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
10 if ($ENV{PERL_CORE}) {
16 use Math::Complex 1.54;
18 use vars qw($VERSION);
22 my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val);
27 'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10,$z0,$z1,$z2);' .
32 if ($^O eq 'unicos') { # For some reason root() produces very inaccurate
33 $eps = 1e-10; # results in Cray UNICOS, and occasionally also
34 } # cos(), sin(), cosh(), sinh(). The division
35 # of doubles is the current suspect.
38 push @script, "{ my \$t=$test; ".q{
39 my $a = Math::Complex->new(1);
42 print "not " unless "$a" eq "3" && "$b" eq "1";
48 next if $_ eq '' || /^\#/;
50 $test_set = 0; # Assume not a test over a set of values
56 set($1, \@set, \@val);
60 $test_set = 1; # Requests we loop over the set...
62 my @args = split(/:/);
65 for ($i = 0; $i < @set; $i++) {
68 # textual value as found in set definition
70 test($zvalue, $target, @args);
73 test($op, undef, @args);
83 push(@script, <<'EOT');
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;
91 push(@script, qq(print "ok $test\\n"}\n));
94 push(@script, <<'EOT');
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;
104 push(@script, qq(print "ok $test\\n"}\n));
107 push(@script, <<'EOT');
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;
117 push(@script, qq(print "ok $test\\n"}\n));
124 my $pi = cplx(pi, 0);
125 my $pii = cplx(0, pi);
126 my $pip2 = cplx(pi/2, 0);
127 my $pip4 = cplx(pi/4, 0);
128 my $zero = cplx(0, 0);
132 push(@script, $constants);
135 # test the divbyzeros
140 push(@script, <<EOT);
142 (\$bad) = (\$@ =~ /(.+)/);
143 print "# $test op = $op divbyzero? \$bad...\n";
144 print 'not ' unless (\$@ =~ /Division by zero/);
146 push(@script, qq(print "ok $test\\n";\n));
150 # test the logofzeros
155 push(@script, <<EOT);
157 (\$bad) = (\$@ =~ /(.+)/);
158 print "# $test op = $op logofzero? \$bad...\n";
159 print 'not ' unless (\$@ =~ /Logarithm of zero/);
161 push(@script, qq(print "ok $test\\n";\n));
169 # 'acoth(-1)', # Log of zero.
177 # 'atanh(-1)', # Log of zero.
183 'atan(cplx(0, 1), cplx(1, 0))',
199 push(@script, <<EOT);
201 (\$bad) = (\$@ =~ /(.+)/);
202 print "# $test op = $op badroot? \$bad...\n";
203 print 'not ' unless (\$@ =~ /root rank must be/);
205 push(@script, qq(print "ok $test\\n";\n));
209 test_broot(qw(-3 -2.1 0 0.99));
211 sub test_display_format {
214 print "# package display_format cartesian?\n";
215 print "not " unless Math::Complex->display_format eq 'cartesian';
220 my \$j = (root(1,3))[1];
222 \$j->display_format('polar');
227 print "# j display_format polar?\n";
228 print "not " unless \$j->display_format eq 'polar';
235 print "not " unless "\$j" eq "[1,2pi/3]";
240 %display_format = \$j->display_format;
245 print "# display_format{style} polar?\n";
246 print "not " unless \$display_format{style} eq 'polar';
252 print "# keys %display_format == 2?\n";
253 print "not " unless keys %display_format == 2;
256 \$j->display_format('style' => 'cartesian', 'format' => '%.5f');
262 print "not " unless "\$j" eq "-0.50000+0.86603i";
