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	47057	use 5.014;
	1		3
4	1	1	4	use strict;
	1		2
	1		17
5	1	1	4	use warnings;
	1		1
	1		26
6
7	1	1	4	no warnings qw(numeric uninitialized);
	1		2
	1		33
8
9	1	1	114	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 $VERSION = '0.15';
17				our ($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 ($x) = @_;
694
695				if (ref($x) eq 'Math::GMPz') {
696				my $d = CORE::int(Math::GMPz::Rmpz_get_d($x));
697				($d < 0 or $d > ULONG_MAX) && return;
698				return $d;
699				}
700
701				if (ref($x) eq 'Math::GMPq') {
702				my $d = CORE::int(Math::GMPq::Rmpq_get_d($x));
703				($d < 0 or $d > ULONG_MAX) && return;
704				return $d;
705				}
706
707				if (ref($x) eq 'Math::MPFR') {
708				if (Math::MPFR::Rmpfr_number_p($x)) {
709				my $d = CORE::int(Math::MPFR::Rmpfr_get_d($x, $ROUND));
710				($d < 0 or $d > ULONG_MAX) && return;
711				return $d;
712				}
713				return;
714				}
715
716				(@_) = _any2mpfr($x);
717				goto &_any2ui;
718				}
719
720				sub _any2si {
721				my ($x) = @_;
722
723				if (ref($x) eq 'Math::GMPz') {
724				my $d = CORE::int(Math::GMPz::Rmpz_get_d($x));
725				($d < LONG_MIN or $d > ULONG_MAX) && return;
726				return $d;
727				}
728
729				if (ref($x) eq 'Math::GMPq') {
730				my $d = CORE::int(Math::GMPq::Rmpq_get_d($x));
731				($d < LONG_MIN or $d > ULONG_MAX) && return;
732				return $d;
733				}
734
735				if (ref($x) eq 'Math::MPFR') {
736				if (Math::MPFR::Rmpfr_number_p($x)) {
737				my $d = CORE::int(Math::MPFR::Rmpfr_get_d($x, $ROUND));
738				($d < LONG_MIN or $d > ULONG_MAX) && return;
739				return $d;
740				}
741				return;
742				}
743
744				(@_) = _any2mpfr($x);
745				goto &_any2si;
746				}
747
748				#
749				## Anything to MPFR (including strings)
750				#
751				sub _star2mpfr {
752				my ($x) = @_;
753
754				$x =
755				ref($x) eq __PACKAGE__ ? $$x
756				: ref($x) ? _star2obj($x)
757				: _str2obj($x);
758
759				ref($x) eq 'Math::MPFR' and return $x;
760
761				(@_) = $x;
762				ref($x) eq 'Math::GMPz' && goto &_mpz2mpfr;
763				ref($x) eq 'Math::GMPq' && goto &_mpq2mpfr;
764				goto &_any2mpfr;
765				}
766
767				#
768				## Anything to GMPz (including strings)
769				#
770				sub _star2mpz {
771				my ($x) = @_;
772
773				$x =
774				ref($x) eq __PACKAGE__ ? $$x
775				: ref($x) ? _star2obj($x)
776				: _str2obj($x);
777
778				ref($x) eq 'Math::GMPz' and return $x;
779
780				(@_) = $x;
781				ref($x) eq 'Math::GMPq' and goto &_mpq2mpz;
782				goto &_any2mpz;
783				}
784
785				#
786				## Anything to MPFR or MPC, in this order (including strings)
787				#
788				sub _star2mpfr_mpc {
789				my ($x) = @_;
790
791				$x =
792				ref($x) eq __PACKAGE__ ? $$x
793				: ref($x) ? _star2obj($x)
794				: _str2obj($x);
795
796				if ( ref($x) eq 'Math::MPFR'
797				or ref($x) eq 'Math::MPC') {
798				return $x;
799				}
800
801				(@_) = $x;
802				ref($x) eq 'Math::GMPz' && goto &_mpz2mpfr;
803				ref($x) eq 'Math::GMPq' && goto &_mpq2mpfr;
804				goto &_any2mpfr; # this should not happen
805				}
806
807				# Anything to a {GMP*, MPFR or MPC} object
808				sub _star2obj {
809				my ($x) = @_;
810
811				ref($x) \|\| goto &_str2obj;
812
813				if (ref($x) eq __PACKAGE__) {
814				$$x;
815				}
816				elsif (
817				ref($x)
818				and ( ref($x) eq 'Math::GMPz'
819				or ref($x) eq 'Math::GMPq'
820				or ref($x) eq 'Math::MPFR'
821				or ref($x) eq 'Math::MPC')
822				) {
823				$x;
824				}
825				else {
826				(@_) = "$x";
827				goto &_str2obj;
828				}
829				}
830
831				sub new {
832				my ($class, $num, $base) = @_;
833
834				my $ref = ref($num);
835
836				# Special string values
837				if (!$ref and (!defined($base) or CORE::int($base) == 10)) {
838				return bless \_str2obj($num), $class;
839				}
840
841				# Special case
842				if (!defined($base) and $ref eq __PACKAGE__) {
843				return $num;
844				}
845
846				# Number with base
847				if (defined($base) and CORE::int($base) != 10) {
848
849				my $int_base = CORE::int($base);
850
851				if ($int_base < 2 or $int_base > 36) {
852				require Carp;
853				Carp::croak("base must be between 2 and 36, got $base");
854				}
855
856				$num = defined($num) ? "$num" : '0';
857
858				if (index($num, '/') != -1) {
859				my $r = Math::GMPq::Rmpq_init();
860				eval { Math::GMPq::Rmpq_set_str($r, $num, $int_base); 1 } // goto &nan;
861
862				if (Math::GMPq::Rmpq_get_str($r, 10) !~ m{^\s[-+]?[0-9]+\s/\s[-+]?[1-9]+[0-9]\s*\z}) {
863				goto &nan;
864				}
865
866				Math::GMPq::Rmpq_canonicalize($r);
867				return bless \$r, $class;
868				}
869				elsif (index($num, '.') != -1) {
870				my $r = Math::MPFR::Rmpfr_init2($PREC);
871				if (Math::MPFR::Rmpfr_set_str($r, $num, $int_base, $ROUND)) {
872				Math::MPFR::Rmpfr_set_nan($r);
873				}
874				return bless \$r, $class;
875				}
876				else {
877				return bless \(eval { Math::GMPz::Rmpz_init_set_str($num, $int_base) } // goto &nan), $class;
878				}
879				}
880
881				bless \_star2obj($num), $class;
882				}
883
884				sub new_si {
885				my ($class, $si) = @_;
886				bless \Math::GMPz::Rmpz_init_set_si($si), $class;
887				}
888
889				sub new_ui {
890				my ($class, $ui) = @_;
891				bless \Math::GMPz::Rmpz_init_set_ui($ui), $class;
892				}
893
894				sub new_z {
895				my ($class, $str, $base) = @_;
896				bless \Math::GMPz::Rmpz_init_set_str($str, $base // 10), $class;
897				}
898
899				sub new_q {
900				my ($class, $num, $den, $base) = @_;
901				my $r = Math::GMPq::Rmpq_init();
902
903				if (defined($den)) {
904				Math::GMPq::Rmpq_set_str($r, "$num/$den", $base // 10);
905				}
906				else {
907				Math::GMPq::Rmpq_set_str($r, "$num", $base // 10);
908				}
909
910				Math::GMPq::Rmpq_canonicalize($r);
911				bless \$r, $class;
912				}
913
914				sub new_f {
915				my ($class, $str, $base) = @_;
916				my $r = Math::MPFR::Rmpfr_init2($PREC);
917				Math::MPFR::Rmpfr_set_str($r, $str, $base // 10, $ROUND);
918				bless \$r, $class;
919				}
920
921				sub new_c {
922				my ($class, $real, $imag, $base) = @_;
923
924				my $c = Math::MPC::Rmpc_init2($PREC);
925
926				if (defined($imag)) {
927				my $re = Math::MPFR::Rmpfr_init2($PREC);
928				my $im = Math::MPFR::Rmpfr_init2($PREC);
929
930				Math::MPFR::Rmpfr_set_str($re, $real, $base // 10, $ROUND);
931				Math::MPFR::Rmpfr_set_str($im, $imag, $base // 10, $ROUND);
932
933				Math::MPC::Rmpc_set_fr_fr($c, $re, $im, $ROUND);
934				}
935				else {
936				Math::MPC::Rmpc_set_str($c, $real, $base // 10, $ROUND);
937				}
938
939				bless \$c, $class;
940				}
941
942				sub _nan {
943				state $nan = do {
944				my $r = Math::MPFR::Rmpfr_init2($PREC);
945				Math::MPFR::Rmpfr_set_nan($r);
946				$r;
947				};
948				}
949
950				sub nan {
951				state $nan = do {
952				my $r = Math::MPFR::Rmpfr_init2($PREC);
953				Math::MPFR::Rmpfr_set_nan($r);
954				bless \$r;
955				};
956				}
957
958				sub _inf {
959				state $inf = do {
960				my $r = Math::MPFR::Rmpfr_init2($PREC);
961				Math::MPFR::Rmpfr_set_inf($r, 1);
962				$r;
963				};
964				}
965
966				sub inf {
967				state $inf = do {
968				my $r = Math::MPFR::Rmpfr_init2($PREC);
969				Math::MPFR::Rmpfr_set_inf($r, 1);
970				bless \$r;
971				};
972				}
973
974				sub _ninf {
975				state $ninf = do {
976				my $r = Math::MPFR::Rmpfr_init2($PREC);
977				Math::MPFR::Rmpfr_set_inf($r, -1);
978				$r;
979				};
980				}
981
982				sub ninf {
983				state $ninf = do {
984				my $r = Math::MPFR::Rmpfr_init2($PREC);
985				Math::MPFR::Rmpfr_set_inf($r, -1);
986				bless \$r;
987				};
988				}
989
990				sub _zero {
991				state $zero = Math::GMPz::Rmpz_init_set_ui(0);
992				}
993
994				sub zero {
995				state $zero = do {
996				my $r = Math::GMPz::Rmpz_init_set_ui(0);
997				bless \$r;
998				};
999				}
1000
1001				sub _one {
1002				state $one = Math::GMPz::Rmpz_init_set_ui(1);
1003				}
1004
1005				sub one {
1006				state $one = do {
1007				my $r = Math::GMPz::Rmpz_init_set_ui(1);
1008				bless \$r;
1009				};
1010				}
1011
1012				sub _mone {
1013				state $mone = Math::GMPz::Rmpz_init_set_si(-1);
1014				}
1015
1016				sub mone {
1017				state $mone = do {
1018				my $r = Math::GMPz::Rmpz_init_set_si(-1);
1019				bless \$r;
1020				};
1021				}
1022
1023				#
1024				## CONSTANTS
1025				#
1026
1027				sub pi {
1028				my $pi = Math::MPFR::Rmpfr_init2($PREC);
1029				Math::MPFR::Rmpfr_const_pi($pi, $ROUND);
1030				bless \$pi;
1031				}
1032
1033				sub tau {
1034				my $tau = Math::MPFR::Rmpfr_init2($PREC);
1035				Math::MPFR::Rmpfr_const_pi($tau, $ROUND);
1036				Math::MPFR::Rmpfr_mul_2ui($tau, $tau, 1, $ROUND);
1037				bless \$tau;
1038				}
1039
1040				sub ln2 {
1041				my $ln2 = Math::MPFR::Rmpfr_init2($PREC);
1042				Math::MPFR::Rmpfr_const_log2($ln2, $ROUND);
1043				bless \$ln2;
1044				}
1045
1046				sub euler {
1047				my $euler = Math::MPFR::Rmpfr_init2($PREC);
1048				Math::MPFR::Rmpfr_const_euler($euler, $ROUND);
1049				bless \$euler;
1050				}
1051
1052				sub catalan {
1053				my $catalan = Math::MPFR::Rmpfr_init2($PREC);
1054				Math::MPFR::Rmpfr_const_catalan($catalan, $ROUND);
1055				bless \$catalan;
1056				}
1057
1058				sub i {
1059				my $i = Math::MPC::Rmpc_init2($PREC);
1060				Math::MPC::Rmpc_set_ui_ui($i, 0, 1, $ROUND);
1061				bless \$i;
1062				}
1063
1064				sub e {
1065				state $one_f = (Math::MPFR::Rmpfr_init_set_ui_nobless(1, $ROUND))[0];
1066				my $e = Math::MPFR::Rmpfr_init2($PREC);
1067				Math::MPFR::Rmpfr_exp($e, $one_f, $ROUND);
1068				bless \$e;
1069				}
1070
1071				sub phi {
1072				state $five4_f = (Math::MPFR::Rmpfr_init_set_d_nobless(1.25, $ROUND))[0];
1073
1074				my $phi = Math::MPFR::Rmpfr_init2($PREC);
1075				Math::MPFR::Rmpfr_sqrt($phi, $five4_f, $ROUND);
1076				Math::MPFR::Rmpfr_add_d($phi, $phi, 0.5, $ROUND);
1077
1078				bless \$phi;
1079				}
1080
1081				#
1082				## OTHER
1083				#
1084
1085				sub stringify { # used in overloading
1086				require Math::AnyNum::stringify;
1087				(@_) = (${$_[0]});
1088				goto &__stringify__;
1089				}
1090
1091				sub numify { # used in overloading
1092				require Math::AnyNum::numify;
1093				(@_) = (${$_[0]});
1094				goto &__numify__;
1095				}
1096
1097				sub boolify { # used in overloading
1098				require Math::AnyNum::boolify;
1099				(@_) = (${$_[0]});
1100				goto &__boolify__;
1101				}
1102
1103				#
1104				## EQUALITY
1105				#
1106
1107				sub eq { # used in overloading
1108				require Math::AnyNum::eq;
1109				my ($x, $y) = @_;
1110
1111				if (ref($y) eq __PACKAGE__) {
1112				(@_) = ($$x, $$y);
1113				goto &__eq__;
1114				}
1115
1116				if (!ref($y)) {
1117				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1118				(@_) = ($$x, $y);
1119				}
1120				else {
1121				(@_) = ($$x, _str2obj($y));
1122				}
1123				goto &__eq__;
1124				}
1125
1126				(@_) = ($$x, _star2obj($y));
1127				goto &__eq__;
1128				}
1129
1130				#
1131				## INEQUALITY
1132				#
1133
1134				sub ne { # used in overloading
1135				require Math::AnyNum::ne;
1136				my ($x, $y) = @_;
1137
1138				if (ref($y) eq __PACKAGE__) {
1139				(@_) = ($$x, $$y);
1140				goto &__ne__;
1141				}
1142
1143				if (!ref($y)) {
1144				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1145				(@_) = ($$x, $y);
1146				}
1147				else {
1148				(@_) = ($$x, _str2obj($y));
1149				}
1150				goto &__ne__;
