3799				sub remdiv ($$) {
3800				require Math::AnyNum::valuation;
3801				my ($x, $y) = @_;
3802
3803				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3804				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3805
3806				bless \((__valuation__($x, $y))[1]);
3807				}
3808
3809				#
3810				## Invmod
3811				#
3812
3813				sub invmod ($$) {
3814				my ($x, $y) = @_;
3815
3816				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3817				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3818
3819				my $r = Math::GMPz::Rmpz_init();
3820				Math::GMPz::Rmpz_invert($r, $x, $y) \|\| (goto &nan);
3821				bless \$r;
3822				}
3823
3824				#
3825				## Powmod
3826				#
3827
3828				sub powmod ($$$) {
3829				my ($x, $y, $z) = @_;
3830
3831				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3832				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3833				$z = (ref($z) eq __PACKAGE__ ? _any2mpz($$z) : _star2mpz($z)) // goto &nan;
3834
3835				Math::GMPz::Rmpz_sgn($z) \|\| goto &nan;
3836
3837				if (Math::GMPz::Rmpz_sgn($y) < 0) {
3838				my $t = Math::GMPz::Rmpz_init();
3839				Math::GMPz::Rmpz_gcd($t, $x, $z);
3840				Math::GMPz::Rmpz_cmp_ui($t, 1) == 0 or goto &nan;
3841				}
3842
3843				my $r = Math::GMPz::Rmpz_init();
3844				Math::GMPz::Rmpz_powm($r, $x, $y, $z);
3845				bless \$r;
3846				}
3847
3848				#
3849				## Faulhaber summation formula
3850				#
3851
3852				sub faulhaber_sum ($$) {
3853				require Math::AnyNum::bernfrac;
3854				my ($n, $p) = @_;
3855
3856				my $native_n = 0; # true when `n` is a native integer
3857
3858				if (!ref($n) and CORE::int($n) eq $n and $n >= 0 and $n <= ULONG_MAX) {
3859				## `n` is a native unsigned integer
3860				$native_n = 1;
3861				}
3862				else {
3863				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3864
3865				# Try to unbox `n` when it fits inside a native unsigned integer
3866				if (Math::GMPz::Rmpz_fits_ulong_p($n)) {
3867				$native_n = 1;
3868				$n = Math::GMPz::Rmpz_get_ui($n);
3869				}
3870				}
3871
3872				if (!ref($p) and CORE::int($p) eq $p and $p >= 0 and $p <= ULONG_MAX) {
3873				## `p` is already a native unsigned integer
3874				}
3875				else {
3876				$p = (ref($p) eq __PACKAGE__ ? _any2ui($$p) : _any2ui(_star2obj($p))) // goto &nan;
3877				}
3878
3879				state @cache; # cache for Bernoulli numbers
3880
3881				my $t = Math::GMPz::Rmpz_init();
3882				my $u = Math::GMPz::Rmpz_init();
3883
3884				my $numerator = Math::GMPz::Rmpz_init_set_ui(0);
3885				my $denominator = Math::GMPz::Rmpz_init_set_ui(1);
3886
3887				foreach my $j (0 .. $p) {
3888
3889				# When `j` is odd and greater than 1, we can skip it.