265 %display_format = \$j->display_format;
270 print "# display_format{format} %.5f?\n";
271 print "not " unless \$display_format{format} eq '%.5f';
277 print "# keys %display_format == 3?\n";
278 print "not " unless keys %display_format == 3;
281 \$j->display_format('format' => undef);
287 print "not " unless "\$j" =~ /^-0(?:\\.5(?:0000\\d+)?|\\.49999\\d+)\\+0.86602540\\d+i\$/;
290 \$j->display_format('style' => 'polar', 'polar_pretty_print' => 0);
296 print "not " unless "\$j" =~ /^\\[1,2\\.09439510\\d+\\]\$/;
299 \$j->display_format('style' => 'polar', 'format' => "%.4g");
305 print "not " unless "\$j" =~ /^\\[1,2\\.094\\]\$/;
308 \$j->display_format('style' => 'cartesian', 'format' => '(%.5g)');
314 print "not " unless "\$j" eq "(-0.5)+(0.86603)i";
320 print "# j display_format cartesian?\n";
321 print "not " unless \$j->display_format eq 'cartesian';
326 test_display_format();
331 print "# remake 2+3i\n";
333 print "not " unless \$z == Math::Complex->make(2,3);
340 \$z = Math::Complex->make('3i');
341 print "not " unless \$z == cplx(0,3);
347 print "# emake [2,3]\n";
348 \$z = Math::Complex->emake('[2,3]');
349 print "not " unless \$z == cplxe(2,3);
355 print "# make (2,3)\n";
356 \$z = Math::Complex->make('(2,3)');
357 print "not " unless \$z == cplx(2,3);
363 print "# emake [2,3pi/8]\n";
364 \$z = Math::Complex->emake('[2,3pi/8]');
365 print "not " unless \$z == cplxe(2,3*\$pi/8);
371 print "# emake [2]\n";
372 \$z = Math::Complex->emake('[2]');
373 print "not " unless \$z == cplxe(2);
379 push @script, <<'EOS';
381 print "# cplx, cplxe, make, emake without arguments\n";
387 print ((\$z0->Re() == 0) ? "ok $test\n" : "not ok $test\n");
392 print ((\$z0->Im() == 0) ? "ok $test\n" : "not ok $test\n");
398 print ((\$z1->rho() == 0) ? "ok $test\n" : "not ok $test\n");
403 print ((\$z1->theta() == 0) ? "ok $test\n" : "not ok $test\n");
408 my \$z2 = Math::Complex->make();
409 print ((\$z2->Re() == 0) ? "ok $test\n" : "not ok $test\n");
414 print ((\$z2->Im() == 0) ? "ok $test\n" : "not ok $test\n");
419 my \$z3 = Math::Complex->emake();
420 print ((\$z3->rho() == 0) ? "ok $test\n" : "not ok $test\n");
425 print ((\$z3->theta() == 0) ? "ok $test\n" : "not ok $test\n");
431 push @script, <<'EOS';
432 print "# atan2() with some real arguments\n";
434 my @real = (-1, 0, 1);
437 next if $x == 0 && $y == 0;
440 print ((Math::Complex::atan2($y, $x) == CORE::atan2($y, $x)) ? "ok $test\n" : "not ok $test\n");
444 push @script, <<'EOS';
445 print "# atan2() with some complex arguments\n";
449 print (abs(atan2(0, cplx(0, 1))) < $eps ? "ok $test\n" : "not ok $test\n");
453 print (abs(atan2(cplx(0, 1), 0) - \$pip2) < $eps ? "ok $test\n" : "not ok $test\n");
457 print (abs(atan2(cplx(0, 1), cplx(0, 1)) - \$pip4) < $eps ? "ok $test\n" : "not ok $test\n");
461 print (abs(atan2(cplx(0, 1), cplx(1, 1)) - cplx(0.553574358897045, 0.402359478108525)) < $eps ? "ok $test\n" : "not ok $test\n");
477 #print @script, "\n";
478 eval join '', @script;
484 push(@script, qq(print "# $op=\n";));
488 my ($op, $z, @args) = @_;
492 $baop = 1 if ($op =~ s/;=$//);