1151				}
1152
1153				(@_) = ($$x, _star2obj($y));
1154				goto &__ne__;
1155				}
1156
1157				#
1158				## COMPARISON
1159				#
1160
1161				sub cmp ($$) {
1162				require Math::AnyNum::cmp;
1163				my ($x, $y) = @_;
1164
1165				if (ref($y) eq __PACKAGE__) {
1166				(@_) = ($$x, $$y);
1167				goto &__cmp__;
1168				}
1169
1170				if (!ref($y)) {
1171				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1172				(@_) = ($$x, $y);
1173				}
1174				else {
1175				(@_) = ($$x, _str2obj($y));
1176				}
1177				goto &__cmp__;
1178				}
1179
1180				(@_) = ($$x, _star2obj($y));
1181				goto &__cmp__;
1182				}
1183
1184				sub acmp ($$) {
1185				require Math::AnyNum::abs;
1186				require Math::AnyNum::cmp;
1187				my ($x, $y) = @_;
1188
1189				if (!ref($y) and CORE::int($y) eq $y and $y >= 0 and $y <= ULONG_MAX) {
1190				## `y` is a native unsigned integer
1191				}
1192				else {
1193				$y = __abs__(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y));
1194				}
1195
1196				__cmp__(__abs__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), $y);
1197				}
1198
1199				#
1200				## GREATER THAN
1201				#
1202
1203				sub gt { # used in overloading
1204				require Math::AnyNum::cmp;
1205				my ($x, $y) = @_;
1206
1207				if (ref($y) eq __PACKAGE__) {
1208				return ((__cmp__($$x, $$y) // return undef) > 0);
1209				}
1210
1211				if (!ref($y)) {
1212				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1213				return ((__cmp__($$x, $y) // return undef) > 0);
1214				}
1215				return ((__cmp__($$x, _str2obj($y)) // return undef) > 0);
1216				}
1217
1218				(__cmp__($$x, _star2obj($y)) // return undef) > 0;
1219				}
1220
1221				#
1222				## EQUAL OR GREATER THAN
1223				#
1224
1225				sub ge { # used in overloading
1226				require Math::AnyNum::cmp;
1227				my ($x, $y) = @_;
1228
1229				if (ref($y) eq __PACKAGE__) {
1230				return ((__cmp__($$x, $$y) // return undef) >= 0);
1231				}
1232
1233				if (!ref($y)) {
1234				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1235				return ((__cmp__($$x, $y) // return undef) >= 0);
1236				}
1237				return ((__cmp__($$x, _str2obj($y)) // return undef) >= 0);
1238				}
1239
1240				(__cmp__($$x, _star2obj($y)) // return undef) >= 0;
1241				}
1242
1243				#
1244				## LESS THAN
1245				#
1246
1247				sub lt { # used in overloading
1248				require Math::AnyNum::cmp;
1249				my ($x, $y) = @_;
1250
1251				if (ref($y) eq __PACKAGE__) {
1252				return ((__cmp__($$x, $$y) // return undef) < 0);
1253				}
1254
1255				if (!ref($y)) {
1256				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1257				return ((__cmp__($$x, $y) // return undef) < 0);
1258				}
1259				return ((__cmp__($$x, _str2obj($y)) // return undef) < 0);
1260				}
1261
1262				(__cmp__($$x, _star2obj($y)) // return undef) < 0;
1263				}
1264
1265				#
1266				## EQUAL OR LESS THAN
1267				#
1268
1269				sub le { # used in overloading
1270				require Math::AnyNum::cmp;
1271				my ($x, $y) = @_;
1272
1273				if (ref($y) eq __PACKAGE__) {
1274				return ((__cmp__($$x, $$y) // return undef) <= 0);
1275				}
1276
1277				if (!ref($y)) {
1278				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1279				return ((__cmp__($$x, $y) // return undef) <= 0);
1280				}
1281				return ((__cmp__($$x, _str2obj($y)) // return undef) <= 0);
1282				}
1283
1284				(__cmp__($$x, _star2obj($y)) // return undef) <= 0;
1285				}
1286
1287				#
1288				## COPY
1289				#
1290
1291				sub _copy {
1292				my ($x) = @_;
1293				my $ref = ref($x);
1294
1295				if ($ref eq 'Math::GMPz') {
1296				Math::GMPz::Rmpz_init_set($x);
1297				}
1298				elsif ($ref eq 'Math::MPFR') {
1299				my $r = Math::MPFR::Rmpfr_init2($PREC);
1300				Math::MPFR::Rmpfr_set($r, $x, $ROUND);
1301				$r;
1302				}
1303				elsif ($ref eq 'Math::GMPq') {
1304				my $r = Math::GMPq::Rmpq_init();
1305				Math::GMPq::Rmpq_set($r, $x);
1306				$r;
1307				}
1308				elsif ($ref eq 'Math::MPC') {
1309				my $r = Math::MPC::Rmpc_init2($PREC);
1310				Math::MPC::Rmpc_set($r, $x, $ROUND);
1311				$r;
1312				}
1313				else {
1314				_str2obj("$x"); # this should not happen
1315				}
1316				}
1317
1318				sub copy ($) {
1319				my ($x) = @_;
1320				bless \_copy($$x);
1321				}
1322
1323				#
1324				## CONVERSION TO INTEGER
1325				#
1326
1327				sub int { # used in overloading
1328				my ($x) = @_;
1329
1330				bless \(
1331				(
1332				ref($x) eq __PACKAGE__
1333				? ref($$x) eq 'Math::GMPz'
1334				? (return $x)
1335				: _any2mpz($$x)
1336				: _star2mpz($x)
1337				) // goto &nan
1338				);
1339				}
1340
1341				#
1342				## CONVERSION TO RATIONAL
1343				#
1344
1345				sub rat ($) {
1346				my ($x) = @_;
1347				if (ref($x) eq __PACKAGE__) {
1348				ref($$x) eq 'Math::GMPq' && return $x;
1349				bless \(_any2mpq($$x) // (goto &nan));
1350				}
1351				else {
1352
1353				# Parse a decimal number as an exact fraction
1354				if ("$x" =~ /^([+-]?+(?=\.?[0-9])[0-9_]*+(?:\.[0-9_]++)?(?:[Ee](?:[+-]?+[0-9_]+))?)\z/) {
1355				my $frac = _str2frac(lc($1));
1356				my $q = Math::GMPq::Rmpq_init();
1357				Math::GMPq::Rmpq_set_str($q, $frac, 10);
1358				Math::GMPq::Rmpq_canonicalize($q) if (index($frac, '/') != -1);
1359				return bless \$q;
1360				}
1361
1362				my $r = __PACKAGE__->new($x);
1363				ref($$r) eq 'Math::GMPq' && return $r;
1364				bless(\_any2mpq($$r) // goto &nan);
1365				}
1366				}
1367
1368				#
1369				## CONVERSION TO FLOATING-POINT
1370				#
1371
1372				sub float ($) {
1373				my ($x) = @_;
1374
1375				bless \(
1376				ref($x) eq __PACKAGE__
1377				? ref($$x) eq 'Math::MPFR'
1378				? (return $x)
1379				: _any2mpfr($$x)
1380				: _star2mpfr($x)
1381				);
1382				}
1383
1384				#
1385				## CONVERSION TO COMPLEX
1386				#
1387
1388				sub complex ($) {
1389				my ($x) = @_;
1390				bless \(
1391				ref($x) eq __PACKAGE__
1392				? ref($$x) eq 'Math::MPC'
1393				? (return $x)
1394				: _any2mpc($$x)
1395				: _any2mpc(_star2obj($x))
1396				);
1397				}
1398
1399				#
1400				## NEGATION
1401				#
1402
1403				sub neg { # used in overloading
1404				require Math::AnyNum::neg;
1405				my ($x) = @_;
1406				bless \__neg__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
1407				}
1408
1409				#
1410				## ABSOLUTE VALUE
1411				#
1412
1413				sub abs { # used in overloading
1414				require Math::AnyNum::abs;
1415				my ($x) = @_;
1416				bless \__abs__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
1417				}
1418
1419				#
1420				## MULTIPLICATIVE INVERSE
1421				#
1422
1423				sub inv ($) {
1424				require Math::AnyNum::inv;
1425				my ($x) = @_;
1426				bless \__inv__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
1427				}
1428
1429				#
1430				## INCREMENTATION BY ONE
1431				#
1432
1433				sub inc ($) {
1434				require Math::AnyNum::inc;
1435				my ($x) = @_;
1436				bless \__inc__($$x);
1437				}
1438
1439				#
1440				## DECREMENTATION BY ONE
1441				#
1442
1443				sub dec ($) {
1444				require Math::AnyNum::dec;
1445				my ($x) = @_;
1446				bless \__dec__($$x);
1447				}
1448
1449				sub conj ($) {
1450				my ($x) = @_;
1451
1452				if (ref($x) ne __PACKAGE__) {
1453				$x = __PACKAGE__->new($x);
1454				}
1455
1456				if (ref($$x) eq 'Math::MPC') {
1457				my $r = Math::MPC::Rmpc_init2($PREC);
1458				Math::MPC::Rmpc_conj($r, $$x, $ROUND);
1459				bless \$r;
1460				}
1461				else {
1462				$x;
1463				}
1464				}
1465
1466				sub real ($) {
1467				my ($x) = @_;
1468
1469				if (ref($x) ne __PACKAGE__) {
1470				$x = __PACKAGE__->new($x);
1471				}
1472
1473				if (ref($$x) eq 'Math::MPC') {
1474				my $r = Math::MPFR::Rmpfr_init2($PREC);
1475				Math::MPC::RMPC_RE($r, $$x);
1476				bless \$r;
1477				}
1478				else {
1479				$x;
1480				}
1481				}
1482
1483				sub imag ($) {
1484				my ($x) = @_;
1485
1486				if (ref($x) ne __PACKAGE__) {
1487				$x = __PACKAGE__->new($x);
1488				}
1489
1490				if (ref($$x) eq 'Math::MPC') {
1491				my $r = Math::MPFR::Rmpfr_init2($PREC);
1492				Math::MPC::RMPC_IM($r, $$x);
1493				bless \$r;
1494				}
1495				else {
1496				goto &zero;
1497				}
1498				}
1499
1500				sub reals ($) {
1501				my ($x) = @_;
1502
1503				if (ref($x) ne __PACKAGE__) {
1504				$x = __PACKAGE__->new($x);
1505				}
1506
1507				($x->real, $x->imag);
1508				}
1509
1510				#
1511				## ADDITION
1512				#
1513
1514				sub add { # used in overloading
1515				require Math::AnyNum::add;
1516				my ($x, $y) = @_;
1517
1518				if (ref($y) eq __PACKAGE__) {
1519				return bless \__add__($$x, $$y);
1520				}
1521
1522				$x = $$x;
1523
1524				if (!ref($y)) {
1525				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1526				if (ref($x) eq 'Math::GMPq') {
1527				my $r = Math::GMPq::Rmpq_init();
1528				$y < 0
1529				? Math::GMPq::Rmpq_set_si($r, $y, 1)
1530				: Math::GMPq::Rmpq_set_ui($r, $y, 1);
1531				Math::GMPq::Rmpq_add($r, $r, $x);
1532				return bless \$r;
1533				}
1534
1535				return bless \__add__($x, $y);
1536				}
1537
1538				return bless \__add__($x, _str2obj($y));
1539				}
1540
1541				bless \__add__($x, _star2obj($y));
1542				}
1543
1544				#
1545				## SUBTRACTION
1546				#
1547
1548				sub sub { # used in overloading
1549				require Math::AnyNum::sub;
1550				my ($x, $y) = @_;
1551
1552				$x =
1553				ref($x) eq __PACKAGE__ ? $$x
1554				: ref($x) ? _star2obj($x)
1555				: _str2obj($x);
1556
1557				if (ref($y) eq __PACKAGE__) {
1558				return bless \__sub__($x, $$y);
1559				}
1560
1561				if (!ref($y)) {
1562				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1563				if (ref($x) eq 'Math::GMPq') {
1564				my $r = Math::GMPq::Rmpq_init();
1565				$y < 0
1566				? Math::GMPq::Rmpq_set_si($r, $y, 1)
1567				: Math::GMPq::Rmpq_set_ui($r, $y, 1);
1568				Math::GMPq::Rmpq_sub($r, $x, $r);
1569				return bless \$r;
1570				}
1571
1572				return bless \__sub__($x, $y);
1573				}
1574
1575				return bless \__sub__($x, _str2obj($y));
1576				}
1577
1578				bless \__sub__($x, _star2obj($y));
1579				}
1580
1581				#
1582				## MULTIPLICATION
1583				#
1584
1585				sub mul { # used in overloading
1586				require Math::AnyNum::mul;
1587				my ($x, $y) = @_;
1588
1589				if (ref($y) eq __PACKAGE__) {
1590				return bless \__mul__($$x, $$y);
1591				}
1592
1593				$x = $$x;
1594
1595				if (!ref($y)) {
1596				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1597				if (ref($x) eq 'Math::GMPq') {
1598				my $r = Math::GMPq::Rmpq_init();
1599				$y < 0
1600				? Math::GMPq::Rmpq_set_si($r, $y, 1)
1601				: Math::GMPq::Rmpq_set_ui($r, $y, 1);
1602				Math::GMPq::Rmpq_mul($r, $r, $x);
1603				return bless \$r;
1604				}
1605
1606				return bless \__mul__($x, $y);
1607				}
1608
1609				return bless \__mul__($x, _str2obj($y));
1610				}
1611
1612				bless \__mul__($x, _star2obj($y));
1613				}
1614
1615				#
1616				## DIVISION
1617				#
1618
1619				sub div { # used in overloading
1620				require Math::AnyNum::div;
1621				my ($x, $y) = @_;
1622
1623				$x =
1624				ref($x) eq __PACKAGE__ ? $$x
1625				: ref($x) ? _star2obj($x)
1626				: _str2obj($x);
1627
1628				if (ref($y) eq __PACKAGE__) {
1629				return bless \__div__($x, $$y);
1630				}
1631
1632				if (!ref($y)) {
1633				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN and CORE::int($y)) {
1634				if (ref($x) eq 'Math::GMPq') {
1635				my $r = Math::GMPq::Rmpq_init();
1636				$y < 0
1637				? Math::GMPq::Rmpq_set_si($r, -1, -$y)
