3 # $RCSfile: complex.t,v $
5 # Regression tests for the Math::Complex pacakge
6 # -- Raphael Manfredi since Sep 1996
7 # -- Jarkko Hietaniemi since Mar 1997
8 # -- Daniel S. Lewart since Sep 1997
17 use vars qw($VERSION);
21 my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val);
26 'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10, $z0,$z1,$z2);' .
31 if ($^O eq 'unicos') { # For some reason root() produces very inaccurate
32 $eps = 1e-10; # results in Cray UNICOS, and occasionally also
33 } # cos(), sin(), cosh(), sinh(). The division
34 # of doubles is the current suspect.
38 next if $_ eq '' || /^\#/;
40 $test_set = 0; # Assume not a test over a set of values
46 set($1, \@set, \@val);
50 $test_set = 1; # Requests we loop over the set...
52 my @args = split(/:/);
55 for ($i = 0; $i < @set; $i++) {
58 # textual value as found in set definition
60 test($zvalue, $target, @args);
63 test($op, undef, @args);
73 push(@script, <<'EOT');
78 print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
79 print 'not ' unless Re($z) == 2 and Im($z) == 3;
81 push(@script, qq(print "ok $test\\n"}\n));
84 push(@script, <<'EOT');
88 print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
89 print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and
90 (arg($z) - pi / 4 ) < $eps and
91 (Re($z) - 3 ) < $eps and
94 push(@script, qq(print "ok $test\\n"}\n));
97 push(@script, <<'EOT');
100 $z->arg(-3 / 4 * pi);
101 print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n";
102 print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and
103 (abs($z) - sqrt(2) ) < $eps and
104 (Re($z) + 1 ) < $eps and
105 (Im($z) + 1 ) < $eps;
107 push(@script, qq(print "ok $test\\n"}\n));
114 my $pi = cplx(pi, 0);
115 my $pii = cplx(0, pi);
116 my $pip2 = cplx(pi/2, 0);
117 my $zero = cplx(0, 0);
120 push(@script, $constants);
123 # test the divbyzeros
128 push(@script, <<EOT);
130 (\$bad) = (\$@ =~ /(.+)/);
131 print "# $test op = $op divbyzero? \$bad...\n";
132 print 'not ' unless (\$@ =~ /Division by zero/);
134 push(@script, qq(print "ok $test\\n";\n));
138 # test the logofzeros
143 push(@script, <<EOT);
145 (\$bad) = (\$@ =~ /(.+)/);
146 print "# $test op = $op logofzero? \$bad...\n";
147 print 'not ' unless (\$@ =~ /Logarithm of zero/);
149 push(@script, qq(print "ok $test\\n";\n));
157 # 'acoth(-1)', # Log of zero.
165 # 'atanh(-1)', # Log of zero.