3890				$j % 2 == 0 or $j == 1 or next;
3891
3892				Math::GMPz::Rmpz_bin_uiui($t, $p + 1, $j); # t = binomial(p+1, j)
3893
3894				#<<<
3895				$native_n
3896				? Math::GMPz::Rmpz_ui_pow_ui($u, $n, $p + 1 - $j) # u = n^(p + 1 - j)
3897				: Math::GMPz::Rmpz_pow_ui( $u, $n, $p + 1 - $j); # ==//==
3898				#>>>
3899
3900				Math::GMPz::Rmpz_mul($t, $t, $u); # t = t * u
3901
3902				# Compute Bernouli(j)
3903				my $bern = ($j <= 100 ? ($cache[$j] //= __bernfrac__($j)) : __bernfrac__($j));
3904
3905				# Bernoulli(j) = 1 for j=0
3906				if (!$j) {
3907				Math::GMPz::Rmpz_add($numerator, $numerator, $t); # numerator = numerator + t
3908				}
3909				else {
3910				#<<<
3911				Math::GMPq::Rmpq_get_num($u, $bern); # u = numerator(bern)
3912				Math::GMPz::Rmpz_mul($t, $t, $u); # t = t * u
3913				Math::GMPq::Rmpq_get_den($u, $bern); # u = denominator(bern)
3914
3915				Math::GMPz::Rmpz_mul( $numerator, $numerator, $u); # numerator = numerator * u
3916				Math::GMPz::Rmpz_addmul($numerator, $denominator, $t); # numerator += denominator * t
3917				Math::GMPz::Rmpz_mul( $denominator, $denominator, $u); # denominator = denominator * u
3918				#>>>
3919				}
3920				}
3921
3922				#<<<
3923				Math::GMPz::Rmpz_mul_ui($denominator, $denominator, $p + 1); # denominator = denominator * (p+1)
3924				Math::GMPz::Rmpz_divexact($numerator, $numerator, $denominator); # numerator = numerator / denominator
3925				#>>>
3926
3927				bless \$numerator;
3928				}
3929
3930				#
3931				## Binomial
3932				#
3933
3934				sub binomial ($$) {
3935				my ($x, $y) = @_;
3936
3937				# `x` and `y` are native unsigned integers
3938				if ( !ref($x)
3939				and !ref($y)
3940				and CORE::int($x) eq $x
3941				and CORE::int($y) eq $y
3942				and $x >= 0
3943				and $y >= 0
3944				and $x <= ULONG_MAX
3945				and $y <= ULONG_MAX) {
3946				my $r = Math::GMPz::Rmpz_init();
3947				Math::GMPz::Rmpz_bin_uiui($r, $x, $y);
3948				return bless \$r;
3949				}
3950
3951				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3952
3953				if (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3954				## `y` is a native integer
3955				}
3956				else {
3957				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // goto &nan;
3958				}
3959
3960				my $r = Math::GMPz::Rmpz_init();
3961
3962				if ($y >= 0 and Math::GMPz::Rmpz_fits_ulong_p($x)) {
3963				Math::GMPz::Rmpz_bin_uiui($r, Math::GMPz::Rmpz_get_ui($x), $y);
3964				}
3965				else {
3966				$y < 0
3967				? Math::GMPz::Rmpz_bin_si($r, $x, $y)
3968				: Math::GMPz::Rmpz_bin_ui($r, $x, $y);
3969				}
3970
3971				bless \$r;
3972				}
3973
3974				#
3975				## AND
3976				#
3977
3978				sub and { # used in overloading
3979				my ($x, $y) = @_;
3980
3981				$x = _any2mpz($$x) // (goto &nan);
3982				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3983
3984				my $r = Math::GMPz::Rmpz_init();
3985				Math::GMPz::Rmpz_and($r, $x, $y);
3986				bless \$r;
3987				}
3988
3989				#
3990				## OR
3991				#
3992
3993				sub or { # used in overloading
3994				my ($x, $y) = @_;
3995
3996				$x = _any2mpz($$x) // (goto &nan);
3997				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3998
3999				my $r = Math::GMPz::Rmpz_init();
4000				Math::GMPz::Rmpz_ior($r, $x, $y);
4001				bless \$r;
4002				}
4003
4004				#
4005				## XOR
4006				#
4007
4008				sub xor { # used in overloading
4009				my ($x, $y) = @_;
4010
4011				$x = _any2mpz($$x) // (goto &nan);