1638				: Math::GMPq::Rmpq_set_ui($r, 1, $y);
1639				Math::GMPq::Rmpq_mul($r, $r, $x);
1640				return bless \$r;
1641				}
1642
1643				return bless \__div__($x, $y);
1644				}
1645
1646				return bless \__div__($x, _str2obj($y));
1647				}
1648
1649				bless \__div__($x, _star2obj($y));
1650				}
1651
1652				#
1653				## INTEGER ADDITION
1654				#
1655
1656				sub iadd ($$) {
1657				my ($x, $y) = @_;
1658
1659				if (!ref($x) and ref($y)) {
1660				($x, $y) = ($y, $x);
1661				}
1662
1663				$x = _star2mpz($x) // goto &nan;
1664
1665				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1666				my $r = Math::GMPz::Rmpz_init();
1667				$y < 0
1668				? Math::GMPz::Rmpz_sub_ui($r, $x, -$y)
1669				: Math::GMPz::Rmpz_add_ui($r, $x, $y);
1670				return bless \$r;
1671				}
1672
1673				$y = _star2mpz($y) // goto &nan;
1674
1675				my $r = Math::GMPz::Rmpz_init();
1676				Math::GMPz::Rmpz_add($r, $x, $y);
1677				bless \$r;
1678				}
1679
1680				#
1681				## INTEGER SUBTRACTION
1682				#
1683
1684				sub isub ($$) {
1685				my ($x, $y) = @_;
1686
1687				$x = _star2mpz($x) // goto &nan;
1688
1689				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1690				my $r = Math::GMPz::Rmpz_init();
1691				$y < 0
1692				? Math::GMPz::Rmpz_add_ui($r, $x, -$y)
1693				: Math::GMPz::Rmpz_sub_ui($r, $x, $y);
1694				return bless \$r;
1695				}
1696
1697				$y = _star2mpz($y) // goto &nan;
1698
1699				my $r = Math::GMPz::Rmpz_init();
1700				Math::GMPz::Rmpz_sub($r, $x, $y);
1701				bless \$r;
1702				}
1703
1704				#
1705				## INTEGER MULTIPLICATION
1706				#
1707
1708				sub imul ($$) {
1709				my ($x, $y) = @_;
1710
1711				if (!ref($x) and ref($y)) {
1712				($x, $y) = ($y, $x);
1713				}
1714
1715				$x = _star2mpz($x) // goto &nan;
1716
1717				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1718				my $r = Math::GMPz::Rmpz_init();
1719				Math::GMPz::Rmpz_mul_ui($r, $x, CORE::abs($y));
1720				Math::GMPz::Rmpz_neg($r, $r) if $y < 0;
1721				return bless \$r;
1722				}
1723
1724				$y = _star2mpz($y) // goto &nan;
1725
1726				my $r = Math::GMPz::Rmpz_init();
1727				Math::GMPz::Rmpz_mul($r, $x, $y);
1728				bless \$r;
1729				}
1730
1731				#
1732				## INTEGER DIVISION
1733				#
1734
1735				sub idiv ($$) {
1736				my ($x, $y) = @_;
1737
1738				$x = _star2mpz($x) // goto &nan;
1739
1740				if (!ref($y) and CORE::int($y) eq $y and CORE::int($y) and CORE::abs($y) <= ULONG_MAX) {
1741				my $r = Math::GMPz::Rmpz_init();
1742				Math::GMPz::Rmpz_tdiv_q_ui($r, $x, CORE::abs($y));
1743				Math::GMPz::Rmpz_neg($r, $r) if $y < 0;
1744				return bless \$r;
1745				}
1746
1747				$y = _star2mpz($y) // goto &nan;
1748
1749				# Detect division by zero
1750				Math::GMPz::Rmpz_sgn($y) \|\| do {
1751				my $sign = Math::GMPz::Rmpz_sgn($x);
1752
1753				if ($sign == 0) { # 0/0
1754				goto &nan;
1755				}
1756				elsif ($sign > 0) { # x/0 where: x > 0
1757				goto &inf;
1758				}
1759				else { # x/0 where: x < 0
1760				goto &ninf;
1761				}
1762				};
1763
1764				my $r = Math::GMPz::Rmpz_init();
1765				Math::GMPz::Rmpz_tdiv_q($r, $x, $y);
1766				bless \$r;
1767				}
1768
1769				#
1770				## POWER
1771				#
1772
1773				sub pow ($$) {
1774				require Math::AnyNum::pow;
1775				my ($x, $y) = @_;
1776
1777				$x =
1778				ref($x) eq __PACKAGE__ ? $$x
1779				: ref($x) ? _star2obj($x)
1780				: _str2obj($x);
1781
1782				if (ref($y) eq __PACKAGE__) {
1783				return bless \__pow__($x, $$y);
1784				}
1785
1786				if (!ref($y)) {
1787				if (CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
1788				return bless \__pow__($x, $y);
1789				}
1790
1791				return bless \__pow__($x, _str2obj($y));
1792				}
1793
1794				bless \__pow__($x, _star2obj($y));
1795				}
1796
1797				#
1798				## INTEGER POWER
1799				#
1800
1801				sub ipow ($$) {
1802				my ($x, $y) = @_;
1803
1804				# Both `x` and `y` are strings
1805				if ( !ref($x)
1806				and !ref($y)
1807				and CORE::int($x) eq $x
1808				and $x >= 0
1809				and $x <= ULONG_MAX
1810				and CORE::int($y) eq $y
1811				and $y >= 0
1812				and $y <= ULONG_MAX) {
1813
1814				my $r = Math::GMPz::Rmpz_init();
1815				Math::GMPz::Rmpz_ui_pow_ui($r, $x, $y);
1816				return bless \$r;
1817				}
1818
1819				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
1820
1821				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1822				## `y` is already a native integer
1823				}
1824				else {
1825				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // (goto &nan);
1826				}
1827
1828				my $r = Math::GMPz::Rmpz_init();
1829				Math::GMPz::Rmpz_pow_ui($r, $x, CORE::abs($y));
1830
1831				if ($y < 0) {
1832				Math::GMPz::Rmpz_sgn($r) \|\| goto &inf; # 0^(-y) = Inf
1833				state $ONE_Z = Math::GMPz::Rmpz_init_set_ui_nobless(1);
1834				Math::GMPz::Rmpz_tdiv_q($r, $ONE_Z, $r);
1835				}
1836
1837				bless \$r;
1838				}
1839
1840				#
1841				## IPOW2
1842				#
1843
1844				sub ipow2 ($) {
1845				my ($n) = @_;
1846
1847				if (ref($n) eq __PACKAGE__) {
1848				$n = _any2si($$n) // goto &nan;
1849				}
1850				elsif ( !ref($n)
1851				and CORE::int($n) eq $n
1852				and $n >= LONG_MIN
1853				and $n <= ULONG_MAX) {
1854				## `n` is already a native integer
1855				}
1856				else {
1857				$n = _any2si(_star2obj($n)) // goto &nan;
1858				}
1859
1860				goto &zero if $n < 0;
1861				state $one = Math::GMPz::Rmpz_init_set_ui_nobless(1);
1862
1863				my $r = Math::GMPz::Rmpz_init();
1864				Math::GMPz::Rmpz_mul_2exp($r, $one, $n);
1865				bless \$r;
1866				}
1867
1868				#
1869				## IPOW10
1870				#
1871
1872				sub ipow10 ($) {
1873				my ($n) = @_;
1874
1875				if (ref($n) eq __PACKAGE__) {
1876				$n = _any2si($$n) // goto &nan;
1877				}
1878				elsif ( !ref($n)
1879				and CORE::int($n) eq $n
1880				and $n >= LONG_MIN
1881				and $n <= ULONG_MAX) {
1882				## $n is a native integer
1883				}
1884				else {
1885				$n = _any2si(_star2obj($n)) // goto &nan;
1886				}
1887
1888				goto &zero if $n < 0;
1889
1890				my $r = Math::GMPz::Rmpz_init();
1891				Math::GMPz::Rmpz_ui_pow_ui($r, 10, $n);
1892				bless \$r;
1893				}
1894
1895				#
1896				## ROOT
1897				#
1898
1899				sub root ($$) {
1900				require Math::AnyNum::pow;
1901				require Math::AnyNum::inv;
1902				my ($x, $y) = @_;
1903
1904				$x =
1905				ref($x) eq __PACKAGE__ ? $$x
1906				: ref($x) ? _star2obj($x)
1907				: _str2obj($x);
1908
1909				$y =
1910				ref($y) eq __PACKAGE__ ? $$y
1911				: ref($y) ? _star2obj($y)
1912				: _str2obj($y);
1913
1914				bless \__pow__($x, __inv__($y));
1915				}
1916
1917				#
1918				## Polygonal root
1919				#
1920
1921				sub polygonal_root ($$) {
1922				require Math::AnyNum::polygonal_root;
1923				bless \__polygonal_root__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
1924				}
1925
1926				#
1927				## Second polygonal root
1928				#
1929
1930				sub polygonal_root2 ($$) {
1931				require Math::AnyNum::polygonal_root;
1932				bless \__polygonal_root__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]), 1);
1933				}
1934
1935				#
1936				## isqrt
1937				#
1938
1939				sub isqrt ($) {
1940				my ($x) = @_;
1941
1942				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
1943
1944				Math::GMPz::Rmpz_sgn($x) < 0 and goto &nan;
1945				my $r = Math::GMPz::Rmpz_init();
1946				Math::GMPz::Rmpz_sqrt($r, $x);
1947				bless \$r;
1948				}
1949
1950				#
1951				## icbrt
1952				#
1953
1954				sub icbrt ($) {
1955				my ($x) = @_;
1956				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
1957				my $r = Math::GMPz::Rmpz_init();
1958				Math::GMPz::Rmpz_root($r, $x, 3);
1959				bless \$r;
1960				}
1961
1962				#
1963				## IROOT
1964				#
1965
1966				sub iroot ($$) {
1967				my ($x, $y) = @_;
1968
1969				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
1970
1971				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
1972				## `y`is native integer
1973				}
1974				elsif (ref($y) eq __PACKAGE__) {
1975				$y = _any2si($$y) // goto &nan;
1976				}
1977				else {
1978				$y = _any2si(_star2obj($y)) // goto &nan;
1979				}
1980
1981				if ($y == 0) {
1982				Math::GMPz::Rmpz_sgn($x) \|\| goto &zero; # 0^Inf = 0
1983
1984				# 1^Inf = 1 ; (-1)^Inf = 1
1985				if (Math::GMPz::Rmpz_cmpabs_ui($x, 1) == 0) {
1986				goto &one;
1987				}
1988
1989				goto &inf;
1990				}
1991
1992				if ($y < 0) {
1993				my $sign = Math::GMPz::Rmpz_sgn($x)
1994				\|\| goto &inf; # 1 / 0^k = Inf
1995
1996				if ($sign < 0) {
1997				goto &nan;
1998				}
1999
2000				if (Math::GMPz::Rmpz_cmp_ui($x, 1) == 0) { # 1 / 1^k = 1
2001				goto &one;
2002				}
2003
2004				goto &zero;
2005				}
2006
2007				if ($y % 2 == 0 and Math::GMPz::Rmpz_sgn($x) < 0) {
2008				goto &nan;
2009				}
2010
2011				my $r = Math::GMPz::Rmpz_init();
2012				$y == 2
2013				? Math::GMPz::Rmpz_sqrt($r, $x)
2014				: Math::GMPz::Rmpz_root($r, $x, $y);
2015				bless \$r;
2016				}
2017
2018				#
2019				## ISQRTREM
2020				#
2021
2022				sub isqrtrem ($) {
2023				my ($x) = @_;
2024
2025				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // return (nan(), nan());
2026
2027				Math::GMPz::Rmpz_sgn($x) < 0
2028				and return (nan(), nan());
2029
2030				my $r = Math::GMPz::Rmpz_init();
2031				my $s = Math::GMPz::Rmpz_init();
2032
2033				Math::GMPz::Rmpz_sqrtrem($r, $s, $x);
2034
2035				((bless \$r), (bless \$s));
2036				}
2037
2038				#
2039				## IROOTREM
2040				#
2041
2042				sub irootrem ($$) {
2043				my ($x, $y) = @_;
2044
2045				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // return (nan(), nan());
2046
2047				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
2048				## `y` is a native integer
2049				}
2050				else {
2051				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // (return (nan(), nan()));
2052				}
2053
2054				if ($y == 0) {
2055
2056				# 0^Inf = 0
2057				if (Math::GMPz::Rmpz_sgn($x) == 0) {
2058				return (zero(), mone());
2059				}
2060
2061				my $r = Math::GMPz::Rmpz_init();
2062				Math::GMPz::Rmpz_sub_ui($r, $x, 1);
2063
2064				# 1^Inf = 1 ; (-1)^Inf = 1
2065				if (Math::GMPz::Rmpz_cmpabs_ui($x, 1) == 0) {
2066				return (one(), (bless \$r));
2067				}
2068
2069				return (inf(), (bless \$r));
2070				}
2071
2072				if ($y < 0) {
2073				my $sgn = Math::GMPz::Rmpz_sgn($x);
2074
2075				# 1 / 0^k = Inf
2076				if ($sgn == 0) {
2077				return (inf(), zero());
2078				}
2079
2080				# 1 / 1^k = 1
2081				if (Math::GMPz::Rmpz_cmp_ui($x, 1) == 0) {
2082				return (one(), zero());
2083				}
2084
2085				# x is negative
2086				if ($sgn < 0) {
2087				return (nan(), nan());
2088				}
2089
2090				return (zero(), ninf());
2091				}
2092
2093				if ($y % 2 == 0 and Math::GMPz::Rmpz_sgn($x) < 0) {
2094				return (nan(), nan());
2095				}
2096
2097				my $r = Math::GMPz::Rmpz_init();
2098				my $s = Math::GMPz::Rmpz_init();
2099
2100				$y == 2
2101				? Math::GMPz::Rmpz_sqrtrem($r, $s, $x)
2102				: Math::GMPz::Rmpz_rootrem($r, $s, $x, $y);
2103
2104				((bless \$r), (bless \$s));
2105				}
2106
2107				#
2108				## MOD
2109				#
2110
2111				sub mod ($$) {
2112				require Math::AnyNum::mod;
2113				my ($x, $y) = @_;
2114
2115				$x =
2116				ref($x) eq __PACKAGE__ ? $$x
2117				: ref($x) ? _star2obj($x)
2118				: _str2obj($x);
2119
2120				if (ref($y) eq __PACKAGE__) {
2121				return bless \__mod__($x, $$y);
2122				}
2123
2124				if (!ref($y)) {
2125