186 push(@script, <<EOT);
188 (\$bad) = (\$@ =~ /(.+)/);
189 print "# $test op = $op badroot? \$bad...\n";
190 print 'not ' unless (\$@ =~ /root rank must be/);
192 push(@script, qq(print "ok $test\\n";\n));
196 test_broot(qw(-3 -2.1 0 0.99));
198 sub test_display_format {
201 print "# package display_format cartesian?\n";
202 print "not " unless Math::Complex->display_format eq 'cartesian';
207 my \$j = (root(1,3))[1];
209 \$j->display_format('polar');
214 print "# j display_format polar?\n";
215 print "not " unless \$j->display_format eq 'polar';
222 print "not " unless "\$j" eq "[1,2pi/3]";
227 %display_format = \$j->display_format;
232 print "# display_format{style} polar?\n";
233 print "not " unless \$display_format{style} eq 'polar';
239 print "# keys %display_format == 2?\n";
240 print "not " unless keys %display_format == 2;
243 \$j->display_format('style' => 'cartesian', 'format' => '%.5f');
249 print "not " unless "\$j" eq "-0.50000+0.86603i";
252 %display_format = \$j->display_format;
257 print "# display_format{format} %.5f?\n";
258 print "not " unless \$display_format{format} eq '%.5f';
264 print "# keys %display_format == 3?\n";
265 print "not " unless keys %display_format == 3;
268 \$j->display_format('format' => undef);
274 print "not " unless "\$j" =~ /^-0(?:\\.5(?:0000\\d+)?|\\.49999\\d+)\\+0.86602540\\d+i\$/;
277 \$j->display_format('style' => 'polar', 'polar_pretty_print' => 0);
283 print "not " unless "\$j" =~ /^\\[1,2\\.09439510\\d+\\]\$/;
286 \$j->display_format('style' => 'cartesian', 'format' => '(%.5g)');
292 print "not " unless "\$j" eq "(-0.5)+(0.86603)i";
298 print "# j display_format cartesian?\n";
299 print "not " unless \$j->display_format eq 'cartesian';
304 test_display_format();
307 eval join '', @script;
313 push(@script, qq(print "# $op=\n";));
317 my ($op, $z, @args) = @_;
321 $baop = 1 if ($op =~ s/;=$//);
322 for ($i = 0; $i < @args; $i++) {
323 $val = value($args[$i]);
324 push @script, "\$z$i = $val;\n";
327 $args = "'$op'"; # Really the value
328 $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
329 push @script, "\$res = $try; ";
330 push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n";
335 $args = "'$args[0]'";
337 $try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1";
338 $args = "'$args[0]', '$args[1]'";
340 push @script, "\$res = $try; ";
341 push @script, "check($test, '$try', \$res, \$z$#args, $args);\n";
342 if (@args > 2 and $baop) { # binary assignment ops
344 # check the op= works
347 my \$za = cplx(ref \$z0 ? \@{\$z0->cartesian} : (\$z0, 0));
349 my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->cartesian} : (\$z1, 0);
351 my \$zb = cplx(\$z1r, \$z1i);
354 my (\$zbr, \$zbi) = \@{\$zb->cartesian};
