1#!./perl
2
3#
4# Regression tests for the Math::Complex pacakge
5# -- Raphael Manfredi	since Sep 1996
6# -- Jarkko Hietaniemi	since Mar 1997
7# -- Daniel S. Lewart	since Sep 1997
8
9use strict;
10use warnings;
11
12use Math::Complex 1.54;
13
14# they are used later in the test and not exported by Math::Complex
15*_stringify_cartesian = \&Math::Complex::_stringify_cartesian;
16*_stringify_polar     = \&Math::Complex::_stringify_polar;
17
18our $vax_float = (pack("d",1) =~ /^[\x80\x10]\x40/);
19our $has_inf   = !$vax_float;
20
21my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val);
22my ($bad, $z);
23
24$test = 0;
25$| = 1;
26my @script = (
27    'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10,$z0,$z1,$z2);' .
28	"\n\n"
29);
30my $eps = 1e-13;
31
32if ($^O eq 'unicos') { 	# For some reason root() produces very inaccurate
33    $eps = 1e-10;	# results in Cray UNICOS, and occasionally also
34}			# cos(), sin(), cosh(), sinh().  The division
35			# of doubles is the current suspect.
36
37$test++;
38push @script, "{ my \$t=$test; ".q{
39    my $a = Math::Complex->new(1);
40    my $b = $a;
41    $a += 2;
42    print "not " unless "$a" eq "3" && "$b" eq "1";
43    print "ok $t\n";
44}."}";
45
46while (<DATA>) {
47	s/^\s+//;
48	next if $_ eq '' || /^\#/;
49	chomp;
50	$test_set = 0;		# Assume not a test over a set of values
51	if (/^&(.+)/) {
52		$op = $1;
53		next;
54	}
55	elsif (/^\{(.+)\}/) {
56		set($1, \@set, \@val);
57		next;
58	}
59	elsif (s/^\|//) {
60		$test_set = 1;	# Requests we loop over the set...
61	}
62	my @args = split(/:/);
63	if ($test_set == 1) {
64		my $i;
65		for ($i = 0; $i < @set; $i++) {
66			# complex number
67			$target = $set[$i];
68			# textual value as found in set definition
69			$zvalue = $val[$i];
70			test($zvalue, $target, @args);
71		}
72	} else {
73		test($op, undef, @args);
74	}
75}
76
77#
78
79sub test_mutators {
80    my $op;
81
82    $test++;
83push(@script, <<'EOT');
84{
85    my $z = cplx(  1,  1);
86    $z->Re(2);
87    $z->Im(3);
88    print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
89    print 'not ' unless Re($z) == 2 and Im($z) == 3;
90EOT
91    push(@script, qq(print "ok $test\\n"}\n));
92
93    $test++;
94push(@script, <<'EOT');
95{
96    my $z = cplx(  1,  1);
97    $z->abs(3 * sqrt(2));
98    print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
99    print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and
100                        (arg($z) - pi / 4     ) < $eps and
101                        (Re($z) - 3           ) < $eps and
102                        (Im($z) - 3           ) < $eps;
103EOT
104    push(@script, qq(print "ok $test\\n"}\n));
105
106    $test++;
107push(@script, <<'EOT');
108{
109    my $z = cplx(  1,  1);
110    $z->arg(-3 / 4 * pi);
111    print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
112    print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and
113                        (abs($z) - sqrt(2)   ) < $eps and
114                        (Re($z) + 1          ) < $eps and
115                        (Im($z) + 1          ) < $eps;
116EOT
117    push(@script, qq(print "ok $test\\n"}\n));
118}
119
120test_mutators();
121
122my $constants = '
123my $i    = cplx(0,  1);
124my $pi   = cplx(pi, 0);
125my $pii  = cplx(0, pi);
126my $pip2 = cplx(pi/2, 0);
127my $pip4 = cplx(pi/4, 0);
128my $zero = cplx(0, 0);
129';
130
131if ($has_inf) {
132    $constants .= <<'EOF';
133my $inf  = 9**9**9;
134EOF
135}
136
137push(@script, $constants);
138
139
140# test the divbyzeros
141
142sub test_dbz {
143    for my $op (@_) {
144	$test++;
145	push(@script, <<EOT);
146	eval '$op';
147	(\$bad) = (\$@ =~ /(.+)/);
148	print "# $test op = $op divbyzero? \$bad...\n";
149	print 'not ' unless (\$@ =~ /Division by zero/);
150EOT
151        push(@script, qq(print "ok $test\\n";\n));
152    }
153}
154
155# test the logofzeros
156
157sub test_loz {
158    for my $op (@_) {
159	$test++;
160	push(@script, <<EOT);
161	eval '$op';
162	(\$bad) = (\$@ =~ /(.+)/);
163	print "# $test op = $op logofzero? \$bad...\n";
164	print 'not ' unless (\$@ =~ /Logarithm of zero/);
165EOT
166        push(@script, qq(print "ok $test\\n";\n));
167    }
168}
169
170test_dbz(
171	 'i/0',
172	 'acot(0)',
173	 'acot(+$i)',
174#	 'acoth(-1)',	# Log of zero.
175	 'acoth(0)',
176	 'acoth(+1)',
177	 'acsc(0)',
178	 'acsch(0)',
179	 'asec(0)',
180	 'asech(0)',
181	 'atan($i)',
182#	 'atanh(-1)',	# Log of zero.