2126				if ( ref($x) ne 'Math::GMPq'
2127				and CORE::int($y) eq $y
2128				and $y > 0
2129				and $y <= ULONG_MAX) {
2130				return bless \__mod__($x, $y);
2131				}
2132
2133				return bless \__mod__($x, _str2obj($y));
2134				}
2135
2136				bless \__mod__($x, _star2obj($y));
2137				}
2138
2139				#
2140				## IMOD
2141				#
2142
2143				sub imod ($$) {
2144				my ($x, $y) = @_;
2145
2146				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
2147
2148				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
2149
2150				CORE::int($y) \|\| goto &nan;
2151
2152				my $neg_y = $y < 0;
2153				$y = -$y if $neg_y;
2154
2155				my $r = Math::GMPz::Rmpz_init();
2156				Math::GMPz::Rmpz_mod_ui($r, $x, $y);
2157
2158				if (!Math::GMPz::Rmpz_sgn($r)) {
2159				## OK
2160				}
2161				elsif ($neg_y) {
2162				Math::GMPz::Rmpz_sub_ui($r, $r, $y);
2163				}
2164
2165				return bless \$r;
2166				}
2167
2168				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
2169
2170				my $sign_y = Math::GMPz::Rmpz_sgn($y) \|\| goto &nan;
2171
2172				my $r = Math::GMPz::Rmpz_init();
2173				Math::GMPz::Rmpz_mod($r, $x, $y);
2174
2175				if (!Math::GMPz::Rmpz_sgn($r)) {
2176				## OK
2177				}
2178				elsif ($sign_y < 0) {
2179				Math::GMPz::Rmpz_add($r, $r, $y);
2180				}
2181
2182				bless \$r;
2183				}
2184
2185				#
2186				## DIVMOD
2187				#
2188
2189				sub divmod ($$) {
2190				my ($x, $y) = @_;
2191
2192				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // return (nan(), nan());
2193				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // return (nan(), nan());
2194
2195				Math::GMPz::Rmpz_sgn($y)
2196				\|\| return (nan(), nan());
2197
2198				my $r = Math::GMPz::Rmpz_init();
2199				my $s = Math::GMPz::Rmpz_init();
2200
2201				Math::GMPz::Rmpz_divmod($r, $s, $x, $y);
2202
2203				((bless \$r), (bless \$s));
2204				}
2205
2206				#
2207				## is_div
2208				#
2209
2210				sub is_div ($$) {
2211				require Math::AnyNum::eq;
2212				(@_) = (${mod($_[0], $_[1])}, 0);
2213				goto &__eq__;
2214				}
2215
2216				#
2217				## SPECIAL
2218				#
2219
2220				sub ln { # used in overloading
2221				require Math::AnyNum::log;
2222				bless \__log__(_star2mpfr_mpc($_[0]));
2223				}
2224
2225				sub log2 ($) {
2226				require Math::AnyNum::log;
2227				bless \__log2__(_star2mpfr_mpc($_[0]));
2228				}
2229
2230				sub log10 ($) {
2231				require Math::AnyNum::log;
2232				bless \__log10__(_star2mpfr_mpc($_[0]));
2233				}
2234
2235				sub length ($) {
2236				my ($z) = _star2mpz($_[0]) // return -1;
2237				CORE::length(Math::GMPz::Rmpz_get_str($z, 10)) - (Math::GMPz::Rmpz_sgn($z) < 0 ? 1 : 0);
2238				}
2239
2240				sub log (_;$) {
2241				require Math::AnyNum::log;
2242				my ($x, $y) = @_;
2243
2244				if (!defined($y)) {
2245				return bless \__log__(_star2mpfr_mpc($x));
2246				}
2247
2248				require Math::AnyNum::div;
2249				bless \__div__(__log__(_star2mpfr_mpc($x)), __log__(_star2mpfr_mpc($y)));
2250				}
2251
2252				#
2253				## ILOG
2254				#
2255
2256				sub ilog2 ($) {
2257				require Math::AnyNum::ilog;
2258				state $two = Math::GMPz::Rmpz_init_set_ui(2);
2259				bless \__ilog__((_star2mpz($_[0]) // goto &nan), $two);
2260				}
2261
2262				sub ilog10 ($) {
2263				require Math::AnyNum::ilog;
2264				state $ten = Math::GMPz::Rmpz_init_set_ui(10);
2265				bless \__ilog__((_star2mpz($_[0]) // goto &nan), $ten);
2266				}
2267
2268				sub ilog ($;$) {
2269				my ($x, $y) = @_;
2270
2271				if (!defined($y)) {
2272				require Math::AnyNum::log;
2273				return bless \(_any2mpz(__log__(_star2mpfr_mpc($x))) // goto &nan);
2274				}
2275
2276				require Math::AnyNum::ilog;
2277				bless \__ilog__((_star2mpz($x) // goto &nan), (_star2mpz($y) // goto &nan));
2278				}
2279
2280				#
2281				## SQRT
2282				#
2283
2284				sub sqrt { # used in overloading
2285				require Math::AnyNum::sqrt;
2286				bless \__sqrt__(_star2mpfr_mpc($_[0]));
2287				}
2288
2289				sub cbrt ($) {
2290				require Math::AnyNum::cbrt;
2291				bless \__cbrt__(_star2mpfr_mpc($_[0]));
2292				}
2293
2294				sub sqr ($) {
2295				require Math::AnyNum::mul;
2296				my ($x) = @_;
2297
2298				$x =
2299				ref($x) eq __PACKAGE__
2300				? $$x
2301				: _star2obj($x);
2302
2303				bless \__mul__($x, $x);
2304				}
2305
2306				sub norm ($) {
2307				require Math::AnyNum::norm;
2308				my ($x) = @_;
2309
2310				$x =
2311				ref($x) eq __PACKAGE__
2312				? $$x
2313				: _star2obj($x);
2314
2315				bless \__norm__($x);
2316				}
2317
2318				sub exp { # used in overloading
2319				require Math::AnyNum::exp;
2320				bless \__exp__(_star2mpfr_mpc($_[0]));
2321				}
2322
2323				sub exp2 ($) {
2324				require Math::AnyNum::pow;
2325				my ($x) = @_;
2326
2327				state $base = Math::GMPz::Rmpz_init_set_ui(2);
2328
2329				if (ref($x) eq __PACKAGE__) {
2330				bless \__pow__($base, $$x);
2331				}
2332				elsif (!ref($x) and CORE::int($x) eq $x and $x >= LONG_MIN and $x <= ULONG_MAX) {
2333				bless \__pow__($base, $x);
2334				}
2335				else {
2336				bless \__pow__($base, _star2obj($x));
2337				}
2338				}
2339
2340				sub exp10 ($) {
2341				require Math::AnyNum::pow;
2342				my ($x) = @_;
2343
2344				state $base = Math::GMPz::Rmpz_init_set_ui(10);
2345
2346				if (ref($x) eq __PACKAGE__) {
2347				bless \__pow__($base, $$x);
2348				}
2349				elsif (!ref($x) and CORE::int($x) eq $x and $x >= LONG_MIN and $x <= ULONG_MAX) {
2350				bless \__pow__($base, $x);
2351				}
2352				else {
2353				bless \__pow__($base, _star2obj($x));
2354				}
2355				}
2356
2357				sub floor ($) {
2358				require Math::AnyNum::floor;
2359				my ($x) = @_;
2360
2361				if (ref($x) ne __PACKAGE__) {
2362				$x = __PACKAGE__->new($x);
2363				}
2364
2365				ref($$x) eq 'Math::GMPz' and return $x; # already an integer
2366				bless \__floor__($$x);
2367				}
2368
2369				sub ceil ($) {
2370				require Math::AnyNum::ceil;
2371				my ($x) = @_;
2372
2373				if (ref($x) ne __PACKAGE__) {
2374				$x = __PACKAGE__->new($x);
2375				}
2376
2377				ref($$x) eq 'Math::GMPz' and return $x; # already an integer
2378				bless \__ceil__($$x);
2379				}
2380
2381				#
2382				## sin / sinh / asin / asinh
2383				#
2384
2385				sub sin { # used in overloading
2386				require Math::AnyNum::sin;
2387				bless \__sin__(_star2mpfr_mpc($_[0]));
2388				}
2389
2390				sub sinh ($) {
2391				require Math::AnyNum::sinh;
2392				bless \__sinh__(_star2mpfr_mpc($_[0]));
2393				}
2394
2395				sub asin ($) {
2396				require Math::AnyNum::asin;
2397				bless \__asin__(_star2mpfr_mpc($_[0]));
2398				}
2399
2400				sub asinh ($) {
2401				require Math::AnyNum::asinh;
2402				bless \__asinh__(_star2mpfr_mpc($_[0]));
2403				}
2404
2405				#
2406				## cos / cosh / acos / acosh
2407				#
2408
2409				sub cos { # used in overloading
2410				require Math::AnyNum::cos;
2411				bless \__cos__(_star2mpfr_mpc($_[0]));
2412				}
2413
2414				sub cosh ($) {
2415				require Math::AnyNum::cosh;
2416				bless \__cosh__(_star2mpfr_mpc($_[0]));
2417				}
2418
2419				sub acos ($) {
2420				require Math::AnyNum::acos;
2421				bless \__acos__(_star2mpfr_mpc($_[0]));
2422				}
2423
2424				sub acosh ($) {
2425				require Math::AnyNum::acosh;
2426				bless \__acosh__(_star2mpfr_mpc($_[0]));
2427				}
2428
2429				#
2430				## tan / tanh / atan / atanh
2431				#
2432
2433				sub tan ($) {
2434				require Math::AnyNum::tan;
2435				bless \__tan__(_star2mpfr_mpc($_[0]));
2436				}
2437
2438				sub tanh ($) {
2439				require Math::AnyNum::tanh;
2440				bless \__tanh__(_star2mpfr_mpc($_[0]));
2441				}
2442
2443				sub atan ($) {
2444				require Math::AnyNum::atan;
2445				bless \__atan__(_star2mpfr_mpc($_[0]));
2446				}
2447
2448				sub atanh ($) {
2449				require Math::AnyNum::atanh;
2450				bless \__atanh__(_star2mpfr_mpc($_[0]));
2451				}
2452
2453				sub atan2 ($$) {
2454				require Math::AnyNum::atan2;
2455				bless \__atan2__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
2456				}
2457
2458				#
2459				## sec / sech / asec / asech
2460				#
2461
2462				sub sec ($) {
2463				require Math::AnyNum::sec;
2464				bless \__sec__(_star2mpfr_mpc($_[0]));
2465				}
2466
2467				sub sech ($) {
2468				require Math::AnyNum::sech;
2469				bless \__sech__(_star2mpfr_mpc($_[0]));
2470				}
2471
2472				sub asec ($) {
2473				require Math::AnyNum::asec;
2474				bless \__asec__(_star2mpfr_mpc($_[0]));
2475				}
2476
2477				sub asech ($) {
2478				require Math::AnyNum::asech;
2479				bless \__asech__(_star2mpfr_mpc($_[0]));
2480				}
2481
2482				#
2483				## csc / csch / acsc / acsch
2484				#
2485
2486				sub csc ($) {
2487				require Math::AnyNum::csc;
2488				bless \__csc__(_star2mpfr_mpc($_[0]));
2489				}
2490
2491				sub csch ($) {
2492				require Math::AnyNum::csch;
2493				bless \__csch__(_star2mpfr_mpc($_[0]));
2494				}
2495
2496				sub acsc ($) {
2497				require Math::AnyNum::acsc;
2498				bless \__acsc__(_star2mpfr_mpc($_[0]));
2499				}
2500
2501				sub acsch ($) {
2502				require Math::AnyNum::acsch;
2503				bless \__acsch__(_star2mpfr_mpc($_[0]));
2504				}
2505
2506				#
2507				## cot / coth / acot / acoth
2508				#
2509
2510				sub cot ($) {
2511				require Math::AnyNum::cot;
2512				bless \__cot__(_star2mpfr_mpc($_[0]));
2513				}
2514
2515				sub coth ($) {
2516				require Math::AnyNum::coth;
2517				bless \__coth__(_star2mpfr_mpc($_[0]));
2518				}
2519
2520				sub acot ($) {
2521				require Math::AnyNum::acot;
2522				bless \__acot__(_star2mpfr_mpc($_[0]));
2523				}
2524
2525				sub acoth ($) {
2526				require Math::AnyNum::acoth;
2527				bless \__acoth__(_star2mpfr_mpc($_[0]));
2528				}
2529
2530				sub deg2rad ($) {
2531				require Math::AnyNum::mul;
2532				my ($x) = @_;
2533				my $f = Math::MPFR::Rmpfr_init2($PREC);
2534				Math::MPFR::Rmpfr_const_pi($f, $ROUND);
2535				Math::MPFR::Rmpfr_div_ui($f, $f, 180, $ROUND);
2536				bless \__mul__(_star2mpfr_mpc($x), $f);
2537				}
2538
2539				sub rad2deg ($) {
2540				require Math::AnyNum::mul;
2541				my ($x) = @_;
2542				my $f = Math::MPFR::Rmpfr_init2($PREC);
2543				Math::MPFR::Rmpfr_const_pi($f, $ROUND);
2544				Math::MPFR::Rmpfr_ui_div($f, 180, $f, $ROUND);
2545				bless \__mul__(_star2mpfr_mpc($x), $f);
2546				}
2547
2548				#
2549				## gamma
2550				#
2551
2552				sub gamma ($) {
2553				my $r = Math::MPFR::Rmpfr_init2($PREC);
2554				Math::MPFR::Rmpfr_gamma($r, _star2mpfr($_[0]), $ROUND);
2555				bless \$r;
2556				}
2557
2558				#
2559				## lgamma
2560				#
2561
2562				sub lgamma ($) {
2563				my $r = Math::MPFR::Rmpfr_init2($PREC);
2564				Math::MPFR::Rmpfr_lgamma($r, _star2mpfr($_[0]), $ROUND);
2565				bless \$r;
2566				}
2567
2568				#
2569				## lngamma
2570				#
2571
2572				sub lngamma ($) {
2573				my $r = Math::MPFR::Rmpfr_init2($PREC);
2574				Math::MPFR::Rmpfr_lngamma($r, _star2mpfr($_[0]), $ROUND);
2575				bless \$r;
2576				}
2577
2578				#
2579				## digamma
2580				#
2581
2582				sub digamma ($) {
2583				my $r = Math::MPFR::Rmpfr_init2($PREC);
2584				Math::MPFR::Rmpfr_digamma($r, _star2mpfr($_[0]), $ROUND);
2585				bless \$r;
2586				}
2587
2588				#
2589				## zeta
2590				#
2591
2592				sub zeta ($) {
2593				my ($x) = @_;
2594
2595				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2596				## $x is an unsigned integer
2597				}
2598				else {
2599				$x = _star2mpfr($x);
2600
2601				# If $x fits inside an unsigned integer, then unpack it.