493 for ($i = 0; $i < @args; $i++) {
494 $val = value($args[$i]);
495 push @script, "\$z$i = $val;\n";
498 $args = "'$op'"; # Really the value
499 $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
500 push @script, "\$res = $try; ";
501 push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n";
506 $args = "'$args[0]'";
508 $try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1";
509 $args = "'$args[0]', '$args[1]'";
511 push @script, "\$res = $try; ";
512 push @script, "check($test, '$try', \$res, \$z$#args, $args);\n";
513 if (@args > 2 and $baop) { # binary assignment ops
515 # check the op= works
518 my \$za = cplx(ref \$z0 ? \@{\$z0->_cartesian} : (\$z0, 0));
520 my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->_cartesian} : (\$z1, 0);
522 my \$zb = cplx(\$z1r, \$z1i);
525 my (\$zbr, \$zbi) = \@{\$zb->_cartesian};
527 check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args);
530 # check that the rhs has not changed
531 push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
532 push @script, qq(print "ok $test\\n";\n);
539 my ($set, $setref, $valref) = @_;
542 my @set = split(/;\s*/, $set);
545 for ($i = 0; $i < @set; $i++) {
546 push(@{$valref}, $set[$i]);
547 my $val = value($set[$i]);
548 push @script, "\$s$i = $val;\n";
549 push @{$setref}, "\$s$i";
555 if (/^\s*\((.*),(.*)\)/) {
556 return "cplx($1,$2)";
558 elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
561 elsif (/^\s*\[(.*),(.*)\]/) {
562 return "cplxe($1,$2)";
564 elsif (/^\s*'(.*)'/) {
566 $ex =~ s/\bz\b/$target/g;
567 $ex =~ s/\br\b/abs($target)/g;
568 $ex =~ s/\bt\b/arg($target)/g;
569 $ex =~ s/\ba\b/Re($target)/g;
570 $ex =~ s/\bb\b/Im($target)/g;
573 elsif (/^\s*"(.*)"/) {
580 my ($test, $try, $got, $expected, @z) = @_;
584 if ("$got" eq "$expected"
586 ($expected =~ /^-?\d/ && $got == $expected)
588 (abs(Math::Complex->make($got) - Math::Complex->make($expected)) < $eps)
590 (abs($got - $expected) < $eps)
594 print "not ok $test\n";
595 my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]";
596 print "# '$try' expected: '$expected' got: '$got' for $args\n";
602 return ($z1 + i*$z2) * ($z1 - i*$z2);
607 return ($z1 + $z2) * ($z1 - $z2);
637 [2,2*pi/3]:(1,0):[2,2*pi/3]
649 [4,pi]:[2,pi/2]:[2,pi/2]
650 [2,pi/2]:[4,pi]:[0.5,-(pi)/2]
655 (2,3):(4,0):(-119,-120)
705 &_stringify_cartesian
717 [6, -2*pi/3]:"[6,-2pi/3]"
718 [0.5, -9*pi/11]:"[0.5,-9pi/11]"
721 { (4,3); [3,2]; (-3,4); (0,2); [2,1] }
724 |'z - ~z':'2*i*Im(z)'
725 |'z * ~z':'abs(z) * abs(z)'
727 { (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
729 |'(root(z, 4))[1] ** 4':'z'
730 |'(root(z, 5))[3] ** 5':'z'
731 |'(root(z, 8))[7] ** 8':'z'
732 |'(root(z, 8, 0)) ** 8':'z'
733 |'(root(z, 8, 7)) ** 8':'z'
735 |'acot(z)':'acotan(z)'
736 |'acsc(z)':'acosec(z)'
737 |'acsc(z)':'asin(1 / z)'
738 |'asec(z)':'acos(1 / z)'
739 |'cbrt(z)':'cbrt(r) * exp(i * t/3)'
741 |'addsq(cos(z), sin(z))':1
742 |'cos(z)':'cosh(i*z)'
743 |'subsq(cosh(z), sinh(z))':1
745 |'cot(z)':'1 / tan(z)'
748 |'csc(z)':'1 / sin(z)'
751 |'exp(z)':'exp(a) * exp(i * b)'