356 check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args);
359 # check that the rhs has not changed
360 push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
361 push @script, qq(print "ok $test\\n";\n);
368 my ($set, $setref, $valref) = @_;
371 my @set = split(/;\s*/, $set);
374 for ($i = 0; $i < @set; $i++) {
375 push(@{$valref}, $set[$i]);
376 my $val = value($set[$i]);
377 push @script, "\$s$i = $val;\n";
378 push @{$setref}, "\$s$i";
384 if (/^\s*\((.*),(.*)\)/) {
385 return "cplx($1,$2)";
387 elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
390 elsif (/^\s*\[(.*),(.*)\]/) {
391 return "cplxe($1,$2)";
393 elsif (/^\s*'(.*)'/) {
395 $ex =~ s/\bz\b/$target/g;
396 $ex =~ s/\br\b/abs($target)/g;
397 $ex =~ s/\bt\b/arg($target)/g;
398 $ex =~ s/\ba\b/Re($target)/g;
399 $ex =~ s/\bb\b/Im($target)/g;
402 elsif (/^\s*"(.*)"/) {
409 my ($test, $try, $got, $expected, @z) = @_;
413 if ("$got" eq "$expected"
415 ($expected =~ /^-?\d/ && $got == $expected)
417 (abs($got - $expected) < $eps)
421 print "not ok $test\n";
422 my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]";
423 print "# '$try' expected: '$expected' got: '$got' for $args\n";
429 return ($z1 + i*$z2) * ($z1 - i*$z2);
434 return ($z1 + $z2) * ($z1 - $z2);
464 [2,2*pi/3]:(1,0):[2,2*pi/3]
476 [4,pi]:[2,pi/2]:[2,pi/2]
477 [2,pi/2]:[4,pi]:[0.5,-(pi)/2]
482 (2,3):(4,0):(-119,-120)
544 [6, -2*pi/3]:"[6,-2pi/3]"
545 [0.5, -9*pi/11]:"[0.5,-9pi/11]"
547 { (4,3); [3,2]; (-3,4); (0,2); [2,1] }
550 |'z - ~z':'2*i*Im(z)'
551 |'z * ~z':'abs(z) * abs(z)'
553 { (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
555 |'(root(z, 4))[1] ** 4':'z'
556 |'(root(z, 5))[3] ** 5':'z'
557 |'(root(z, 8))[7] ** 8':'z'
559 |'acot(z)':'acotan(z)'
560 |'acsc(z)':'acosec(z)'
561 |'acsc(z)':'asin(1 / z)'
562 |'asec(z)':'acos(1 / z)'
563 |'cbrt(z)':'cbrt(r) * exp(i * t/3)'
565 |'addsq(cos(z), sin(z))':1
566 |'cos(z)':'cosh(i*z)'
567 |'subsq(cosh(z), sinh(z))':1
569 |'cot(z)':'1 / tan(z)'
572 |'csc(z)':'1 / sin(z)'
575 |'exp(z)':'exp(a) * exp(i * b)'
578 |'log(z)':'log(r) + i*t'
579 |'log10(z)':'log(z) / log(10)'
580 |'logn(z, 2)':'log(z) / log(2)'
581 |'logn(z, 3)':'log(z) / log(3)'
583 |'sec(z)':'1 / cos(z)'
585 |'sin(i * z)':'i * sinh(z)'
586 |'sqrt(z) * sqrt(z)':'z'
587 |'sqrt(z)':'sqrt(r) * exp(i * t/2)'
589 |'z**z':'exp(z * log(z))'
591 { (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
593 |'cosh(acosh(z))':'z'
594 |'coth(acoth(z))':'z'
595 |'coth(z)':'1 / tanh(z)'
596 |'coth(z)':'cotanh(z)'
597 |'csch(acsch(z))':'z'
598 |'csch(z)':'1 / sinh(z)'
599 |'csch(z)':'cosech(z)'
600 |'sech(asech(z))':'z'
601 |'sech(z)':'1 / cosh(z)'
602 |'sinh(asinh(z))':'z'
603 |'tanh(atanh(z))':'z'
605 { (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