183	 'atanh(+1)',
184	 'cot(0)',
185	 'coth(0)',
186	 'csc(0)',
187	 'csch(0)',
188	 'atan(cplx(0, 1), cplx(1, 0))',
189	);
190
191test_loz(
192	 'log($zero)',
193	 'atan(-$i)',
194	 'acot(-$i)',
195	 'atanh(-1)',
196	 'acoth(-1)',
197	);
198
199# test the bad roots
200
201sub test_broot {
202    for my $op (@_) {
203	$test++;
204	push(@script, <<EOT);
205	eval 'root(2, $op)';
206	(\$bad) = (\$@ =~ /(.+)/);
207	print "# $test op = $op badroot? \$bad...\n";
208	print 'not ' unless (\$@ =~ /root rank must be/);
209EOT
210        push(@script, qq(print "ok $test\\n";\n));
211    }
212}
213
214test_broot(qw(-3 -2.1 0 0.99));
215
216sub test_display_format {
217    $test++;
218    push @script, <<EOS;
219    print "# package display_format cartesian?\n";
220    print "not " unless Math::Complex->display_format eq 'cartesian';
221    print "ok $test\n";
222EOS
223
224    push @script, <<EOS;
225    my \$j = (root(1,3))[1];
226
227    \$j->display_format('polar');
228EOS
229
230    $test++;
231    push @script, <<EOS;
232    print "# j display_format polar?\n";
233    print "not " unless \$j->display_format eq 'polar';
234    print "ok $test\n";
235EOS
236
237    $test++;
238    push @script, <<EOS;
239    print "# j = \$j\n";
240    print "not " unless "\$j" eq "[1,2pi/3]";
241    print "ok $test\n";
242
243    my %display_format;
244
245    %display_format = \$j->display_format;
246EOS
247
248    $test++;
249    push @script, <<EOS;
250    print "# display_format{style} polar?\n";
251    print "not " unless \$display_format{style} eq 'polar';
252    print "ok $test\n";
253EOS
254
255    $test++;
256    push @script, <<EOS;
257    print "# keys %display_format == 2?\n";
258    print "not " unless keys %display_format == 2;
259    print "ok $test\n";
260
261    \$j->display_format('style' => 'cartesian', 'format' => '%.5f');
262EOS
263
264    $test++;
265    push @script, <<EOS;
266    print "# j = \$j\n";
267    print "not " unless "\$j" eq "-0.50000+0.86603i";
268    print "ok $test\n";
269
270    %display_format = \$j->display_format;
271EOS
272
273    $test++;
274    push @script, <<EOS;
275    print "# display_format{format} %.5f?\n";
276    print "not " unless \$display_format{format} eq '%.5f';
277    print "ok $test\n";
278EOS
279
280    $test++;
281    push @script, <<EOS;
282    print "# keys %display_format == 3?\n";
283    print "not " unless keys %display_format == 3;
284    print "ok $test\n";
285
286    \$j->display_format('format' => undef);
287EOS
288
289    $test++;
290    push @script, <<EOS;
291    print "# j = \$j\n";
292    print "not " unless "\$j" =~ /^-0(?:\\.5(?:0000\\d+)?|\\.49999\\d+)\\+0.86602540\\d+i\$/;
293    print "ok $test\n";
294
295    \$j->display_format('style' => 'polar', 'polar_pretty_print' => 0);
296EOS
297
298    $test++;
299    push @script, <<EOS;
300    print "# j = \$j\n";
301    print "not " unless "\$j" =~ /^\\[1,2\\.09439510\\d+\\]\$/;
302    print "ok $test\n";
303
304    \$j->display_format('style' => 'polar', 'format' => "%.4g");
305EOS
306
307    $test++;
308    push @script, <<EOS;
309    print "# j = \$j\n";
310    print "not " unless "\$j" =~ /^\\[1,2\\.094\\]\$/;
311    print "ok $test\n";
312
313    \$j->display_format('style' => 'cartesian', 'format' => '(%.5g)');
314EOS
315
316    $test++;
317    push @script, <<EOS;
318    print "# j = \$j\n";
319    print "not " unless "\$j" eq "(-0.5)+(0.86603)i";
320    print "ok $test\n";
321EOS
322
323    $test++;
324    push @script, <<EOS;
325    print "# j display_format cartesian?\n";
326    print "not " unless \$j->display_format eq 'cartesian';
327    print "ok $test\n";
328EOS
329}
330
331test_display_format();
332
333sub test_remake {
334    $test++;
335    push @script, <<EOS;
336    print "# remake 2+3i\n";
337    \$z = cplx('2+3i');
338    print "not " unless \$z == Math::Complex->make(2,3);
339    print "ok $test\n";
340EOS
341
342    $test++;
343    push @script, <<EOS;
344    print "# make 3i\n";
345    \$z = Math::Complex->make('3i');
346    print "not " unless \$z == cplx(0,3);
347    print "ok $test\n";
348EOS
349
350    $test++;