2602				if ( Math::MPFR::Rmpfr_integer_p($x)
2603				and Math::MPFR::Rmpfr_fits_ulong_p($x, $ROUND)) {
2604				$x = Math::MPFR::Rmpfr_get_ui($x, $ROUND);
2605				}
2606				}
2607
2608				my $r = Math::MPFR::Rmpfr_init2($PREC);
2609
2610				ref($x)
2611				? Math::MPFR::Rmpfr_zeta($r, $x, $ROUND)
2612				: Math::MPFR::Rmpfr_zeta_ui($r, $x, $ROUND);
2613
2614				bless \$r;
2615				}
2616
2617				#
2618				## eta
2619				#
2620
2621				sub eta ($) {
2622				require Math::AnyNum::eta;
2623				bless \__eta__(_star2mpfr($_[0]));
2624				}
2625
2626				#
2627				## beta
2628				#
2629				sub beta ($$) {
2630				require Math::AnyNum::beta;
2631				bless \__beta__(_star2mpfr($_[0]), _star2mpfr($_[1]));
2632				}
2633
2634				#
2635				## Airy function (Ai)
2636				#
2637
2638				sub Ai ($) {
2639				my $r = Math::MPFR::Rmpfr_init2($PREC);
2640				Math::MPFR::Rmpfr_ai($r, _star2mpfr($_[0]), $ROUND);
2641				bless \$r;
2642				}
2643
2644				#
2645				## Exponential integral (Ei)
2646				#
2647
2648				sub Ei ($) {
2649				my $r = Math::MPFR::Rmpfr_init2($PREC);
2650				Math::MPFR::Rmpfr_eint($r, _star2mpfr($_[0]), $ROUND);
2651				bless \$r;
2652				}
2653
2654				#
2655				## Logarithmic integral (Li)
2656				#
2657				sub Li ($) {
2658				my $r = Math::MPFR::Rmpfr_init2($PREC);
2659				Math::MPFR::Rmpfr_log($r, _star2mpfr($_[0]), $ROUND);
2660				Math::MPFR::Rmpfr_eint($r, $r, $ROUND);
2661				bless \$r;
2662				}
2663
2664				#
2665				## Dilogarithm function (Li_2)
2666				#
2667				sub Li2 ($) {
2668				my $r = Math::MPFR::Rmpfr_init2($PREC);
2669				Math::MPFR::Rmpfr_li2($r, _star2mpfr($_[0]), $ROUND);
2670				bless \$r;
2671				}
2672
2673				#
2674				## Error function
2675				#
2676				sub erf ($) {
2677				my $r = Math::MPFR::Rmpfr_init2($PREC);
2678				Math::MPFR::Rmpfr_erf($r, _star2mpfr($_[0]), $ROUND);
2679				bless \$r;
2680				}
2681
2682				#
2683				## Complementary error function
2684				#
2685				sub erfc ($) {
2686				my $r = Math::MPFR::Rmpfr_init2($PREC);
2687				Math::MPFR::Rmpfr_erfc($r, _star2mpfr($_[0]), $ROUND);
2688				bless \$r;
2689				}
2690
2691				#
2692				## Lambert W
2693				#
2694
2695				sub LambertW ($) {
2696				require Math::AnyNum::LambertW;
2697				bless \__LambertW__(_star2mpfr_mpc($_[0]));
2698				}
2699
2700				#
2701				## lgrt -- logarithmic root
2702				#
2703
2704				sub lgrt ($) {
2705				require Math::AnyNum::lgrt;
2706				bless \__lgrt__(_star2mpfr_mpc($_[0]));
2707				}
2708
2709				#
2710				## agm
2711				#
2712				sub agm ($$) {
2713				require Math::AnyNum::agm;
2714				bless \__agm__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
2715				}
2716
2717				#
2718				## hypot
2719				#
2720
2721				sub hypot ($$) {
2722				require Math::AnyNum::hypot;
2723				bless \__hypot__(_star2mpfr_mpc($_[0]), _star2mpfr_mpc($_[1]));
2724				}
2725
2726				#
2727				## BesselJ
2728				#
2729
2730				sub BesselJ ($$) {
2731				require Math::AnyNum::BesselJ;
2732				my ($x, $y) = @_;
2733
2734				$x = ref($x) eq __PACKAGE__ ? _any2mpfr($$x) : _star2mpfr($x);
2735
2736				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
2737				return bless \__BesselJ__($x, $y);
2738				}
2739
2740				bless \__BesselJ__($x, _star2mpz($y) // (goto &nan));
2741				}
2742
2743				#
2744				## BesselY
2745				#
2746
2747				sub BesselY ($$) {
2748				require Math::AnyNum::BesselY;
2749				my ($x, $y) = @_;
2750
2751				$x = ref($x) eq __PACKAGE__ ? _any2mpfr($$x) : _star2mpfr($x);
2752
2753				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
2754				return bless \__BesselY__($x, $y);
2755				}
2756
2757				bless \__BesselY__($x, _star2mpz($y) // (goto &nan));
2758				}
2759
2760				#
2761				## ROUND
2762				#
2763
2764				sub round ($;$) {
2765				require Math::AnyNum::round;
2766				my ($x, $y) = @_;
2767
2768				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
2769
2770				if (!defined($y)) {
2771				return bless \__round__($x, 0);
2772				}
2773
2774				if (!ref($y)) {
2775				if (CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
2776				## `y` is a native integer
2777				}
2778				else {
2779				$y = _any2si(_str2obj($y)) // (goto &nan);
2780				}
2781				}
2782				elsif (ref($y) eq __PACKAGE__) {
2783				$y = _any2si($$y) // (goto &nan);
2784				}
2785				else {
2786				$y = _any2si(_star2obj($y)) // (goto &nan);
2787				}
2788
2789				bless \__round__($x, $y);
2790				}
2791
2792				#
2793				## RAND / IRAND
2794				#
2795
2796				{
2797				my $srand = srand();
2798
2799				{
2800				state $state = Math::MPFR::Rmpfr_randinit_mt_nobless();
2801				Math::MPFR::Rmpfr_randseed_ui($state, $srand);
2802
2803				sub rand (;$;$) {
2804				require Math::AnyNum::mul;
2805				my ($x, $y) = @_;
2806
2807				if (@_ == 0) {
2808				$x = one();
2809				}
2810
2811				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
2812
2813				if (!defined($y)) {
2814				my $rand = Math::MPFR::Rmpfr_init2($PREC);
2815				Math::MPFR::Rmpfr_urandom($rand, $state, $ROUND);
2816				return bless \__mul__($rand, $x);
2817				}
2818
2819				require Math::AnyNum::sub;
2820				require Math::AnyNum::add;
2821
2822				$y = ref($y) eq __PACKAGE__ ? $$y : _star2obj($y);
2823
2824				my $rand = Math::MPFR::Rmpfr_init2($PREC);
2825				Math::MPFR::Rmpfr_urandom($rand, $state, $ROUND);
2826				$rand = __mul__($rand, __sub__($y, $x));
2827				bless \__add__($rand, $x);
2828				}
2829
2830				sub seed ($) {
2831				my $z = _star2mpz($_[0]) // do {
2832				require Carp;
2833				Carp::croak("seed(): invalid seed value <<$_[0]>> (expected an integer)");
2834				};
2835				Math::MPFR::Rmpfr_randseed($state, $z);
2836				bless \$z;
2837				}
2838				}
2839
2840				{
2841				state $state = Math::GMPz::zgmp_randinit_mt_nobless();
2842				Math::GMPz::zgmp_randseed_ui($state, $srand);
2843
2844				sub irand ($;$) {
2845				require Math::AnyNum::irand;
2846				my ($x, $y) = @_;
2847
2848				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
2849
2850				if (!defined($y)) {
2851				return bless \__irand__($x, undef, $state);
2852				}
2853
2854				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
2855				bless \__irand__($x, $y, $state);
2856				}
2857
2858				sub iseed ($) {
2859				my $z = _star2mpz($_[0]) // do {
2860				require Carp;
2861				Carp::croak("iseed(): invalid seed value <<$_[0]>> (expected an integer)");
2862				};
2863				Math::GMPz::zgmp_randseed($state, $z);
2864				bless \$z;
2865				}
2866				}
2867				}
2868
2869				#
2870				## Fibonacci
2871				#
2872				sub fibonacci ($) {
2873				my ($x) = @_;
2874
2875				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2876				## `x` is a native unsigned integer
2877				}
2878				elsif (ref($x) eq __PACKAGE__) {
2879				$x = _any2ui($$x) // goto &nan;
2880				}
2881				else {
2882				$x = _any2ui(_star2obj($x)) // goto &nan;
2883				}
2884
2885				my $r = Math::GMPz::Rmpz_init();
2886				Math::GMPz::Rmpz_fib_ui($r, $x);
2887				bless \$r;
2888				}
2889
2890				#
2891				## Lucas
2892				#
2893				sub lucas ($) {
2894				my ($x) = @_;
2895
2896				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2897				## `x` is a native unsigned integer
2898				}
2899				elsif (ref($x) eq __PACKAGE__) {
2900				$x = _any2ui($$x) // goto &nan;
2901				}
2902				else {
2903				$x = _any2ui(_star2obj($x)) // goto &nan;
2904				}
2905
2906				my $r = Math::GMPz::Rmpz_init();
2907				Math::GMPz::Rmpz_lucnum_ui($r, $x);
2908				bless \$r;
2909				}
2910
2911				#
2912				## Primorial
2913				#
2914				sub primorial ($) {
2915				my ($x) = @_;
2916
2917				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2918				## `x` is a native unsigned integer
2919				}
2920				elsif (ref($x) eq __PACKAGE__) {
2921				$x = _any2ui($$x) // goto &nan;
2922				}
2923				else {
2924				$x = _any2ui(_star2obj($x)) // goto &nan;
2925				}
2926
2927				my $r = Math::GMPz::Rmpz_init();
2928				Math::GMPz::Rmpz_primorial_ui($r, $x);
2929				bless \$r;
2930				}
2931
2932				#
2933				## bernfrac
2934				#
2935
2936				sub bernfrac ($) {
2937				require Math::AnyNum::bernfrac;
2938				my ($x) = @_;
2939
2940				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2941				## `x` is a native unsigned integer
2942				}
2943				elsif (ref($x) eq __PACKAGE__) {
2944				$x = _any2ui($$x) // goto &nan;
2945				}
2946				else {
2947				$x = _any2ui(_star2obj($x)) // goto &nan;
2948				}
2949
2950				bless \__bernfrac__($x);
2951				}
2952
2953				#
2954				## harmfrac
2955				#
2956
2957				sub harmfrac ($) {
2958				require Math::AnyNum::harmfrac;
2959				my ($x) = @_;
2960
2961				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2962				## `x` is a native unsigned integer
2963				}
2964				elsif (ref($x) eq __PACKAGE__) {
2965				$x = _any2ui($$x) // goto &nan;
2966				}
2967				else {
2968				$x = _any2ui(_star2obj($x)) // goto &nan;
2969				}
2970
2971				bless \__harmfrac__($x);
2972				}
2973
2974				#
2975				## bernreal
2976				#
2977
2978				sub bernreal ($) {
2979				require Math::AnyNum::bernreal;
2980				my ($x) = @_;
2981
2982				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
2983				## `x` is a native unsigned integer
2984				}
2985				elsif (ref($x) eq __PACKAGE__) {
2986				$x = _any2ui($$x) // goto &nan;
2987				}
2988				else {
2989				$x = _any2ui(_star2obj($x)) // goto &nan;
2990				}
2991
2992				bless \__bernreal__($x);
2993				}
2994
2995				#
2996				## harmreal
2997				#
2998
2999				sub harmreal ($) {
3000				require Math::AnyNum::harmreal;
3001				bless \__harmreal__(_star2mpfr($_[0]) // (goto &nan));
3002				}
3003
3004				#
3005				## Factorial
3006				#
3007				sub factorial ($) {
3008				my ($x) = @_;
3009
3010				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3011				## `x` is a native unsigned integer
3012				}
3013				elsif (ref($x) eq __PACKAGE__) {
3014				$x = _any2ui($$x) // goto &nan;
3015				}
3016				else {
3017				$x = _any2ui(_star2obj($x)) // goto &nan;
3018				}
3019
3020				my $r = Math::GMPz::Rmpz_init();
3021				Math::GMPz::Rmpz_fac_ui($r, $x);
3022				bless \$r;
3023				}
3024
3025				#
3026				## Double-factorial
3027				#
3028
3029				sub dfactorial ($) {
3030				my ($x) = @_;
3031
3032				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3033				## `x` is a native unsigned integer
3034				}
3035				elsif (ref($x) eq __PACKAGE__) {
3036				$x = _any2ui($$x) // goto &nan;
3037				}
3038				else {
3039				$x = _any2ui(_star2obj($x)) // goto &nan;
3040				}
3041
3042				my $r = Math::GMPz::Rmpz_init();
3043				Math::GMPz::Rmpz_2fac_ui($r, $x);
3044				bless \$r;
3045				}
3046
3047				#
3048				## M-factorial
3049				#
3050
3051				sub mfactorial ($$) {
3052				my ($x, $y) = @_;
3053
3054				if (!ref($x) and CORE::int($x) eq $x and $x >= 0 and $x <= ULONG_MAX) {
3055				## `x` is an unsigned native integer
3056				}
3057				elsif (ref($x) eq __PACKAGE__) {
3058				$x = _any2ui($$x) // goto &nan;
3059				}
3060				else {
3061				$x = _any2ui(_star2obj($x)) // goto &nan;
3062				}
3063
3064				if (!ref($y) and CORE::int($y) eq $y and $y >= 0 and $y <= ULONG_MAX) {
3065				## `y` is an unsigned native integer
3066				}
3067				elsif (ref($y) eq __PACKAGE__) {
3068				$y = _any2ui($$y) // goto &nan;
3069				}
3070				else {
3071				$y = _any2ui(_star2obj($y)) // goto &nan;
3072				}
3073
3074				my $r = Math::GMPz::Rmpz_init();
3075				Math::GMPz::Rmpz_mfac_uiui($r, $x, $y);
3076				bless \$r;
3077				}
3078
3079				#
3080				## falling_factorial(x, +y) = binomial(x, y) * y!