607 |'acos(cos(z)) ** 2':'z * z'
608 |'acosh(cosh(z)) ** 2':'z * z'
609 |'acoth(z)':'acotanh(z)'
610 |'acoth(z)':'atanh(1 / z)'
611 |'acsch(z)':'acosech(z)'
612 |'acsch(z)':'asinh(1 / z)'
613 |'asech(z)':'acosh(1 / z)'
615 |'asinh(sinh(z))':'z'
617 |'atanh(tanh(z))':'z'
620 (-2.0,0):( 0.69314718055995, 3.14159265358979)
621 (-1.0,0):( 0 , 3.14159265358979)
622 (-0.5,0):( -0.69314718055995, 3.14159265358979)
623 ( 0.5,0):( -0.69314718055995, 0 )
625 ( 2.0,0):( 0.69314718055995, 0 )
628 ( 2, 3):( 1.28247467873077, 0.98279372324733)
629 (-2, 3):( 1.28247467873077, 2.15879893034246)
630 (-2,-3):( 1.28247467873077, -2.15879893034246)
631 ( 2,-3):( 1.28247467873077, -0.98279372324733)
634 (-2.0,0):( -0.90929742682568, 0 )
635 (-1.0,0):( -0.84147098480790, 0 )
636 (-0.5,0):( -0.47942553860420, 0 )
638 ( 0.5,0):( 0.47942553860420, 0 )
639 ( 1.0,0):( 0.84147098480790, 0 )
640 ( 2.0,0):( 0.90929742682568, 0 )
643 ( 2, 3):( 9.15449914691143, -4.16890695996656)
644 (-2, 3):( -9.15449914691143, -4.16890695996656)
645 (-2,-3):( -9.15449914691143, 4.16890695996656)
646 ( 2,-3):( 9.15449914691143, 4.16890695996656)
649 (-2.0,0):( -0.41614683654714, 0 )
650 (-1.0,0):( 0.54030230586814, 0 )
651 (-0.5,0):( 0.87758256189037, 0 )
653 ( 0.5,0):( 0.87758256189037, 0 )
654 ( 1.0,0):( 0.54030230586814, 0 )
655 ( 2.0,0):( -0.41614683654714, 0 )
658 ( 2, 3):( -4.18962569096881, -9.10922789375534)
659 (-2, 3):( -4.18962569096881, 9.10922789375534)
660 (-2,-3):( -4.18962569096881, -9.10922789375534)
661 ( 2,-3):( -4.18962569096881, 9.10922789375534)
664 (-2.0,0):( 2.18503986326152, 0 )
665 (-1.0,0):( -1.55740772465490, 0 )
666 (-0.5,0):( -0.54630248984379, 0 )
668 ( 0.5,0):( 0.54630248984379, 0 )
669 ( 1.0,0):( 1.55740772465490, 0 )
670 ( 2.0,0):( -2.18503986326152, 0 )
673 ( 2, 3):( -0.00376402564150, 1.00323862735361)
674 (-2, 3):( 0.00376402564150, 1.00323862735361)
675 (-2,-3):( 0.00376402564150, -1.00323862735361)
676 ( 2,-3):( -0.00376402564150, -1.00323862735361)
679 (-2.0,0):( -2.40299796172238, 0 )
680 (-1.0,0):( 1.85081571768093, 0 )
681 (-0.5,0):( 1.13949392732455, 0 )
683 ( 0.5,0):( 1.13949392732455, 0 )
684 ( 1.0,0):( 1.85081571768093, 0 )
685 ( 2.0,0):( -2.40299796172238, 0 )
688 ( 2, 3):( -0.04167496441114, 0.09061113719624)
689 (-2, 3):( -0.04167496441114, -0.09061113719624)
690 (-2,-3):( -0.04167496441114, 0.09061113719624)
691 ( 2,-3):( -0.04167496441114, -0.09061113719624)
694 (-2.0,0):( -1.09975017029462, 0 )
695 (-1.0,0):( -1.18839510577812, 0 )
696 (-0.5,0):( -2.08582964293349, 0 )
697 ( 0.5,0):( 2.08582964293349, 0 )
698 ( 1.0,0):( 1.18839510577812, 0 )
699 ( 2.0,0):( 1.09975017029462, 0 )
702 ( 2, 3):( 0.09047320975321, 0.04120098628857)
703 (-2, 3):( -0.09047320975321, 0.04120098628857)