351    push @script, <<EOS;
352    print "# emake [2,3]\n";
353    \$z = Math::Complex->emake('[2,3]');
354    print "not " unless \$z == cplxe(2,3);
355    print "ok $test\n";
356EOS
357
358    $test++;
359    push @script, <<EOS;
360    print "# make (2,3)\n";
361    \$z = Math::Complex->make('(2,3)');
362    print "not " unless \$z == cplx(2,3);
363    print "ok $test\n";
364EOS
365
366    $test++;
367    push @script, <<EOS;
368    print "# emake [2,3pi/8]\n";
369    \$z = Math::Complex->emake('[2,3pi/8]');
370    print "not " unless \$z == cplxe(2,3*\$pi/8);
371    print "ok $test\n";
372EOS
373
374    $test++;
375    push @script, <<EOS;
376    print "# emake [2]\n";
377    \$z = Math::Complex->emake('[2]');
378    print "not " unless \$z == cplxe(2);
379    print "ok $test\n";
380EOS
381}
382
383sub test_no_args {
384    push @script, <<'EOS';
385{
386    print "# cplx, cplxe, make, emake without arguments\n";
387EOS
388
389    $test++;
390    push @script, <<EOS;
391    my \$z0 = cplx();
392    print ((\$z0->Re()  == 0) ? "ok $test\n" : "not ok $test\n");
393EOS
394
395    $test++;
396    push @script, <<EOS;
397    print ((\$z0->Im()  == 0) ? "ok $test\n" : "not ok $test\n");
398EOS
399
400    $test++;
401    push @script, <<EOS;
402    my \$z1 = cplxe();
403    print ((\$z1->rho()   == 0) ? "ok $test\n" : "not ok $test\n");
404EOS
405
406    $test++;
407    push @script, <<EOS;
408    print ((\$z1->theta() == 0) ? "ok $test\n" : "not ok $test\n");
409EOS
410
411    $test++;
412    push @script, <<EOS;
413    my \$z2 = Math::Complex->make();
414    print ((\$z2->Re()  == 0) ? "ok $test\n" : "not ok $test\n");
415EOS
416
417    $test++;
418    push @script, <<EOS;
419    print ((\$z2->Im()  == 0) ? "ok $test\n" : "not ok $test\n");
420EOS
421
422    $test++;
423    push @script, <<EOS;
424    my \$z3 = Math::Complex->emake();
425    print ((\$z3->rho()   == 0) ? "ok $test\n" : "not ok $test\n");
426EOS
427
428    $test++;
429    push @script, <<EOS;
430    print ((\$z3->theta() == 0) ? "ok $test\n" : "not ok $test\n");
431}
432EOS
433}
434
435sub test_atan2 {
436    push @script, <<'EOS';
437print "# atan2() with some real arguments\n";
438EOS
439    my @real = (-1, 0, 1);
440    for my $x (@real) {
441	for my $y (@real) {
442	    next if $x == 0 && $y == 0;
443	    $test++;
444	    push @script, <<EOS;
445print ((Math::Complex::atan2($y, $x) == CORE::atan2($y, $x)) ? "ok $test\n" : "not ok $test\n");
446EOS
447        }
448    }
449    push @script, <<'EOS';
450    print "# atan2() with some complex arguments\n";
451EOS
452    $test++;
453    push @script, <<EOS;
454    print (abs(atan2(0, cplx(0, 1))) < $eps ? "ok $test\n" : "not ok $test\n");
455EOS
456    $test++;
457    push @script, <<EOS;
458    print (abs(atan2(cplx(0, 1), 0) - \$pip2) < $eps ? "ok $test\n" : "not ok $test\n");
459EOS
460    $test++;
461    push @script, <<EOS;
462    print (abs(atan2(cplx(0, 1), cplx(0, 1)) - \$pip4) < $eps ? "ok $test\n" : "not ok $test\n");
463EOS
464    $test++;
465    push @script, <<EOS;
466    print (abs(atan2(cplx(0, 1), cplx(1, 1)) - cplx(0.553574358897045, 0.402359478108525)) < $eps ? "ok $test\n" : "not ok $test\n");
467EOS
468}
469
470sub test_decplx {
471}
472
473test_remake();
474
475test_no_args();
476
477test_atan2();
478
479test_decplx();
480
481print "1..$test\n";
482#print @script, "\n";
483eval join '', @script;
484die $@ if $@;
485
486sub abop {
487	my ($op) = @_;
488
489	push(@script, qq(print "# $op=\n";));
490}
491
492sub test {
493	my ($op, $z, @args) = @_;
494	my ($baop) = 0;
495	$test++;
496	my $i;
497	$baop = 1 if ($op =~ s/;=$//);
498	for ($i = 0; $i < @args; $i++) {
499		$val = value($args[$i]);
500		push @script, "\$z$i = $val;\n";
501	}
502	if (defined $z) {
503		$args = "'$op'";		# Really the value
504		$try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
505		push @script, "\$res = $try; ";
506		push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n";
507	} else {
508		my ($try, $args);
509		if (@args == 2) {
510			$try = "$op \$z0";
511			$args = "'$args[0]'";
512		} else {