3081				## falling_factorial(x, -y) = 1/falling_factorial(x + y, y)
3082				#
3083				sub falling_factorial ($$) {
3084				my ($x, $y) = @_;
3085
3086				$x = _star2mpz($x) // goto &nan;
3087
3088				if (ref($y) eq __PACKAGE__) {
3089				$y = _any2si($$y) // goto &nan;
3090				}
3091				elsif (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3092				## `y` is a native integer
3093				}
3094				else {
3095				$y = _any2si(_star2obj($y)) // goto &nan;
3096				}
3097
3098				my $r = Math::GMPz::Rmpz_init_set($x);
3099
3100				if ($y < 0) {
3101				Math::GMPz::Rmpz_add_ui($r, $r, CORE::abs($y));
3102				}
3103
3104				Math::GMPz::Rmpz_fits_ulong_p($r)
3105				? Math::GMPz::Rmpz_bin_uiui($r, Math::GMPz::Rmpz_get_ui($r), CORE::abs($y))
3106				: Math::GMPz::Rmpz_bin_ui($r, $r, CORE::abs($y));
3107
3108				Math::GMPz::Rmpz_sgn($r) \|\| do {
3109				$y < 0
3110				? (goto &nan)
3111				: (goto &zero);
3112				};
3113
3114				state $t = Math::GMPz::Rmpz_init_nobless();
3115				Math::GMPz::Rmpz_fac_ui($t, CORE::abs($y));
3116				Math::GMPz::Rmpz_mul($r, $r, $t);
3117
3118				if ($y < 0) {
3119				my $q = Math::GMPq::Rmpq_init();
3120				Math::GMPq::Rmpq_set_z($q, $r);
3121				Math::GMPq::Rmpq_inv($q, $q);
3122				return bless \$q;
3123				}
3124
3125				bless \$r;
3126				}
3127
3128				#
3129				## rising_factorial(x, +y) = binomial(x + y - 1, y) * y!
3130				## rising_factorial(x, -y) = 1/rising_factorial(x - y, y)
3131				#
3132				sub rising_factorial ($$) {
3133				my ($x, $y) = @_;
3134
3135				$x = _star2mpz($x) // goto &nan;
3136
3137				if (ref($y) eq __PACKAGE__) {
3138				$y = _any2si($$y) // goto &nan;
3139				}
3140				elsif (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3141				## `y` is a native integer
3142				}
3143				else {
3144				$y = _any2si(_star2obj($y)) // goto &nan;
3145				}
3146
3147				my $r = Math::GMPz::Rmpz_init_set($x);
3148				Math::GMPz::Rmpz_add_ui($r, $r, CORE::abs($y));
3149				Math::GMPz::Rmpz_sub_ui($r, $r, 1);
3150
3151				if ($y < 0) {
3152				Math::GMPz::Rmpz_sub_ui($r, $r, CORE::abs($y));
3153				}
3154
3155				Math::GMPz::Rmpz_fits_ulong_p($r)
3156				? Math::GMPz::Rmpz_bin_uiui($r, Math::GMPz::Rmpz_get_ui($r), CORE::abs($y))
3157				: Math::GMPz::Rmpz_bin_ui($r, $r, CORE::abs($y));
3158
3159				Math::GMPz::Rmpz_sgn($r) \|\| do {
3160				$y < 0
3161				? (goto &nan)
3162				: (goto &zero);
3163				};
3164
3165				state $t = Math::GMPz::Rmpz_init_nobless();
3166				Math::GMPz::Rmpz_fac_ui($t, CORE::abs($y));
3167				Math::GMPz::Rmpz_mul($r, $r, $t);
3168
3169				if ($y < 0) {
3170				my $q = Math::GMPq::Rmpq_init();
3171				Math::GMPq::Rmpq_set_z($q, $r);
3172				Math::GMPq::Rmpq_inv($q, $q);
3173				return bless \$q;
3174				}
3175
3176				bless \$r;
3177				}
3178
3179				#
3180				## Greatest common multiple
3181				#
3182
3183				sub gcd ($$) {
3184				my ($x, $y) = @_;
3185
3186				if (ref($y) and !ref($x)) {
3187				($x, $y) = ($y, $x);
3188				}
3189
3190				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
3191
3192				my $r = Math::GMPz::Rmpz_init();
3193
3194				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
3195				Math::GMPz::Rmpz_gcd_ui($r, $x, CORE::abs($y));
3196				}
3197				else {
3198				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3199				Math::GMPz::Rmpz_gcd($r, $x, $y);
3200				}
3201
3202				bless \$r;
3203				}
3204
3205				#
3206				## Least common multiple
3207				#
3208
3209				sub lcm ($$) {
3210				my ($x, $y) = @_;
3211
3212				if (ref($y) and !ref($x)) {
3213				($x, $y) = ($y, $x);
3214				}
3215
3216				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // (goto &nan);
3217
3218				my $r = Math::GMPz::Rmpz_init();
3219
3220				if (!ref($y) and CORE::int($y) eq $y and CORE::abs($y) <= ULONG_MAX) {
3221				Math::GMPz::Rmpz_lcm_ui($r, $x, CORE::abs($y));
3222				}
3223				else {
3224				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3225				Math::GMPz::Rmpz_lcm($r, $x, $y);
3226				}
3227
3228				bless \$r;
3229				}
3230
3231				#
3232				## Next prime after `x`.
3233				#
3234
3235				sub next_prime ($) {
3236				my ($x) = @_;
3237				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3238				my $r = Math::GMPz::Rmpz_init();
3239				Math::GMPz::Rmpz_nextprime($r, $x);
3240				bless \$r;
3241				}
3242
3243				#
3244				## Is prime?
3245				#
3246
3247				sub is_prime ($;$) {
3248				require Math::AnyNum::is_int;
3249				my ($x, $y) = @_;
3250
3251				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3252
3253				__is_int__($x) \|\| return 0;
3254
3255				$y = defined($y) ? (CORE::abs(CORE::int($y)) \|\| 20) : 20;
3256				Math::GMPz::Rmpz_probab_prime_p(_any2mpz($x) // (return 0), $y);
3257				}
3258
3259				#
3260				## Is `x` coprime to `y`?
3261				#
3262
3263				sub is_coprime ($$) {
3264				require Math::AnyNum::is_int;
3265				my ($x, $y) = @_;
3266
3267				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3268
3269				__is_int__($x) \|\| return 0;
3270
3271				$x = _any2mpz($x) // return 0;
3272
3273				if (!ref($y) and CORE::int($y) eq $y and $y >= 0 and $y <= ULONG_MAX) {
3274				## `y` is a native integer
3275				}
3276				else {
3277				$y = ref($y) eq __PACKAGE__ ? $$y : _star2obj($y);
3278				__is_int__($y) \|\| return 0;
3279				$y = _any2mpz($y) // return 0;
3280				}
3281
3282				state $t = Math::GMPz::Rmpz_init_nobless();
3283
3284				ref($y)
3285				? Math::GMPz::Rmpz_gcd($t, $x, $y)
3286				: Math::GMPz::Rmpz_gcd_ui($t, $x, $y);
3287
3288				Math::GMPz::Rmpz_cmp_ui($t, 1) == 0;
3289				}
3290
3291				#
3292				## Is integer?
3293				#
3294
3295				sub is_int ($) {
3296				require Math::AnyNum::is_int;
3297				my ($x) = @_;
3298				__is_int__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
3299				}
3300
3301				#
3302				## Is rational?
3303				#
3304
3305				sub is_rat ($) {
3306				my ($x) = @_;
3307				my $ref = ref(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
3308				$ref eq 'Math::GMPz' or $ref eq 'Math::GMPq';
3309				}
3310
3311				#
3312				## Numerator of a number
3313				#
3314
3315				sub numerator ($) {
3316				my ($x) = @_;
3317
3318				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3319				{
3320				my $ref = ref($r);
3321				ref($r) eq 'Math::GMPz' && return $x; # is an integer
3322
3323				if (ref($r) eq 'Math::GMPq') {
3324				my $z = Math::GMPz::Rmpz_init();
3325				Math::GMPq::Rmpq_get_num($z, $r);
3326				return bless \$z;
3327				}
3328
3329				$r = _any2mpq($r) // goto &nan;
3330				redo;
3331				}
3332				}
3333
3334				#
3335				## Denominator of a number
3336				#
3337
3338				sub denominator ($) {
3339				my ($x) = @_;
3340
3341				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3342				{
3343				my $ref = ref($r);
3344				ref($r) eq 'Math::GMPz' && goto &one; # is an integer
3345
3346				if (ref($r) eq 'Math::GMPq') {
3347				my $z = Math::GMPz::Rmpz_init();
3348				Math::GMPq::Rmpq_get_den($z, $r);
3349				return bless \$z;
3350				}
3351				$r = _any2mpq($r) // goto &nan;
3352				redo;
3353				}
3354				}
3355
3356				#
3357				## (numerator, denominator)
3358				#
3359
3360				sub nude ($) {
3361				my ($x) = @_;
3362
3363				if (ref($x) ne __PACKAGE__) {
3364				$x = __PACKAGE__->new($x);
3365				}
3366
3367				($x->numerator, $x->denominator);
3368				}
3369
3370				#
3371				## Sign of a number
3372				#
3373
3374				sub sgn ($) {
3375				require Math::AnyNum::sgn;
3376				my ($x) = @_;
3377				my $r = __sgn__(ref($x) eq __PACKAGE__ ? $$x : _star2obj($x));
3378				ref($r) ? (bless \$r) : $r;
3379				}
3380
3381				#
3382				## Is a real number?