4012				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
4013
4014				my $r = Math::GMPz::Rmpz_init();
4015				Math::GMPz::Rmpz_xor($r, $x, $y);
4016				bless \$r;
4017				}
4018
4019				#
4020				## NOT
4021				#
4022
4023				sub not { # used in overloading
4024				my ($x) = @_;
4025				$x = _any2mpz($$x) // (goto &nan);
4026				my $r = Math::GMPz::Rmpz_init();
4027				Math::GMPz::Rmpz_com($r, $x);
4028				bless \$r;
4029				}
4030
4031				#
4032				## LEFT SHIFT
4033				#
4034
4035				sub lsft { # used in overloading
4036				my ($x, $y) = @_;
4037
4038				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
4039
4040				if (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
4041				## `y` is a native integer
4042				}
4043				else {
4044				$y = (ref($y) eq __PACKAGE__ ? _any2si($$y) : _any2si(_star2obj($y))) // goto &nan;
4045				}
4046
4047				my $r = Math::GMPz::Rmpz_init();
4048
4049				$y < 0
4050				? Math::GMPz::Rmpz_div_2exp($r, $x, -$y)
4051				: Math::GMPz::Rmpz_mul_2exp($r, $x, $y);
4052
4053				bless \$r;
4054				}
4055
4056				#
4057				## RIGHT SHIFT
4058				#
4059
4060				sub rsft { # used in overloading
4061				my ($x, $y) = @_;
4062
4063				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
4064
4065				if (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
4066				## `y` is a native integer
4067				}
4068				else {
4069				$y = (ref($y) eq __PACKAGE__ ? _any2si($$y) : _any2si(_star2obj($y))) // goto &nan;
4070				}
4071
4072				my $r = Math::GMPz::Rmpz_init();
4073
4074				$y < 0
4075				? Math::GMPz::Rmpz_mul_2exp($r, $x, -$y)
4076				: Math::GMPz::Rmpz_div_2exp($r, $x, $y);
4077
4078				bless \$r;
4079				}
4080
4081				#
4082				## POPCOUNT
4083				#
4084
4085				sub popcount ($) {
4086				my ($x) = @_;
4087
4088				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _any2mpz(_star2obj($x))) // return -1;
4089
4090				if (Math::GMPz::Rmpz_sgn($x) < 0) {
4091				my $t = Math::GMPz::Rmpz_init();
4092				Math::GMPz::Rmpz_neg($t, $x);
4093				$x = $t;
4094				}
4095
4096				Math::GMPz::Rmpz_popcount($x);
4097				}
4098
4099				#
4100				## Conversions
4101				#
4102
4103				sub as_bin ($) {
4104				Math::GMPz::Rmpz_get_str((_star2mpz($_[0]) // return undef), 2);
4105				}
4106
4107				sub as_oct ($) {
4108				Math::GMPz::Rmpz_get_str((_star2mpz($_[0]) // return undef), 8);
4109				}
4110
4111				sub as_hex ($) {
4112				Math::GMPz::Rmpz_get_str((_star2mpz($_[0]) // return undef), 16);
4113				}
4114
4115				sub as_int ($;$) {
4116				my ($x, $y) = @_;
4117
4118				my $base = 10;
4119				if (defined($y)) {
4120
4121				if (!ref($y) and CORE::int($y) eq $y) {
4122				$base = $y;
4123				}
4124				elsif (ref($y) eq __PACKAGE__) {
4125				$base = _any2ui($$y) // 0;
4126				}
4127				else {
4128				$base = _any2ui(_star2mpz($y) // return undef) // 0;
4129				}
4130
4131				if ($base < 2 or $base > 36) {
4132				require Carp;
4133				Carp::croak("base must be between 2 and 36, got $y");
4134				}
4135				}
4136
4137				Math::GMPz::Rmpz_get_str((_star2mpz($x) // return undef), $base);
4138				}
4139
4140				sub as_frac ($;$) {
4141				my ($x, $y) = @_;
4142
4143				my $base = 10;
4144				if (defined($y)) {
4145
4146				if (!ref($y) and CORE::int($y) eq $y) {
4147				$base = $y;
4148				}
4149				elsif (ref($y) eq __PACKAGE__) {
4150				$base = _any2ui($$y) // 0;
4151				}
4152				else {
4153				$base = _any2ui(_star2mpz($y) // return undef) // 0;
4154				}