513			$try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1";
514			$args = "'$args[0]', '$args[1]'";
515		}
516		push @script, "\$res = $try; ";
517		push @script, "check($test, '$try', \$res, \$z$#args, $args);\n";
518		if (@args > 2 and $baop) { # binary assignment ops
519			$test++;
520			# check the op= works
521			push @script, <<EOB;
522{
523	my \$za = cplx(ref \$z0 ? \@{\$z0->_cartesian} : (\$z0, 0));
524
525	my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->_cartesian} : (\$z1, 0);
526
527	my \$zb = cplx(\$z1r, \$z1i);
528
529	\$za $op= \$zb;
530	my (\$zbr, \$zbi) = \@{\$zb->_cartesian};
531
532	check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args);
533EOB
534			$test++;
535			# check that the rhs has not changed
536			push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
537			push @script, qq(print "ok $test\\n";\n);
538			push @script, "}\n";
539		}
540	}
541}
542
543sub set {
544	my ($set, $setref, $valref) = @_;
545	@{$setref} = ();
546	@{$valref} = ();
547	my @set = split(/;\s*/, $set);
548	my @res;
549	my $i;
550	for ($i = 0; $i < @set; $i++) {
551		push(@{$valref}, $set[$i]);
552		my $val = value($set[$i]);
553		push @script, "\$s$i = $val;\n";
554		push @{$setref}, "\$s$i";
555	}
556}
557
558sub value {
559	local ($_) = @_;
560	if (/^\s*\((.*),(.*)\)/) {
561		return "cplx($1,$2)";
562	}
563	elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
564		return "cplx($1,0)";
565	}
566	elsif (/^\s*\[(.*),(.*)\]/) {
567		return "cplxe($1,$2)";
568	}
569	elsif (/^\s*'(.*)'/) {
570		my $ex = $1;
571		$ex =~ s/\bz\b/$target/g;
572		$ex =~ s/\br\b/abs($target)/g;
573		$ex =~ s/\bt\b/arg($target)/g;
574		$ex =~ s/\ba\b/Re($target)/g;
575		$ex =~ s/\bb\b/Im($target)/g;
576		return $ex;
577	}
578	elsif (/^\s*"(.*)"/) {
579		return "\"$1\"";
580	}
581	return $_;
582}
583
584sub check {
585	my ($test, $try, $got, $expected, @z) = @_;
586
587	print "# @_\n";
588
589	if ("$got" eq "$expected"
590	    ||
591	    ($expected =~ /^-?\d/ && $got == $expected)
592	    ||
593	    (abs(Math::Complex->make($got) - Math::Complex->make($expected)) < $eps)
594	    ||
595	    (abs($got - $expected) < $eps)
596	    ) {
597		print "ok $test\n";
598	} else {
599		print "not ok $test\n";
600		my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]";
601		print "# '$try' expected: '$expected' got: '$got' for $args\n";
602	}
603}
604
605sub addsq {
606    my ($z1, $z2) = @_;
607    return ($z1 + i*$z2) * ($z1 - i*$z2);
608}
609
610sub subsq {
611    my ($z1, $z2) = @_;
612    return ($z1 + $z2) * ($z1 - $z2);
613}
614
615__END__
616&+;=
617(3,4):(3,4):(6,8)
618(-3,4):(3,-4):(0,0)
619(3,4):-3:(0,4)
6201:(4,2):(5,2)
621[2,0]:[2,pi]:(0,0)
622
623&++
624(2,1):(3,1)
625
626&-;=
627(2,3):(-2,-3)
628[2,pi/2]:[2,-(pi)/2]
6292:[2,0]:(0,0)
630[3,0]:2:(1,0)
6313:(4,5):(-1,-5)
632(4,5):3:(1,5)
633(2,1):(3,5):(-1,-4)
634
635&--
636(1,2):(0,2)
637[2,pi]:[3,pi]
638
639&*;=
640(0,1):(0,1):(-1,0)
641(4,5):(1,0):(4,5)
642[2,2*pi/3]:(1,0):[2,2*pi/3]
6432:(0,1):(0,2)
644(0,1):3:(0,3)
645(0,1):(4,1):(-1,4)
646(2,1):(4,-1):(9,2)
647
648&/;=
649(3,4):(3,4):(1,0)
650(4,-5):1:(4,-5)
6511:(0,1):(0,-1)
652(0,6):(0,2):(3,0)
653(9,2):(4,-1):(2,1)
654[4,pi]:[2,pi/2]:[2,pi/2]
655[2,pi/2]:[4,pi]:[0.5,-(pi)/2]
656
657&**;=
658(2,0):(3,0):(8,0)
659(3,0):(2,0):(9,0)
660(2,3):(4,0):(-119,-120)
661(0,0):(1,0):(0,0)
662(0,0):(2,3):(0,0)
663(1,0):(0,0):(1,0)
664(1,0):(1,0):(1,0)
665(1,0):(2,3):(1,0)
666(2,3):(0,0):(1,0)
667(2,3):(1,0):(2,3)
668(0,0):(0,0):(1,0)
669
670&Re
671(3,4):3
672(-3,4):-3
673[1,pi/2]:0
674
675&Im
676(3,4):4
677(3,-4):-4
678[1,pi/2]:1
679
680&abs
681(3,4):5
682(-3,4):5
683
684&arg
685[2,0]:0
686[-2,0]:pi
687
688&~
689(4,5):(4,-5)
690(-3,4):(-3,-4)
691[2,pi/2]:[2,-(pi)/2]
692
693&<
694(3,4):(1,2):0
695(3,4):(3,2):0
696(3,4):(3,8):1
697(4,4):(5,129):1
698
699&==
700(3,4):(4,5):0
701(3,4):(3,5):0
702(3,4):(2,4):0
703(3,4):(3,4):1
704
705&sqrt
706-9:(0,3)
707(-100,0):(0,10)
708(16,-30):(5,-3)
709
710&_stringify_cartesian
711(-100,0):"-100"
712(0,1):"i"
713(4,-3):"4-3i"
714(4,0):"4"
715(-4,0):"-4"
716(-2,4):"-2+4i"
717(-2,-1):"-2-i"
718
719&_stringify_polar
720[-1, 0]:"[1,pi]"