3383				#
3384
3385				sub is_real ($) {
3386				my ($x) = @_;
3387
3388				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3389				{
3390				my $ref = ref($r);
3391
3392				$ref eq 'Math::GMPz' && return 1;
3393				$ref eq 'Math::GMPq' && return 1;
3394				$ref eq 'Math::MPFR' && return Math::MPFR::Rmpfr_number_p($r);
3395
3396				$r = _any2mpfr($r);
3397				redo;
3398				}
3399				}
3400
3401				#
3402				## Is an imaginary number?
3403				#
3404
3405				sub is_imag ($) {
3406				my ($x) = @_;
3407
3408				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3409				ref($r) eq 'Math::MPC' or return 0;
3410
3411				my $f = Math::MPFR::Rmpfr_init2($PREC);
3412				Math::MPC::RMPC_RE($f, $r);
3413				Math::MPFR::Rmpfr_zero_p($f) \|\| return 0; # is complex
3414				Math::MPC::RMPC_IM($f, $r);
3415				!Math::MPFR::Rmpfr_zero_p($f);
3416				}
3417
3418				#
3419				## Is a complex number?
3420				#
3421
3422				sub is_complex ($) {
3423				my ($x) = @_;
3424
3425				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3426				ref($r) eq 'Math::MPC' or return 0;
3427
3428				my $f = Math::MPFR::Rmpfr_init2($PREC);
3429				Math::MPC::RMPC_IM($f, $r);
3430				Math::MPFR::Rmpfr_zero_p($f) && return 0; # is real
3431				Math::MPC::RMPC_RE($f, $r);
3432				!Math::MPFR::Rmpfr_zero_p($f);
3433				}
3434
3435				#
3436				## Is positive infinity?
3437				#
3438
3439				sub is_inf ($) {
3440				my ($x) = @_;
3441
3442				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3443				{
3444				my $ref = ref($r);
3445
3446				$ref eq 'Math::GMPz' && return 0;
3447				$ref eq 'Math::GMPq' && return 0;
3448				$ref eq 'Math::MPFR' && return (Math::MPFR::Rmpfr_inf_p($r) and Math::MPFR::Rmpfr_sgn($r) > 0);
3449
3450				$r = _any2mpfr($r);
3451				redo;
3452				}
3453				}
3454
3455				#
3456				## Is negative infinity?
3457				#
3458
3459				sub is_ninf ($) {
3460				my ($x) = @_;
3461
3462				my $r = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3463				{
3464				my $ref = ref($r);
3465
3466				$ref eq 'Math::GMPz' && return 0;
3467				$ref eq 'Math::GMPq' && return 0;
3468				$ref eq 'Math::MPFR' && return (Math::MPFR::Rmpfr_inf_p($r) and Math::MPFR::Rmpfr_sgn($r) < 0);
3469
3470				$r = _any2mpfr($r);
3471				redo;
3472				}
3473				}
3474
3475				#
3476				## Is Not-A-Number?
3477				#
3478
3479				sub is_nan ($) {
3480				my ($x) = @_;
3481
3482				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3483
3484				ref($x) eq 'Math::GMPz' && return 0;
3485				ref($x) eq 'Math::GMPq' && return 0;
3486				ref($x) eq 'Math::MPFR' && return Math::MPFR::Rmpfr_nan_p($x);
3487
3488				my $t = Math::MPFR::Rmpfr_init2($PREC);
3489
3490				Math::MPC::RMPC_RE($t, $x);
3491				Math::MPFR::Rmpfr_nan_p($t) && return 1;
3492
3493				Math::MPC::RMPC_IM($t, $x);
3494				Math::MPFR::Rmpfr_nan_p($t) && return 1;
3495
3496				return 0;
3497				}
3498
3499				#
3500				## Is an even integer?
3501				#
3502
3503				sub is_even ($) {
3504				require Math::AnyNum::is_int;
3505				my ($x) = @_;
3506
3507				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3508
3509				__is_int__($x)
3510				&& Math::GMPz::Rmpz_even_p(_any2mpz($x) // (return 0));
3511				}
3512
3513				#
3514				## Is an odd integer?
3515				#
3516
3517				sub is_odd ($) {
3518				require Math::AnyNum::is_int;
3519				my ($x) = @_;
3520
3521				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3522
3523				__is_int__($x)
3524				&& Math::GMPz::Rmpz_odd_p(_any2mpz($x) // (return 0));
3525				}
3526
3527				#
3528				## Is zero?
3529				#
3530
3531				sub is_zero ($) {
3532				require Math::AnyNum::eq;
3533				my ($x) = @_;
3534				(@_) = ((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 0);
3535				goto &__eq__;
3536				}
3537
3538				#
3539				## Is one?
3540				#
3541
3542				sub is_one ($) {
3543				require Math::AnyNum::eq;
3544				my ($x) = @_;
3545				(@_) = ((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 1);
3546				goto &__eq__;
3547				}
3548
3549				#
3550				## Is minus one?
3551				#
3552
3553				sub is_mone ($) {
3554				require Math::AnyNum::eq;
3555				my ($x) = @_;
3556				(@_) = ((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), -1);
3557				goto &__eq__;
3558				}
3559
3560				#
3561				## Is positive?
3562				#
3563
3564				sub is_pos ($) {
3565				require Math::AnyNum::cmp;
3566				my ($x) = @_;
3567				(__cmp__((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 0) // return undef) > 0;
3568				}
3569
3570				#
3571				## Is negative?
3572				#
3573
3574				sub is_neg ($) {
3575				require Math::AnyNum::cmp;
3576				my ($x) = @_;
3577				(__cmp__((ref($x) eq __PACKAGE__ ? $$x : _star2obj($x)), 0) // return undef) < 0;
3578				}
3579
3580				#
3581				## Is square?
3582				#
3583
3584				sub is_square ($) {
3585				require Math::AnyNum::is_int;
3586				my ($x, $y) = @_;
3587
3588				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3589
3590				__is_int__($x)
3591				&& Math::GMPz::Rmpz_perfect_square_p(_any2mpz($x) // (return 0));
3592				}
3593
3594				#
3595				## Is a polygonal number?
3596				#
3597
3598				sub is_polygonal ($$) {
3599				require Math::AnyNum::is_int;
3600				require Math::AnyNum::is_polygonal;
3601				my ($n, $k) = @_;
3602
3603				$n = ref($n) eq __PACKAGE__ ? $$n : _star2obj($n);
3604
3605				$n = (__is_int__($n) ? _any2mpz($n) : return 0) // return 0;
3606				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // return 0;
3607
3608				__is_polygonal__($n, $k);
3609				}
3610
3611				#
3612				## Is a second polygonal number?
3613				#
3614
3615				sub is_polygonal2 ($$) {
3616				require Math::AnyNum::is_int;
3617				require Math::AnyNum::is_polygonal;
3618				my ($n, $k) = @_;
3619
3620				$n = ref($n) eq __PACKAGE__ ? $$n : _star2obj($n);
3621
3622				$n = (__is_int__($n) ? _any2mpz($n) : return 0) // return 0;
3623				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // return 0;
3624
3625				__is_polygonal__($n, $k, 1);
3626				}
3627
3628				#
3629				## Integer polygonal root
3630				#
3631
3632				sub ipolygonal_root ($$) {
3633				require Math::AnyNum::ipolygonal_root;
3634				my ($n, $k) = @_;
3635
3636				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3637				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // goto &nan;
3638
3639				bless \__ipolygonal_root__($n, $k);
3640				}
3641
3642				#
3643				## Second integer polygonal root
3644				#
3645
3646				sub ipolygonal_root2 ($$) {
3647				require Math::AnyNum::ipolygonal_root;
3648				my ($n, $k) = @_;
3649
3650				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3651				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // goto &nan;
3652
3653				bless \__ipolygonal_root__($n, $k, 1);
3654				}
3655
3656				#
3657				## n-th k-gonal number
3658				#
3659
3660				sub polygonal ($$) {
3661				my ($n, $k) = @_;
3662
3663				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3664
3665				if (!ref($k) and CORE::int($k) eq $k and $k >= 0 and $k <= ULONG_MAX) {
3666				## `k` is a native unsigned integer
3667				}
3668				else {
3669				$k = (ref($k) eq __PACKAGE__ ? _any2mpz($$k) : _star2mpz($k)) // goto &nan;
3670				}
3671
3672				#
3673				## polygonal(n, k) = n * (kn - k - 2n + 4) / 2
3674				#
3675
3676				my $r = Math::GMPz::Rmpz_init();
3677
3678				if (!ref($k)) { # `k` is a native unsigned integer
3679				Math::GMPz::Rmpz_mul_ui($r, $n, $k); # r = n*k
3680				Math::GMPz::Rmpz_sub_ui($r, $r, $k); # r = r-k
3681				}
3682				else {
3683				Math::GMPz::Rmpz_mul($r, $n, $k); # r = n*k
3684				Math::GMPz::Rmpz_sub($r, $r, $k); # r = r-k
3685				}
3686
3687				Math::GMPz::Rmpz_sub($r, $r, $n); # r = r-n
3688				Math::GMPz::Rmpz_sub($r, $r, $n); # r = r-n
3689				Math::GMPz::Rmpz_add_ui($r, $r, 4); # r = r+4
3690				Math::GMPz::Rmpz_mul($r, $r, $n); # r = r*n
3691				Math::GMPz::Rmpz_div_2exp($r, $r, 1); # r = r/2
3692
3693				bless \$r;
3694				}
3695
3696				#
3697				## is_power
3698				#
3699
3700				sub is_power ($;$) {
3701				require Math::AnyNum::is_power;
3702				require Math::AnyNum::is_int;
3703				my ($x, $y) = @_;
3704
3705				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
3706
3707				__is_int__($x) \|\| return 0;
3708				$x = _any2mpz($x) // goto &nan;
3709
3710				if (!defined($y)) {
3711				return Math::GMPz::Rmpz_perfect_power_p($x);
3712				}
3713
3714				if (!ref($y) and CORE::int($y) eq $y and $y <= ULONG_MAX and $y >= LONG_MIN) {
3715				## `y` is a native integer
3716				}
3717				else {
3718				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // return 0;
3719				}
3720
3721				__is_power__($x, $y);
3722				}
3723
3724				#
3725				## kronecker
3726				#
3727
3728				sub kronecker ($$) {
3729				my ($x, $y) = @_;
3730
3731				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3732				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3733
3734				Math::GMPz::Rmpz_kronecker($x, $y);
3735				}
3736
3737				#
3738				## valuation
3739				#
3740
3741				sub valuation ($$) {
3742				require Math::AnyNum::valuation;
3743				my ($x, $y) = @_;
3744
3745				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3746				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3747
3748				(__valuation__($x, $y))[0];
3749				}
3750
3751				#
3752				## remdiv
3753				#
3754
3755				sub remdiv ($$) {
3756				require Math::AnyNum::valuation;
3757				my ($x, $y) = @_;
3758
3759				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3760				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3761
3762				bless \((__valuation__($x, $y))[1]);
3763				}
3764
3765				#
3766				## Invmod
3767				#
3768
3769				sub invmod ($$) {
3770				my ($x, $y) = @_;
3771
3772				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3773				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3774
3775				my $r = Math::GMPz::Rmpz_init();
3776				Math::GMPz::Rmpz_invert($r, $x, $y) \|\| (goto &nan);
3777				bless \$r;
3778				}
3779
3780				#
3781				## Powmod
3782				#
3783
3784				sub powmod ($$$) {
3785				my ($x, $y, $z) = @_;
3786
3787				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3788				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // goto &nan;
3789				$z = (ref($z) eq __PACKAGE__ ? _any2mpz($$z) : _star2mpz($z)) // goto &nan;
3790
3791				Math::GMPz::Rmpz_sgn($z) \|\| goto &nan;
3792
3793				if (Math::GMPz::Rmpz_sgn($y) < 0) {
3794				my $t = Math::GMPz::Rmpz_init();
3795				Math::GMPz::Rmpz_gcd($t, $x, $z);
3796				Math::GMPz::Rmpz_cmp_ui($t, 1) == 0 or goto &nan;
3797				}
3798
3799				my $r = Math::GMPz::Rmpz_init();
3800				Math::GMPz::Rmpz_powm($r, $x, $y, $z);
3801				bless \$r;
3802				}
3803
3804				#
3805				## Faulhaber summation formula
3806				#
3807
3808				sub faulhaber_sum ($$) {
3809				require Math::AnyNum::bernfrac;
3810				my ($n, $p) = @_;
3811
3812				my $native_n = 0; # true when `n` is a native integer
3813
3814				if (!ref($n) and CORE::int($n) eq $n and $n >= 0 and $n <= ULONG_MAX) {
3815				## `n` is a native unsigned integer
3816				$native_n = 1;
3817				}
3818				else {
3819				$n = (ref($n) eq __PACKAGE__ ? _any2mpz($$n) : _star2mpz($n)) // goto &nan;
3820
3821				# Try to unbox `n` when it fits inside a native unsigned integer
3822				if (Math::GMPz::Rmpz_fits_ulong_p($n)) {
3823				$native_n = 1;
3824				$n = Math::GMPz::Rmpz_get_ui($n);
3825				}
3826				}
3827
3828				if (!ref($p) and CORE::int($p) eq $p and $p >= 0 and $p <= ULONG_MAX) {
3829				## `p` is already a native unsigned integer
3830				}
3831				else {
3832				$p = (ref($p) eq __PACKAGE__ ? _any2ui($$p) : _any2ui(_star2obj($p))) // goto &nan;
3833				}
3834
3835				state @cache; # cache for Bernoulli numbers
3836
3837				my $t = Math::GMPz::Rmpz_init();
3838				my $u = Math::GMPz::Rmpz_init();
3839
3840				my $numerator = Math::GMPz::Rmpz_init_set_ui(0);
3841				my $denominator = Math::GMPz::Rmpz_init_set_ui(1);
3842
3843				foreach my $j (0 .. $p) {
3844
3845				# When `j` is odd and greater than 1, we can skip it.