754 |'log(z)':'log(r) + i*t'
755 |'log10(z)':'log(z) / log(10)'
756 |'logn(z, 2)':'log(z) / log(2)'
757 |'logn(z, 3)':'log(z) / log(3)'
759 |'sec(z)':'1 / cos(z)'
761 |'sin(i * z)':'i * sinh(z)'
762 |'sqrt(z) * sqrt(z)':'z'
763 |'sqrt(z)':'sqrt(r) * exp(i * t/2)'
765 |'z**z':'exp(z * log(z))'
767 { (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
769 |'cosh(acosh(z))':'z'
770 |'coth(acoth(z))':'z'
771 |'coth(z)':'1 / tanh(z)'
772 |'coth(z)':'cotanh(z)'
773 |'csch(acsch(z))':'z'
774 |'csch(z)':'1 / sinh(z)'
775 |'csch(z)':'cosech(z)'
776 |'sech(asech(z))':'z'
777 |'sech(z)':'1 / cosh(z)'
778 |'sinh(asinh(z))':'z'
779 |'tanh(atanh(z))':'z'
781 { (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
783 |'acos(cos(z)) ** 2':'z * z'
784 |'acosh(cosh(z)) ** 2':'z * z'
785 |'acoth(z)':'acotanh(z)'
786 |'acoth(z)':'atanh(1 / z)'
787 |'acsch(z)':'acosech(z)'
788 |'acsch(z)':'asinh(1 / z)'
789 |'asech(z)':'acosh(1 / z)'
791 |'asinh(sinh(z))':'z'
793 |'atanh(tanh(z))':'z'
796 (-2.0,0):( 0.69314718055995, 3.14159265358979)
797 (-1.0,0):( 0 , 3.14159265358979)
798 (-0.5,0):( -0.69314718055995, 3.14159265358979)
799 ( 0.5,0):( -0.69314718055995, 0 )
801 ( 2.0,0):( 0.69314718055995, 0 )
804 ( 2, 3):( 1.28247467873077, 0.98279372324733)
805 (-2, 3):( 1.28247467873077, 2.15879893034246)
806 (-2,-3):( 1.28247467873077, -2.15879893034246)
807 ( 2,-3):( 1.28247467873077, -0.98279372324733)
810 (-2.0,0):( -0.90929742682568, 0 )
811 (-1.0,0):( -0.84147098480790, 0 )
812 (-0.5,0):( -0.47942553860420, 0 )
814 ( 0.5,0):( 0.47942553860420, 0 )
815 ( 1.0,0):( 0.84147098480790, 0 )
816 ( 2.0,0):( 0.90929742682568, 0 )
819 ( 2, 3):( 9.15449914691143, -4.16890695996656)
820 (-2, 3):( -9.15449914691143, -4.16890695996656)
821 (-2,-3):( -9.15449914691143, 4.16890695996656)
822 ( 2,-3):( 9.15449914691143, 4.16890695996656)
825 (-2.0,0):( -0.41614683654714, 0 )
826 (-1.0,0):( 0.54030230586814, 0 )
827 (-0.5,0):( 0.87758256189037, 0 )
829 ( 0.5,0):( 0.87758256189037, 0 )
830 ( 1.0,0):( 0.54030230586814, 0 )
831 ( 2.0,0):( -0.41614683654714, 0 )
834 ( 2, 3):( -4.18962569096881, -9.10922789375534)
835 (-2, 3):( -4.18962569096881, 9.10922789375534)
836 (-2,-3):( -4.18962569096881, -9.10922789375534)
837 ( 2,-3):( -4.18962569096881, 9.10922789375534)
840 (-2.0,0):( 2.18503986326152, 0 )
841 (-1.0,0):( -1.55740772465490, 0 )
842 (-0.5,0):( -0.54630248984379, 0 )
844 ( 0.5,0):( 0.54630248984379, 0 )
845 ( 1.0,0):( 1.55740772465490, 0 )
846 ( 2.0,0):( -2.18503986326152, 0 )
849 ( 2, 3):( -0.00376402564150, 1.00323862735361)
850 (-2, 3):( 0.00376402564150, 1.00323862735361)
851 (-2,-3):( 0.00376402564150, -1.00323862735361)
852 ( 2,-3):( -0.00376402564150, -1.00323862735361)
855 (-2.0,0):( -2.40299796172238, 0 )
856 (-1.0,0):( 1.85081571768093, 0 )
857 (-0.5,0):( 1.13949392732455, 0 )
859 ( 0.5,0):( 1.13949392732455, 0 )
860 ( 1.0,0):( 1.85081571768093, 0 )
861 ( 2.0,0):( -2.40299796172238, 0 )
864 ( 2, 3):( -0.04167496441114, 0.09061113719624)
865 (-2, 3):( -0.04167496441114, -0.09061113719624)
866 (-2,-3):( -0.04167496441114, 0.09061113719624)