721[1, pi/3]:"[1,pi/3]"
722[6, -2*pi/3]:"[6,-2pi/3]"
723[0.5, -9*pi/11]:"[0.5,-9pi/11]"
724[1, 0.5]:"[1, 0.5]"
725
726{ (4,3); [3,2]; (-3,4); (0,2); [2,1] }
727
728|'z + ~z':'2*Re(z)'
729|'z - ~z':'2*i*Im(z)'
730|'z * ~z':'abs(z) * abs(z)'
731
732{ (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
733
734|'(root(z, 4))[1] ** 4':'z'
735|'(root(z, 5))[3] ** 5':'z'
736|'(root(z, 8))[7] ** 8':'z'
737|'(root(z, 8, 0)) ** 8':'z'
738|'(root(z, 8, 7)) ** 8':'z'
739|'abs(z)':'r'
740|'acot(z)':'acotan(z)'
741|'acsc(z)':'acosec(z)'
742|'acsc(z)':'asin(1 / z)'
743|'asec(z)':'acos(1 / z)'
744|'cbrt(z)':'cbrt(r) * exp(i * t/3)'
745|'cos(acos(z))':'z'
746|'addsq(cos(z), sin(z))':1
747|'cos(z)':'cosh(i*z)'
748|'subsq(cosh(z), sinh(z))':1
749|'cot(acot(z))':'z'
750|'cot(z)':'1 / tan(z)'
751|'cot(z)':'cotan(z)'
752|'csc(acsc(z))':'z'
753|'csc(z)':'1 / sin(z)'
754|'csc(z)':'cosec(z)'
755|'exp(log(z))':'z'
756|'exp(z)':'exp(a) * exp(i * b)'
757|'ln(z)':'log(z)'
758|'log(exp(z))':'z'
759|'log(z)':'log(r) + i*t'
760|'log10(z)':'log(z) / log(10)'
761|'logn(z, 2)':'log(z) / log(2)'
762|'logn(z, 3)':'log(z) / log(3)'
763|'sec(asec(z))':'z'
764|'sec(z)':'1 / cos(z)'
765|'sin(asin(z))':'z'
766|'sin(i * z)':'i * sinh(z)'
767|'sqrt(z) * sqrt(z)':'z'
768|'sqrt(z)':'sqrt(r) * exp(i * t/2)'
769|'tan(atan(z))':'z'
770|'z**z':'exp(z * log(z))'
771
772{ (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
773
774|'cosh(acosh(z))':'z'
775|'coth(acoth(z))':'z'
776|'coth(z)':'1 / tanh(z)'
777|'coth(z)':'cotanh(z)'
778|'csch(acsch(z))':'z'
779|'csch(z)':'1 / sinh(z)'
780|'csch(z)':'cosech(z)'
781|'sech(asech(z))':'z'
782|'sech(z)':'1 / cosh(z)'
783|'sinh(asinh(z))':'z'
784|'tanh(atanh(z))':'z'
785
786{ (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
787
788|'acos(cos(z)) ** 2':'z * z'
789|'acosh(cosh(z)) ** 2':'z * z'
790|'acoth(z)':'acotanh(z)'
791|'acoth(z)':'atanh(1 / z)'
792|'acsch(z)':'acosech(z)'
793|'acsch(z)':'asinh(1 / z)'
794|'asech(z)':'acosh(1 / z)'
795|'asin(sin(z))':'z'
796|'asinh(sinh(z))':'z'
797|'atan(tan(z))':'z'
798|'atanh(tanh(z))':'z'
799
800&log
801(-2.0,0):(   0.69314718055995,  3.14159265358979)
802(-1.0,0):(   0               ,  3.14159265358979)
803(-0.5,0):(  -0.69314718055995,  3.14159265358979)
804( 0.5,0):(  -0.69314718055995,  0               )
805( 1.0,0):(   0               ,  0               )
806( 2.0,0):(   0.69314718055995,  0               )
807
808&log
809( 2, 3):(    1.28247467873077,  0.98279372324733)
810(-2, 3):(    1.28247467873077,  2.15879893034246)
811(-2,-3):(    1.28247467873077, -2.15879893034246)
812( 2,-3):(    1.28247467873077, -0.98279372324733)
813
814&sin
815(-2.0,0):(  -0.90929742682568,  0               )
816(-1.0,0):(  -0.84147098480790,  0               )
817(-0.5,0):(  -0.47942553860420,  0               )
818( 0.0,0):(   0               ,  0               )
819( 0.5,0):(   0.47942553860420,  0               )
820( 1.0,0):(   0.84147098480790,  0               )
821( 2.0,0):(   0.90929742682568,  0               )
822
823&sin
824( 2, 3):(  9.15449914691143, -4.16890695996656)
825(-2, 3):( -9.15449914691143, -4.16890695996656)
826(-2,-3):( -9.15449914691143,  4.16890695996656)
827( 2,-3):(  9.15449914691143,  4.16890695996656)
828
829&cos
830(-2.0,0):(  -0.41614683654714,  0               )
831(-1.0,0):(   0.54030230586814,  0               )
832(-0.5,0):(   0.87758256189037,  0               )
833( 0.0,0):(   1               ,  0               )
834( 0.5,0):(   0.87758256189037,  0               )
835( 1.0,0):(   0.54030230586814,  0               )
836( 2.0,0):(  -0.41614683654714,  0               )
837
838&cos
839( 2, 3):( -4.18962569096881, -9.10922789375534)
840(-2, 3):( -4.18962569096881,  9.10922789375534)
841(-2,-3):( -4.18962569096881, -9.10922789375534)
842( 2,-3):( -4.18962569096881,  9.10922789375534)
843
844&tan
845(-2.0,0):(   2.18503986326152,  0               )
846(-1.0,0):(  -1.55740772465490,  0               )
847(-0.5,0):(  -0.54630248984379,  0               )
848( 0.0,0):(   0               ,  0               )
849( 0.5,0):(   0.54630248984379,  0               )