704 (-2,-3):( -0.09047320975321, -0.04120098628857)
705 ( 2,-3):( 0.09047320975321, -0.04120098628857)
708 (-2.0,0):( 0.45765755436029, 0 )
709 (-1.0,0):( -0.64209261593433, 0 )
710 (-0.5,0):( -1.83048772171245, 0 )
711 ( 0.5,0):( 1.83048772171245, 0 )
712 ( 1.0,0):( 0.64209261593433, 0 )
713 ( 2.0,0):( -0.45765755436029, 0 )
716 ( 2, 3):( -0.00373971037634, -0.99675779656936)
717 (-2, 3):( 0.00373971037634, -0.99675779656936)
718 (-2,-3):( 0.00373971037634, 0.99675779656936)
719 ( 2,-3):( -0.00373971037634, 0.99675779656936)
722 (-2.0,0):( -1.57079632679490, 1.31695789692482)
723 (-1.0,0):( -1.57079632679490, 0 )
724 (-0.5,0):( -0.52359877559830, 0 )
726 ( 0.5,0):( 0.52359877559830, 0 )
727 ( 1.0,0):( 1.57079632679490, 0 )
728 ( 2.0,0):( 1.57079632679490, -1.31695789692482)
731 ( 2, 3):( 0.57065278432110, 1.98338702991654)
732 (-2, 3):( -0.57065278432110, 1.98338702991654)
733 (-2,-3):( -0.57065278432110, -1.98338702991654)
734 ( 2,-3):( 0.57065278432110, -1.98338702991654)
737 (-2.0,0):( 3.14159265358979, -1.31695789692482)
738 (-1.0,0):( 3.14159265358979, 0 )
739 (-0.5,0):( 2.09439510239320, 0 )
740 ( 0.0,0):( 1.57079632679490, 0 )
741 ( 0.5,0):( 1.04719755119660, 0 )
743 ( 2.0,0):( 0 , 1.31695789692482)
746 ( 2, 3):( 1.00014354247380, -1.98338702991654)
747 (-2, 3):( 2.14144911111600, -1.98338702991654)
748 (-2,-3):( 2.14144911111600, 1.98338702991654)
749 ( 2,-3):( 1.00014354247380, 1.98338702991654)
752 (-2.0,0):( -1.10714871779409, 0 )
753 (-1.0,0):( -0.78539816339745, 0 )
754 (-0.5,0):( -0.46364760900081, 0 )
756 ( 0.5,0):( 0.46364760900081, 0 )
757 ( 1.0,0):( 0.78539816339745, 0 )
758 ( 2.0,0):( 1.10714871779409, 0 )
761 ( 2, 3):( 1.40992104959658, 0.22907268296854)
762 (-2, 3):( -1.40992104959658, 0.22907268296854)
763 (-2,-3):( -1.40992104959658, -0.22907268296854)
764 ( 2,-3):( 1.40992104959658, -0.22907268296854)
767 (-2.0,0):( 2.09439510239320, 0 )
768 (-1.0,0):( 3.14159265358979, 0 )
769 (-0.5,0):( 3.14159265358979, -1.31695789692482)
770 ( 0.5,0):( 0 , 1.31695789692482)
772 ( 2.0,0):( 1.04719755119660, 0 )
775 ( 2, 3):( 1.42041072246703, 0.23133469857397)
776 (-2, 3):( 1.72118193112276, 0.23133469857397)
777 (-2,-3):( 1.72118193112276, -0.23133469857397)
778 ( 2,-3):( 1.42041072246703, -0.23133469857397)
781 (-2.0,0):( -0.52359877559830, 0 )
782 (-1.0,0):( -1.57079632679490, 0 )
783 (-0.5,0):( -1.57079632679490, 1.31695789692482)
784 ( 0.5,0):( 1.57079632679490, -1.31695789692482)
785 ( 1.0,0):( 1.57079632679490, 0 )
786 ( 2.0,0):( 0.52359877559830, 0 )
789 ( 2, 3):( 0.15038560432786, -0.23133469857397)
790 (-2, 3):( -0.15038560432786, -0.23133469857397)
791 (-2,-3):( -0.15038560432786, 0.23133469857397)
792 ( 2,-3):( 0.15038560432786, 0.23133469857397)