867 ( 2,-3):( -0.04167496441114, -0.09061113719624)
870 (-2.0,0):( -1.09975017029462, 0 )
871 (-1.0,0):( -1.18839510577812, 0 )
872 (-0.5,0):( -2.08582964293349, 0 )
873 ( 0.5,0):( 2.08582964293349, 0 )
874 ( 1.0,0):( 1.18839510577812, 0 )
875 ( 2.0,0):( 1.09975017029462, 0 )
878 ( 2, 3):( 0.09047320975321, 0.04120098628857)
879 (-2, 3):( -0.09047320975321, 0.04120098628857)
880 (-2,-3):( -0.09047320975321, -0.04120098628857)
881 ( 2,-3):( 0.09047320975321, -0.04120098628857)
884 (-2.0,0):( 0.45765755436029, 0 )
885 (-1.0,0):( -0.64209261593433, 0 )
886 (-0.5,0):( -1.83048772171245, 0 )
887 ( 0.5,0):( 1.83048772171245, 0 )
888 ( 1.0,0):( 0.64209261593433, 0 )
889 ( 2.0,0):( -0.45765755436029, 0 )
892 ( 2, 3):( -0.00373971037634, -0.99675779656936)
893 (-2, 3):( 0.00373971037634, -0.99675779656936)
894 (-2,-3):( 0.00373971037634, 0.99675779656936)
895 ( 2,-3):( -0.00373971037634, 0.99675779656936)
898 (-2.0,0):( -1.57079632679490, 1.31695789692482)
899 (-1.0,0):( -1.57079632679490, 0 )
900 (-0.5,0):( -0.52359877559830, 0 )
902 ( 0.5,0):( 0.52359877559830, 0 )
903 ( 1.0,0):( 1.57079632679490, 0 )
904 ( 2.0,0):( 1.57079632679490, -1.31695789692482)
907 ( 2, 3):( 0.57065278432110, 1.98338702991654)
908 (-2, 3):( -0.57065278432110, 1.98338702991654)
909 (-2,-3):( -0.57065278432110, -1.98338702991654)
910 ( 2,-3):( 0.57065278432110, -1.98338702991654)
913 (-2.0,0):( 3.14159265358979, -1.31695789692482)
914 (-1.0,0):( 3.14159265358979, 0 )
915 (-0.5,0):( 2.09439510239320, 0 )
916 ( 0.0,0):( 1.57079632679490, 0 )
917 ( 0.5,0):( 1.04719755119660, 0 )
919 ( 2.0,0):( 0 , 1.31695789692482)
922 ( 2, 3):( 1.00014354247380, -1.98338702991654)
923 (-2, 3):( 2.14144911111600, -1.98338702991654)
924 (-2,-3):( 2.14144911111600, 1.98338702991654)
925 ( 2,-3):( 1.00014354247380, 1.98338702991654)
928 (-2.0,0):( -1.10714871779409, 0 )
929 (-1.0,0):( -0.78539816339745, 0 )
930 (-0.5,0):( -0.46364760900081, 0 )
932 ( 0.5,0):( 0.46364760900081, 0 )
933 ( 1.0,0):( 0.78539816339745, 0 )
934 ( 2.0,0):( 1.10714871779409, 0 )
937 ( 2, 3):( 1.40992104959658, 0.22907268296854)
938 (-2, 3):( -1.40992104959658, 0.22907268296854)
939 (-2,-3):( -1.40992104959658, -0.22907268296854)
940 ( 2,-3):( 1.40992104959658, -0.22907268296854)
943 (-2.0,0):( 2.09439510239320, 0 )
944 (-1.0,0):( 3.14159265358979, 0 )
945 (-0.5,0):( 3.14159265358979, -1.31695789692482)
946 ( 0.5,0):( 0 , 1.31695789692482)
948 ( 2.0,0):( 1.04719755119660, 0 )
951 ( 2, 3):( 1.42041072246703, 0.23133469857397)
952 (-2, 3):( 1.72118193112276, 0.23133469857397)
953 (-2,-3):( 1.72118193112276, -0.23133469857397)
954 ( 2,-3):( 1.42041072246703, -0.23133469857397)
957 (-2.0,0):( -0.52359877559830, 0 )
958 (-1.0,0):( -1.57079632679490, 0 )
959 (-0.5,0):( -1.57079632679490, 1.31695789692482)
960 ( 0.5,0):( 1.57079632679490, -1.31695789692482)
961 ( 1.0,0):( 1.57079632679490, 0 )
962 ( 2.0,0):( 0.52359877559830, 0 )
965 ( 2, 3):( 0.15038560432786, -0.23133469857397)
966 (-2, 3):( -0.15038560432786, -0.23133469857397)
967 (-2,-3):( -0.15038560432786, 0.23133469857397)
968 ( 2,-3):( 0.15038560432786, 0.23133469857397)