4155
4156				if ($base < 2 or $base > 36) {
4157				require Carp;
4158				Carp::croak("base must be between 2 and 36, got $y");
4159				}
4160				}
4161
4162				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
4163
4164				my $ref = ref($x);
4165				if ( $ref eq 'Math::GMPq'
4166				or $ref eq 'Math::GMPz') {
4167				my $frac = (
4168				$ref eq 'Math::GMPq'
4169				? Math::GMPq::Rmpq_get_str($x, $base)
4170				: Math::GMPz::Rmpz_get_str($x, $base)
4171				);
4172				$frac .= '/1' if (index($frac, '/') == -1);
4173				return $frac;
4174				}
4175
4176				$x = _any2mpq($x) // return undef;
4177
4178				my $frac = Math::GMPq::Rmpq_get_str($x, $base);
4179				$frac .= '/1' if (index($frac, '/') == -1);
4180				$frac;
4181				}
4182
4183				sub as_dec ($;$) {
4184				my ($x, $y) = @_;
4185				require Math::AnyNum::stringify;
4186
4187				my $prec = $PREC;
4188				if (defined($y)) {
4189				if (!ref($y) and CORE::int($y) eq $y) {
4190				$prec = $y;
4191				}
4192				elsif (ref($y) eq __PACKAGE__) {
4193				$prec = _any2ui($$y) // 0;
4194				}
4195				else {
4196				$prec = _any2ui(_star2mpz($y) // return undef) // 0;
4197				}
4198
4199				$prec <<= 2;
4200
4201				state $min_prec = Math::MPFR::RMPFR_PREC_MIN();
4202				state $max_prec = Math::MPFR::RMPFR_PREC_MAX();
4203
4204				if ($prec < $min_prec or $prec > $max_prec) {
4205				require Carp;
4206				Carp::croak("precision must be between $min_prec and $max_prec, got ", $prec >> 2);
4207				}
4208				}
4209
4210				local $PREC = $prec;
4211				__stringify__(_star2mpfr_mpc($x));
4212				}
4213
4214				sub rat_approx ($) {
4215				require Math::AnyNum::stringify;
4216				my ($x) = @_;
4217
4218				$x = _star2mpfr($x);
4219
4220				Math::MPFR::Rmpfr_number_p($x) \|\| goto &nan;
4221
4222				my $t = Math::MPFR::Rmpfr_init2($PREC); # temporary variable
4223				my $r = Math::MPFR::Rmpfr_init2($PREC);
4224
4225				Math::MPFR::Rmpfr_set($r, $x, $ROUND);
4226
4227				my $num2cfrac = sub {
4228				my ($callback, $n) = @_;
4229
4230				while (1) {
4231				Math::MPFR::Rmpfr_floor($t, $r);
4232
4233				my $z = Math::GMPz::Rmpz_init();
4234				Math::MPFR::Rmpfr_get_z($z, $t, Math::MPFR::MPFR_RNDZ);
4235
4236				$callback->($z) && return 1;
4237
4238				Math::MPFR::Rmpfr_sub($r, $r, $t, $ROUND);
4239				Math::MPFR::Rmpfr_zero_p($r) && last;
4240				Math::MPFR::Rmpfr_ui_div($r, 1, $r, $ROUND);
4241				}
4242				};
4243
4244				my $q = Math::GMPq::Rmpq_init();
4245
4246				my $cfrac2num = sub {
4247				my (@f) = @_;
4248
4249				Math::GMPq::Rmpq_set_ui($q, 0, 1);
4250
4251				for (1 .. $#f) {
4252				Math::GMPq::Rmpq_add_z($q, $q, CORE::pop(@f));
4253				Math::GMPq::Rmpq_inv($q, $q);
4254				}
4255
4256				Math::GMPq::Rmpq_add_z($q, $q, $f[0]);
4257				};
4258
4259				my @cfrac;
4260				my $s = __stringify__($x);
4261				my $u = Math::MPFR::Rmpfr_init2($PREC); # temporary variable
4262
4263				#<<<
4264				$num2cfrac->(
4265				sub {
4266				my ($n) = @_;
4267				CORE::push(@cfrac, $n);
4268				$cfrac2num->(@cfrac);
4269				Math::MPFR::Rmpfr_set_q($u, $q, $ROUND);
4270				CORE::index(__stringify__($u), $s) == 0;
4271				}, $x
4272				);
4273				#>>>
4274
4275				bless \$q;
4276				}
4277
4278				sub digits ($;$) {
4279				my ($x, $y) = @_;
4280				my $str = as_int($x, $y) // return ();
4281				my @digits = split(//, $str);
4282				shift(@digits) if $digits[0] eq '-';
4283				(@digits);
4284				}
4285
4286				1; # End of Math::AnyNum