850( 1.0,0):(   1.55740772465490,  0               )
851( 2.0,0):(  -2.18503986326152,  0               )
852
853&tan
854( 2, 3):( -0.00376402564150,  1.00323862735361)
855(-2, 3):(  0.00376402564150,  1.00323862735361)
856(-2,-3):(  0.00376402564150, -1.00323862735361)
857( 2,-3):( -0.00376402564150, -1.00323862735361)
858
859&sec
860(-2.0,0):(  -2.40299796172238,  0               )
861(-1.0,0):(   1.85081571768093,  0               )
862(-0.5,0):(   1.13949392732455,  0               )
863( 0.0,0):(   1               ,  0               )
864( 0.5,0):(   1.13949392732455,  0               )
865( 1.0,0):(   1.85081571768093,  0               )
866( 2.0,0):(  -2.40299796172238,  0               )
867
868&sec
869( 2, 3):( -0.04167496441114,  0.09061113719624)
870(-2, 3):( -0.04167496441114, -0.09061113719624)
871(-2,-3):( -0.04167496441114,  0.09061113719624)
872( 2,-3):( -0.04167496441114, -0.09061113719624)
873
874&csc
875(-2.0,0):(  -1.09975017029462,  0               )
876(-1.0,0):(  -1.18839510577812,  0               )
877(-0.5,0):(  -2.08582964293349,  0               )
878( 0.5,0):(   2.08582964293349,  0               )
879( 1.0,0):(   1.18839510577812,  0               )
880( 2.0,0):(   1.09975017029462,  0               )
881
882&csc
883( 2, 3):(  0.09047320975321,  0.04120098628857)
884(-2, 3):( -0.09047320975321,  0.04120098628857)
885(-2,-3):( -0.09047320975321, -0.04120098628857)
886( 2,-3):(  0.09047320975321, -0.04120098628857)
887
888&cot
889(-2.0,0):(   0.45765755436029,  0               )
890(-1.0,0):(  -0.64209261593433,  0               )
891(-0.5,0):(  -1.83048772171245,  0               )
892( 0.5,0):(   1.83048772171245,  0               )
893( 1.0,0):(   0.64209261593433,  0               )
894( 2.0,0):(  -0.45765755436029,  0               )
895
896&cot
897( 2, 3):( -0.00373971037634, -0.99675779656936)
898(-2, 3):(  0.00373971037634, -0.99675779656936)
899(-2,-3):(  0.00373971037634,  0.99675779656936)
900( 2,-3):( -0.00373971037634,  0.99675779656936)
901
902&asin
903(-2.0,0):(  -1.57079632679490,  1.31695789692482)
904(-1.0,0):(  -1.57079632679490,  0               )
905(-0.5,0):(  -0.52359877559830,  0               )
906( 0.0,0):(   0               ,  0               )
907( 0.5,0):(   0.52359877559830,  0               )
908( 1.0,0):(   1.57079632679490,  0               )
909( 2.0,0):(   1.57079632679490, -1.31695789692482)
910
911&asin
912( 2, 3):(  0.57065278432110,  1.98338702991654)
913(-2, 3):( -0.57065278432110,  1.98338702991654)
914(-2,-3):( -0.57065278432110, -1.98338702991654)
915( 2,-3):(  0.57065278432110, -1.98338702991654)
916
917&acos
918(-2.0,0):(   3.14159265358979, -1.31695789692482)
919(-1.0,0):(   3.14159265358979,  0               )
920(-0.5,0):(   2.09439510239320,  0               )
921( 0.0,0):(   1.57079632679490,  0               )
922( 0.5,0):(   1.04719755119660,  0               )
923( 1.0,0):(   0               ,  0               )
924( 2.0,0):(   0               ,  1.31695789692482)
925
926&acos
927( 2, 3):(  1.00014354247380, -1.98338702991654)
928(-2, 3):(  2.14144911111600, -1.98338702991654)
929(-2,-3):(  2.14144911111600,  1.98338702991654)
930( 2,-3):(  1.00014354247380,  1.98338702991654)
931
932&atan
933(-2.0,0):(  -1.10714871779409,  0               )
934(-1.0,0):(  -0.78539816339745,  0               )
935(-0.5,0):(  -0.46364760900081,  0               )
936( 0.0,0):(   0               ,  0               )
937( 0.5,0):(   0.46364760900081,  0               )
938( 1.0,0):(   0.78539816339745,  0               )
939( 2.0,0):(   1.10714871779409,  0               )
940
941&atan
942( 2, 3):(  1.40992104959658,  0.22907268296854)
943(-2, 3):( -1.40992104959658,  0.22907268296854)
944(-2,-3):( -1.40992104959658, -0.22907268296854)
945( 2,-3):(  1.40992104959658, -0.22907268296854)
946
947&asec
948(-2.0,0):(   2.09439510239320,  0               )
949(-1.0,0):(   3.14159265358979,  0               )
950(-0.5,0):(   3.14159265358979, -1.31695789692482)
951( 0.5,0):(   0               ,  1.31695789692482)
952( 1.0,0):(   0               ,  0               )
953( 2.0,0):(   1.04719755119660,  0               )