795 (-2.0,0):( -0.46364760900081, 0 )
796 (-1.0,0):( -0.78539816339745, 0 )
797 (-0.5,0):( -1.10714871779409, 0 )
798 ( 0.5,0):( 1.10714871779409, 0 )
799 ( 1.0,0):( 0.78539816339745, 0 )
800 ( 2.0,0):( 0.46364760900081, 0 )
803 ( 2, 3):( 0.16087527719832, -0.22907268296854)
804 (-2, 3):( -0.16087527719832, -0.22907268296854)
805 (-2,-3):( -0.16087527719832, 0.22907268296854)
806 ( 2,-3):( 0.16087527719832, 0.22907268296854)
809 (-2.0,0):( -3.62686040784702, 0 )
810 (-1.0,0):( -1.17520119364380, 0 )
811 (-0.5,0):( -0.52109530549375, 0 )
813 ( 0.5,0):( 0.52109530549375, 0 )
814 ( 1.0,0):( 1.17520119364380, 0 )
815 ( 2.0,0):( 3.62686040784702, 0 )
818 ( 2, 3):( -3.59056458998578, 0.53092108624852)
819 (-2, 3):( 3.59056458998578, 0.53092108624852)
820 (-2,-3):( 3.59056458998578, -0.53092108624852)
821 ( 2,-3):( -3.59056458998578, -0.53092108624852)
824 (-2.0,0):( 3.76219569108363, 0 )
825 (-1.0,0):( 1.54308063481524, 0 )
826 (-0.5,0):( 1.12762596520638, 0 )
828 ( 0.5,0):( 1.12762596520638, 0 )
829 ( 1.0,0):( 1.54308063481524, 0 )
830 ( 2.0,0):( 3.76219569108363, 0 )
833 ( 2, 3):( -3.72454550491532, 0.51182256998738)
834 (-2, 3):( -3.72454550491532, -0.51182256998738)
835 (-2,-3):( -3.72454550491532, 0.51182256998738)
836 ( 2,-3):( -3.72454550491532, -0.51182256998738)
839 (-2.0,0):( -0.96402758007582, 0 )
840 (-1.0,0):( -0.76159415595576, 0 )
841 (-0.5,0):( -0.46211715726001, 0 )
843 ( 0.5,0):( 0.46211715726001, 0 )
844 ( 1.0,0):( 0.76159415595576, 0 )
845 ( 2.0,0):( 0.96402758007582, 0 )
848 ( 2, 3):( 0.96538587902213, -0.00988437503832)
849 (-2, 3):( -0.96538587902213, -0.00988437503832)
850 (-2,-3):( -0.96538587902213, 0.00988437503832)
851 ( 2,-3):( 0.96538587902213, 0.00988437503832)
854 (-2.0,0):( 0.26580222883408, 0 )
855 (-1.0,0):( 0.64805427366389, 0 )
856 (-0.5,0):( 0.88681888397007, 0 )
858 ( 0.5,0):( 0.88681888397007, 0 )
859 ( 1.0,0):( 0.64805427366389, 0 )
860 ( 2.0,0):( 0.26580222883408, 0 )
863 ( 2, 3):( -0.26351297515839, -0.03621163655877)
864 (-2, 3):( -0.26351297515839, 0.03621163655877)
865 (-2,-3):( -0.26351297515839, -0.03621163655877)
866 ( 2,-3):( -0.26351297515839, 0.03621163655877)
869 (-2.0,0):( -0.27572056477178, 0 )
870 (-1.0,0):( -0.85091812823932, 0 )
871 (-0.5,0):( -1.91903475133494, 0 )
872 ( 0.5,0):( 1.91903475133494, 0 )
873 ( 1.0,0):( 0.85091812823932, 0 )
874 ( 2.0,0):( 0.27572056477178, 0 )
877 ( 2, 3):( -0.27254866146294, -0.04030057885689)
878 (-2, 3):( 0.27254866146294, -0.04030057885689)
879 (-2,-3):( 0.27254866146294, 0.04030057885689)
880 ( 2,-3):( -0.27254866146294, 0.04030057885689)
883 (-2.0,0):( -1.03731472072755, 0 )
884 (-1.0,0):( -1.31303528549933, 0 )
885 (-0.5,0):( -2.16395341373865, 0 )
886 ( 0.5,0):( 2.16395341373865, 0 )