3846				$j % 2 == 0 or $j == 1 or next;
3847
3848				Math::GMPz::Rmpz_bin_uiui($t, $p + 1, $j); # t = binomial(p+1, j)
3849
3850				#<<<
3851				$native_n
3852				? Math::GMPz::Rmpz_ui_pow_ui($u, $n, $p + 1 - $j) # u = n^(p + 1 - j)
3853				: Math::GMPz::Rmpz_pow_ui( $u, $n, $p + 1 - $j); # ==//==
3854				#>>>
3855
3856				Math::GMPz::Rmpz_mul($t, $t, $u); # t = t * u
3857
3858				# Compute Bernouli(j)
3859				my $bern = ($j <= 100 ? ($cache[$j] //= __bernfrac__($j)) : __bernfrac__($j));
3860
3861				# Bernoulli(j) = 1 for j=0
3862				if (!$j) {
3863				Math::GMPz::Rmpz_add($numerator, $numerator, $t); # numerator = numerator + t
3864				}
3865				else {
3866				#<<<
3867				Math::GMPq::Rmpq_get_num($u, $bern); # u = numerator(bern)
3868				Math::GMPz::Rmpz_mul($t, $t, $u); # t = t * u
3869				Math::GMPq::Rmpq_get_den($u, $bern); # u = denominator(bern)
3870
3871				Math::GMPz::Rmpz_mul( $numerator, $numerator, $u); # numerator = numerator * u
3872				Math::GMPz::Rmpz_addmul($numerator, $denominator, $t); # numerator += denominator * t
3873				Math::GMPz::Rmpz_mul( $denominator, $denominator, $u); # denominator = denominator * u
3874				#>>>
3875				}
3876				}
3877
3878				#<<<
3879				Math::GMPz::Rmpz_mul_ui($denominator, $denominator, $p + 1); # denominator = denominator * (p+1)
3880				Math::GMPz::Rmpz_divexact($numerator, $numerator, $denominator); # numerator = numerator / denominator
3881				#>>>
3882
3883				bless \$numerator;
3884				}
3885
3886				#
3887				## Binomial
3888				#
3889
3890				sub binomial ($$) {
3891				my ($x, $y) = @_;
3892
3893				# `x` and `y` are native unsigned integers
3894				if ( !ref($x)
3895				and !ref($y)
3896				and CORE::int($x) eq $x
3897				and CORE::int($y) eq $y
3898				and $x >= 0
3899				and $y >= 0
3900				and $x <= ULONG_MAX
3901				and $y <= ULONG_MAX) {
3902				my $r = Math::GMPz::Rmpz_init();
3903				Math::GMPz::Rmpz_bin_uiui($r, $x, $y);
3904				return bless \$r;
3905				}
3906
3907				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3908
3909				if (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3910				## `y` is a native integer
3911				}
3912				else {
3913				$y = _any2si(ref($y) eq __PACKAGE__ ? $$y : _star2obj($y)) // goto &nan;
3914				}
3915
3916				my $r = Math::GMPz::Rmpz_init();
3917
3918				if ($y >= 0 and Math::GMPz::Rmpz_fits_ulong_p($x)) {
3919				Math::GMPz::Rmpz_bin_uiui($r, Math::GMPz::Rmpz_get_ui($x), $y);
3920				}
3921				else {
3922				$y < 0
3923				? Math::GMPz::Rmpz_bin_si($r, $x, $y)
3924				: Math::GMPz::Rmpz_bin_ui($r, $x, $y);
3925				}
3926
3927				bless \$r;
3928				}
3929
3930				#
3931				## AND
3932				#
3933
3934				sub and { # used in overloading
3935				my ($x, $y) = @_;
3936
3937				$x = _any2mpz($$x) // (goto &nan);
3938				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3939
3940				my $r = Math::GMPz::Rmpz_init();
3941				Math::GMPz::Rmpz_and($r, $x, $y);
3942				bless \$r;
3943				}
3944
3945				#
3946				## OR
3947				#
3948
3949				sub or { # used in overloading
3950				my ($x, $y) = @_;
3951
3952				$x = _any2mpz($$x) // (goto &nan);
3953				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3954
3955				my $r = Math::GMPz::Rmpz_init();
3956				Math::GMPz::Rmpz_ior($r, $x, $y);
3957				bless \$r;
3958				}
3959
3960				#
3961				## XOR
3962				#
3963
3964				sub xor { # used in overloading
3965				my ($x, $y) = @_;
3966
3967				$x = _any2mpz($$x) // (goto &nan);
3968				$y = (ref($y) eq __PACKAGE__ ? _any2mpz($$y) : _star2mpz($y)) // (goto &nan);
3969
3970				my $r = Math::GMPz::Rmpz_init();
3971				Math::GMPz::Rmpz_xor($r, $x, $y);
3972				bless \$r;
3973				}
3974
3975				#
3976				## NOT
3977				#
3978
3979				sub not { # used in overloading
3980				my ($x) = @_;
3981				$x = _any2mpz($$x) // (goto &nan);
3982				my $r = Math::GMPz::Rmpz_init();
3983				Math::GMPz::Rmpz_com($r, $x);
3984				bless \$r;
3985				}
3986
3987				#
3988				## LEFT SHIFT
3989				#
3990
3991				sub lsft { # used in overloading
3992				my ($x, $y) = @_;
3993
3994				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
3995
3996				if (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
3997				## `y` is a native integer
3998				}
3999				else {
4000				$y = (ref($y) eq __PACKAGE__ ? _any2si($$y) : _any2si(_star2obj($y))) // goto &nan;
4001				}
4002
4003				my $r = Math::GMPz::Rmpz_init();
4004
4005				$y < 0
4006				? Math::GMPz::Rmpz_div_2exp($r, $x, -$y)
4007				: Math::GMPz::Rmpz_mul_2exp($r, $x, $y);
4008
4009				bless \$r;
4010				}
4011
4012				#
4013				## RIGHT SHIFT
4014				#
4015
4016				sub rsft { # used in overloading
4017				my ($x, $y) = @_;
4018
4019				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _star2mpz($x)) // goto &nan;
4020
4021				if (!ref($y) and CORE::int($y) eq $y and $y >= LONG_MIN and $y <= ULONG_MAX) {
4022				## `y` is a native integer
4023				}
4024				else {
4025				$y = (ref($y) eq __PACKAGE__ ? _any2si($$y) : _any2si(_star2obj($y))) // goto &nan;
4026				}
4027
4028				my $r = Math::GMPz::Rmpz_init();
4029
4030				$y < 0
4031				? Math::GMPz::Rmpz_mul_2exp($r, $x, -$y)
4032				: Math::GMPz::Rmpz_div_2exp($r, $x, $y);
4033
4034				bless \$r;
4035				}
4036
4037				#
4038				## POPCOUNT
4039				#
4040
4041				sub popcount ($) {
4042				my ($x) = @_;
4043
4044				$x = (ref($x) eq __PACKAGE__ ? _any2mpz($$x) : _any2mpz(_star2obj($x))) // return -1;
4045
4046				if (Math::GMPz::Rmpz_sgn($x) < 0) {
4047				my $t = Math::GMPz::Rmpz_init();
4048				Math::GMPz::Rmpz_neg($t, $x);
4049				$x = $t;
4050				}
4051
4052				Math::GMPz::Rmpz_popcount($x);
4053				}
4054
4055				#
4056				## Conversions
4057				#
4058
4059				sub as_bin ($) {
4060				Math::GMPz::Rmpz_get_str((_star2mpz($_[0]) // return undef), 2);
4061				}
4062
4063				sub as_oct ($) {
4064				Math::GMPz::Rmpz_get_str((_star2mpz($_[0]) // return undef), 8);
4065				}
4066
4067				sub as_hex ($) {
4068				Math::GMPz::Rmpz_get_str((_star2mpz($_[0]) // return undef), 16);
4069				}
4070
4071				sub as_int ($;$) {
4072				my ($x, $y) = @_;
4073
4074				my $base = 10;
4075				if (defined($y)) {
4076
4077				if (!ref($y) and CORE::int($y) eq $y) {
4078				$base = $y;
4079				}
4080				elsif (ref($y) eq __PACKAGE__) {
4081				$base = _any2ui($$y) // 0;
4082				}
4083				else {
4084				$base = _any2ui(_star2mpz($y) // return undef) // 0;
4085				}
4086
4087				if ($base < 2 or $base > 36) {
4088				require Carp;
4089				Carp::croak("base must be between 2 and 36, got $y");
4090				}
4091				}
4092
4093				Math::GMPz::Rmpz_get_str((_star2mpz($x) // return undef), $base);
4094				}
4095
4096				sub as_frac ($;$) {
4097				my ($x, $y) = @_;
4098
4099				my $base = 10;
4100				if (defined($y)) {
4101
4102				if (!ref($y) and CORE::int($y) eq $y) {
4103				$base = $y;
4104				}
4105				elsif (ref($y) eq __PACKAGE__) {
4106				$base = _any2ui($$y) // 0;
4107				}
4108				else {
4109				$base = _any2ui(_star2mpz($y) // return undef) // 0;
4110				}
4111
4112				if ($base < 2 or $base > 36) {
4113				require Carp;
4114				Carp::croak("base must be between 2 and 36, got $y");
4115				}
4116				}
4117
4118				$x = ref($x) eq __PACKAGE__ ? $$x : _star2obj($x);
4119
4120				my $ref = ref($x);
4121				if ( $ref eq 'Math::GMPq'
4122				or $ref eq 'Math::GMPz') {
4123				my $frac = (
4124				$ref eq 'Math::GMPq'
4125				? Math::GMPq::Rmpq_get_str($x, $base)
4126				: Math::GMPz::Rmpz_get_str($x, $base)
4127				);
4128				$frac .= '/1' if (index($frac, '/') == -1);
4129				return $frac;
4130				}
4131
4132				$x = _any2mpq($x) // return undef;
4133
4134				my $frac = Math::GMPq::Rmpq_get_str($x, $base);
4135				$frac .= '/1' if (index($frac, '/') == -1);
4136				$frac;
4137				}
4138
4139				sub as_dec ($;$) {
4140				my ($x, $y) = @_;
4141				require Math::AnyNum::stringify;
4142
4143				my $prec = $PREC;
4144				if (defined($y)) {
4145				if (!ref($y) and CORE::int($y) eq $y) {
4146				$prec = $y;
4147				}
4148				elsif (ref($y) eq __PACKAGE__) {
4149				$prec = _any2ui($$y) // 0;
4150				}
4151				else {
4152				$prec = _any2ui(_star2mpz($y) // return undef) // 0;
4153				}
4154
4155				$prec <<= 2;
4156
4157				state $min_prec = Math::MPFR::RMPFR_PREC_MIN();
4158				state $max_prec = Math::MPFR::RMPFR_PREC_MAX();
4159
4160				if ($prec < $min_prec or $prec > $max_prec) {
4161				require Carp;
4162				Carp::croak("precision must be between $min_prec and $max_prec, got ", $prec >> 2);
4163				}
4164				}
4165
4166				local $PREC = $prec;
4167				__stringify__(_star2mpfr_mpc($x));
4168				}
4169
4170				sub rat_approx ($) {
4171				require Math::AnyNum::stringify;
4172				my ($x) = @_;
4173
4174				$x = _star2mpfr($x);
4175
4176				Math::MPFR::Rmpfr_number_p($x) \|\| goto &nan;
4177
4178				my $t = Math::MPFR::Rmpfr_init2($PREC); # temporary variable
4179				my $r = Math::MPFR::Rmpfr_init2($PREC);
4180
4181				Math::MPFR::Rmpfr_set($r, $x, $ROUND);
4182
4183				my $num2cfrac = sub {
4184				my ($callback, $n) = @_;
4185
4186				while (1) {
4187				Math::MPFR::Rmpfr_floor($t, $r);
4188
4189				my $z = Math::GMPz::Rmpz_init();
4190				Math::MPFR::Rmpfr_get_z($z, $t, Math::MPFR::MPFR_RNDZ);
4191
4192				$callback->($z) && return 1;
4193
4194				Math::MPFR::Rmpfr_sub($r, $r, $t, $ROUND);
4195				Math::MPFR::Rmpfr_zero_p($r) && last;
4196				Math::MPFR::Rmpfr_ui_div($r, 1, $r, $ROUND);
4197				}
4198				};
4199
4200				my $q = Math::GMPq::Rmpq_init();
4201
4202				my $cfrac2num = sub {
4203				my (@f) = @_;
4204
4205				Math::GMPq::Rmpq_set_ui($q, 0, 1);
4206
4207				for (1 .. $#f) {
4208				Math::GMPq::Rmpq_add_z($q, $q, CORE::pop(@f));
4209				Math::GMPq::Rmpq_inv($q, $q);
4210				}
4211
4212				Math::GMPq::Rmpq_add_z($q, $q, $f[0]);
4213				};
4214
4215				my @cfrac;
4216				my $s = __stringify__($x);
4217				my $u = Math::MPFR::Rmpfr_init2($PREC); # temporary variable
4218
4219				#<<<
4220				$num2cfrac->(
4221				sub {
4222				my ($n) = @_;
4223				CORE::push(@cfrac, $n);
4224				$cfrac2num->(@cfrac);
4225				Math::MPFR::Rmpfr_set_q($u, $q, $ROUND);
4226				CORE::index(__stringify__($u), $s) == 0;
4227				}, $x
4228				);
4229				#>>>
4230
4231				bless \$q;
4232				}
4233
4234				sub digits ($;$) {
4235				my ($x, $y) = @_;
4236				my $str = as_int($x, $y) // return ();
4237				my @digits = split(//, $str);
4238				shift(@digits) if $digits[0] eq '-';
4239				(@digits);
4240				}
4241
4242				1; # End of Math::AnyNum