954
955&asec
956( 2, 3):(  1.42041072246703,  0.23133469857397)
957(-2, 3):(  1.72118193112276,  0.23133469857397)
958(-2,-3):(  1.72118193112276, -0.23133469857397)
959( 2,-3):(  1.42041072246703, -0.23133469857397)
960
961&acsc
962(-2.0,0):(  -0.52359877559830,  0               )
963(-1.0,0):(  -1.57079632679490,  0               )
964(-0.5,0):(  -1.57079632679490,  1.31695789692482)
965( 0.5,0):(   1.57079632679490, -1.31695789692482)
966( 1.0,0):(   1.57079632679490,  0               )
967( 2.0,0):(   0.52359877559830,  0               )
968
969&acsc
970( 2, 3):(  0.15038560432786, -0.23133469857397)
971(-2, 3):( -0.15038560432786, -0.23133469857397)
972(-2,-3):( -0.15038560432786,  0.23133469857397)
973( 2,-3):(  0.15038560432786,  0.23133469857397)
974
975&acot
976(-2.0,0):(  -0.46364760900081,  0               )
977(-1.0,0):(  -0.78539816339745,  0               )
978(-0.5,0):(  -1.10714871779409,  0               )
979( 0.5,0):(   1.10714871779409,  0               )
980( 1.0,0):(   0.78539816339745,  0               )
981( 2.0,0):(   0.46364760900081,  0               )
982
983&acot
984( 2, 3):(  0.16087527719832, -0.22907268296854)
985(-2, 3):( -0.16087527719832, -0.22907268296854)
986(-2,-3):( -0.16087527719832,  0.22907268296854)
987( 2,-3):(  0.16087527719832,  0.22907268296854)
988
989&sinh
990(-2.0,0):(  -3.62686040784702,  0               )
991(-1.0,0):(  -1.17520119364380,  0               )
992(-0.5,0):(  -0.52109530549375,  0               )
993( 0.0,0):(   0               ,  0               )
994( 0.5,0):(   0.52109530549375,  0               )
995( 1.0,0):(   1.17520119364380,  0               )
996( 2.0,0):(   3.62686040784702,  0               )
997
998&sinh
999( 2, 3):( -3.59056458998578,  0.53092108624852)
1000(-2, 3):(  3.59056458998578,  0.53092108624852)
1001(-2,-3):(  3.59056458998578, -0.53092108624852)
1002( 2,-3):( -3.59056458998578, -0.53092108624852)
1003
1004&cosh
1005(-2.0,0):(   3.76219569108363,  0               )
1006(-1.0,0):(   1.54308063481524,  0               )
1007(-0.5,0):(   1.12762596520638,  0               )
1008( 0.0,0):(   1               ,  0               )
1009( 0.5,0):(   1.12762596520638,  0               )
1010( 1.0,0):(   1.54308063481524,  0               )
1011( 2.0,0):(   3.76219569108363,  0               )
1012
1013&cosh
1014( 2, 3):( -3.72454550491532,  0.51182256998738)
1015(-2, 3):( -3.72454550491532, -0.51182256998738)
1016(-2,-3):( -3.72454550491532,  0.51182256998738)
1017( 2,-3):( -3.72454550491532, -0.51182256998738)
1018
1019&tanh
1020(-2.0,0):(  -0.96402758007582,  0               )
1021(-1.0,0):(  -0.76159415595576,  0               )
1022(-0.5,0):(  -0.46211715726001,  0               )
1023( 0.0,0):(   0               ,  0               )
1024( 0.5,0):(   0.46211715726001,  0               )
1025( 1.0,0):(   0.76159415595576,  0               )
1026( 2.0,0):(   0.96402758007582,  0               )
1027
1028&tanh
1029( 2, 3):(  0.96538587902213, -0.00988437503832)
1030(-2, 3):( -0.96538587902213, -0.00988437503832)
1031(-2,-3):( -0.96538587902213,  0.00988437503832)
1032( 2,-3):(  0.96538587902213,  0.00988437503832)
1033
1034&sech
1035(-2.0,0):(   0.26580222883408,  0               )
1036(-1.0,0):(   0.64805427366389,  0               )
1037(-0.5,0):(   0.88681888397007,  0               )
1038( 0.0,0):(   1               ,  0               )
1039( 0.5,0):(   0.88681888397007,  0               )
1040( 1.0,0):(   0.64805427366389,  0               )
1041( 2.0,0):(   0.26580222883408,  0               )
1042
1043&sech
1044( 2, 3):( -0.26351297515839, -0.03621163655877)
1045(-2, 3):( -0.26351297515839,  0.03621163655877)
1046(-2,-3):( -0.26351297515839, -0.03621163655877)
1047( 2,-3):( -0.26351297515839,  0.03621163655877)
1048
1049&csch
1050(-2.0,0):(  -0.27572056477178,  0               )
1051(-1.0,0):(  -0.85091812823932,  0               )
1052(-0.5,0):(  -1.91903475133494,  0               )
1053( 0.5,0):(   1.91903475133494,  0               )
1054( 1.0,0):(   0.85091812823932,  0               )
1055( 2.0,0):(   0.27572056477178,  0               )