887 ( 1.0,0):( 1.31303528549933, 0 )
888 ( 2.0,0):( 1.03731472072755, 0 )
891 ( 2, 3):( 1.03574663776500, 0.01060478347034)
892 (-2, 3):( -1.03574663776500, 0.01060478347034)
893 (-2,-3):( -1.03574663776500, -0.01060478347034)
894 ( 2,-3):( 1.03574663776500, -0.01060478347034)
897 (-2.0,0):( -1.44363547517881, 0 )
898 (-1.0,0):( -0.88137358701954, 0 )
899 (-0.5,0):( -0.48121182505960, 0 )
901 ( 0.5,0):( 0.48121182505960, 0 )
902 ( 1.0,0):( 0.88137358701954, 0 )
903 ( 2.0,0):( 1.44363547517881, 0 )
906 ( 2, 3):( 1.96863792579310, 0.96465850440760)
907 (-2, 3):( -1.96863792579310, 0.96465850440761)
908 (-2,-3):( -1.96863792579310, -0.96465850440761)
909 ( 2,-3):( 1.96863792579310, -0.96465850440760)
912 (-2.0,0):( 1.31695789692482, 3.14159265358979)
913 (-1.0,0):( 0, 3.14159265358979)
914 (-0.5,0):( 0, 2.09439510239320)
915 ( 0.0,0):( 0, 1.57079632679490)
916 ( 0.5,0):( 0, 1.04719755119660)
918 ( 2.0,0):( 1.31695789692482, 0 )
921 ( 2, 3):( 1.98338702991654, 1.00014354247380)
922 (-2, 3):( 1.98338702991653, 2.14144911111600)
923 (-2,-3):( 1.98338702991653, -2.14144911111600)
924 ( 2,-3):( 1.98338702991654, -1.00014354247380)
927 (-2.0,0):( -0.54930614433405, 1.57079632679490)
928 (-0.5,0):( -0.54930614433405, 0 )
930 ( 0.5,0):( 0.54930614433405, 0 )
931 ( 2.0,0):( 0.54930614433405, 1.57079632679490)
934 ( 2, 3):( 0.14694666622553, 1.33897252229449)
935 (-2, 3):( -0.14694666622553, 1.33897252229449)
936 (-2,-3):( -0.14694666622553, -1.33897252229449)
937 ( 2,-3):( 0.14694666622553, -1.33897252229449)
940 (-2.0,0):( 0 , 2.09439510239320)
941 (-1.0,0):( 0 , 3.14159265358979)
942 (-0.5,0):( 1.31695789692482, 3.14159265358979)
943 ( 0.5,0):( 1.31695789692482, 0 )
945 ( 2.0,0):( 0 , 1.04719755119660)
948 ( 2, 3):( 0.23133469857397, -1.42041072246703)
949 (-2, 3):( 0.23133469857397, -1.72118193112276)
950 (-2,-3):( 0.23133469857397, 1.72118193112276)
951 ( 2,-3):( 0.23133469857397, 1.42041072246703)
954 (-2.0,0):( -0.48121182505960, 0 )
955 (-1.0,0):( -0.88137358701954, 0 )
956 (-0.5,0):( -1.44363547517881, 0 )
957 ( 0.5,0):( 1.44363547517881, 0 )
958 ( 1.0,0):( 0.88137358701954, 0 )
959 ( 2.0,0):( 0.48121182505960, 0 )
962 ( 2, 3):( 0.15735549884499, -0.22996290237721)
963 (-2, 3):( -0.15735549884499, -0.22996290237721)
964 (-2,-3):( -0.15735549884499, 0.22996290237721)
965 ( 2,-3):( 0.15735549884499, 0.22996290237721)
968 (-2.0,0):( -0.54930614433405, 0 )
969 (-0.5,0):( -0.54930614433405, 1.57079632679490)
970 ( 0.5,0):( 0.54930614433405, 1.57079632679490)
971 ( 2.0,0):( 0.54930614433405, 0 )
974 ( 2, 3):( 0.14694666622553, -0.23182380450040)
975 (-2, 3):( -0.14694666622553, -0.23182380450040)
976 (-2,-3):( -0.14694666622553, 0.23182380450040)
977 ( 2,-3):( 0.14694666622553, 0.23182380450040)