1056
1057&csch
1058( 2, 3):( -0.27254866146294, -0.04030057885689)
1059(-2, 3):(  0.27254866146294, -0.04030057885689)
1060(-2,-3):(  0.27254866146294,  0.04030057885689)
1061( 2,-3):( -0.27254866146294,  0.04030057885689)
1062
1063&coth
1064(-2.0,0):(  -1.03731472072755,  0               )
1065(-1.0,0):(  -1.31303528549933,  0               )
1066(-0.5,0):(  -2.16395341373865,  0               )
1067( 0.5,0):(   2.16395341373865,  0               )
1068( 1.0,0):(   1.31303528549933,  0               )
1069( 2.0,0):(   1.03731472072755,  0               )
1070
1071&coth
1072( 2, 3):(  1.03574663776500,  0.01060478347034)
1073(-2, 3):( -1.03574663776500,  0.01060478347034)
1074(-2,-3):( -1.03574663776500, -0.01060478347034)
1075( 2,-3):(  1.03574663776500, -0.01060478347034)
1076
1077&asinh
1078(-2.0,0):(  -1.44363547517881,  0               )
1079(-1.0,0):(  -0.88137358701954,  0               )
1080(-0.5,0):(  -0.48121182505960,  0               )
1081( 0.0,0):(   0               ,  0               )
1082( 0.5,0):(   0.48121182505960,  0               )
1083( 1.0,0):(   0.88137358701954,  0               )
1084( 2.0,0):(   1.44363547517881,  0               )
1085
1086&asinh
1087( 2, 3):(  1.96863792579310,  0.96465850440760)
1088(-2, 3):( -1.96863792579310,  0.96465850440761)
1089(-2,-3):( -1.96863792579310, -0.96465850440761)
1090( 2,-3):(  1.96863792579310, -0.96465850440760)
1091
1092&acosh
1093(-2.0,0):(   1.31695789692482,  3.14159265358979)
1094(-1.0,0):(   0,                 3.14159265358979)
1095(-0.5,0):(   0,                 2.09439510239320)
1096( 0.0,0):(   0,                 1.57079632679490)
1097( 0.5,0):(   0,                 1.04719755119660)
1098( 1.0,0):(   0               ,  0               )
1099( 2.0,0):(   1.31695789692482,  0               )
1100
1101&acosh
1102( 2, 3):(  1.98338702991654,  1.00014354247380)
1103(-2, 3):(  1.98338702991653,  2.14144911111600)
1104(-2,-3):(  1.98338702991653, -2.14144911111600)
1105( 2,-3):(  1.98338702991654, -1.00014354247380)
1106
1107&atanh
1108(-2.0,0):(  -0.54930614433405,  1.57079632679490)
1109(-0.5,0):(  -0.54930614433405,  0               )
1110( 0.0,0):(   0               ,  0               )
1111( 0.5,0):(   0.54930614433405,  0               )
1112( 2.0,0):(   0.54930614433405,  1.57079632679490)
1113
1114&atanh
1115( 2, 3):(  0.14694666622553,  1.33897252229449)
1116(-2, 3):( -0.14694666622553,  1.33897252229449)
1117(-2,-3):( -0.14694666622553, -1.33897252229449)
1118( 2,-3):(  0.14694666622553, -1.33897252229449)
1119
1120&asech
1121(-2.0,0):(   0               , 2.09439510239320)
1122(-1.0,0):(   0               , 3.14159265358979)
1123(-0.5,0):(   1.31695789692482, 3.14159265358979)
1124( 0.5,0):(   1.31695789692482, 0               )
1125( 1.0,0):(   0               , 0               )
1126( 2.0,0):(   0               , 1.04719755119660)
1127
1128&asech
1129( 2, 3):(  0.23133469857397, -1.42041072246703)
1130(-2, 3):(  0.23133469857397, -1.72118193112276)
1131(-2,-3):(  0.23133469857397,  1.72118193112276)
1132( 2,-3):(  0.23133469857397,  1.42041072246703)
1133
1134&acsch
1135(-2.0,0):(  -0.48121182505960, 0               )
1136(-1.0,0):(  -0.88137358701954, 0               )
1137(-0.5,0):(  -1.44363547517881, 0               )
1138( 0.5,0):(   1.44363547517881, 0               )
1139( 1.0,0):(   0.88137358701954, 0               )
1140( 2.0,0):(   0.48121182505960, 0               )
1141
1142&acsch
1143( 2, 3):(  0.15735549884499, -0.22996290237721)
1144(-2, 3):( -0.15735549884499, -0.22996290237721)
1145(-2,-3):( -0.15735549884499,  0.22996290237721)
1146( 2,-3):(  0.15735549884499,  0.22996290237721)
1147
1148&acoth
1149(-2.0,0):(  -0.54930614433405, 0               )
1150(-0.5,0):(  -0.54930614433405, 1.57079632679490)
1151( 0.5,0):(   0.54930614433405, 1.57079632679490)
1152( 2.0,0):(   0.54930614433405, 0               )
1153
1154&acoth
1155( 2, 3):(  0.14694666622553, -0.23182380450040)
1156(-2, 3):( -0.14694666622553, -0.23182380450040)
1157(-2,-3):( -0.14694666622553,  0.23182380450040)
1158( 2,-3):(  0.14694666622553,  0.23182380450040)
1159
1160# eof
1161