971 (-2.0,0):( -0.46364760900081, 0 )
972 (-1.0,0):( -0.78539816339745, 0 )
973 (-0.5,0):( -1.10714871779409, 0 )
974 ( 0.5,0):( 1.10714871779409, 0 )
975 ( 1.0,0):( 0.78539816339745, 0 )
976 ( 2.0,0):( 0.46364760900081, 0 )
979 ( 2, 3):( 0.16087527719832, -0.22907268296854)
980 (-2, 3):( -0.16087527719832, -0.22907268296854)
981 (-2,-3):( -0.16087527719832, 0.22907268296854)
982 ( 2,-3):( 0.16087527719832, 0.22907268296854)
985 (-2.0,0):( -3.62686040784702, 0 )
986 (-1.0,0):( -1.17520119364380, 0 )
987 (-0.5,0):( -0.52109530549375, 0 )
989 ( 0.5,0):( 0.52109530549375, 0 )
990 ( 1.0,0):( 1.17520119364380, 0 )
991 ( 2.0,0):( 3.62686040784702, 0 )
994 ( 2, 3):( -3.59056458998578, 0.53092108624852)
995 (-2, 3):( 3.59056458998578, 0.53092108624852)
996 (-2,-3):( 3.59056458998578, -0.53092108624852)
997 ( 2,-3):( -3.59056458998578, -0.53092108624852)
1000 (-2.0,0):( 3.76219569108363, 0 )
1001 (-1.0,0):( 1.54308063481524, 0 )
1002 (-0.5,0):( 1.12762596520638, 0 )
1004 ( 0.5,0):( 1.12762596520638, 0 )
1005 ( 1.0,0):( 1.54308063481524, 0 )
1006 ( 2.0,0):( 3.76219569108363, 0 )
1009 ( 2, 3):( -3.72454550491532, 0.51182256998738)
1010 (-2, 3):( -3.72454550491532, -0.51182256998738)
1011 (-2,-3):( -3.72454550491532, 0.51182256998738)
1012 ( 2,-3):( -3.72454550491532, -0.51182256998738)
1015 (-2.0,0):( -0.96402758007582, 0 )
1016 (-1.0,0):( -0.76159415595576, 0 )
1017 (-0.5,0):( -0.46211715726001, 0 )
1019 ( 0.5,0):( 0.46211715726001, 0 )
1020 ( 1.0,0):( 0.76159415595576, 0 )
1021 ( 2.0,0):( 0.96402758007582, 0 )
1024 ( 2, 3):( 0.96538587902213, -0.00988437503832)
1025 (-2, 3):( -0.96538587902213, -0.00988437503832)
1026 (-2,-3):( -0.96538587902213, 0.00988437503832)
1027 ( 2,-3):( 0.96538587902213, 0.00988437503832)
1030 (-2.0,0):( 0.26580222883408, 0 )
1031 (-1.0,0):( 0.64805427366389, 0 )
1032 (-0.5,0):( 0.88681888397007, 0 )
1034 ( 0.5,0):( 0.88681888397007, 0 )
1035 ( 1.0,0):( 0.64805427366389, 0 )
1036 ( 2.0,0):( 0.26580222883408, 0 )
1039 ( 2, 3):( -0.26351297515839, -0.03621163655877)
1040 (-2, 3):( -0.26351297515839, 0.03621163655877)
1041 (-2,-3):( -0.26351297515839, -0.03621163655877)
1042 ( 2,-3):( -0.26351297515839, 0.03621163655877)
1045 (-2.0,0):( -0.27572056477178, 0 )
1046 (-1.0,0):( -0.85091812823932, 0 )
1047 (-0.5,0):( -1.91903475133494, 0 )
1048 ( 0.5,0):( 1.91903475133494, 0 )
1049 ( 1.0,0):( 0.85091812823932, 0 )
1050 ( 2.0,0):( 0.27572056477178, 0 )
1053 ( 2, 3):( -0.27254866146294, -0.04030057885689)
1054 (-2, 3):( 0.27254866146294, -0.04030057885689)
1055 (-2,-3):( 0.27254866146294, 0.04030057885689)
1056 ( 2,-3):( -0.27254866146294, 0.04030057885689)
1059 (-2.0,0):( -1.03731472072755, 0 )
1060 (-1.0,0):( -1.31303528549933, 0 )
1061 (-0.5,0):( -2.16395341373865, 0 )
1062 ( 0.5,0):( 2.16395341373865, 0 )
1063 ( 1.0,0):( 1.31303528549933, 0 )
1064 ( 2.0,0):( 1.03731472072755, 0 )
1067 ( 2, 3):( 1.03574663776500, 0.01060478347034)
1068 (-2, 3):( -1.03574663776500, 0.01060478347034)
1069 (-2,-3):( -1.03574663776500, -0.01060478347034)
1070 ( 2,-3):( 1.03574663776500, -0.01060478347034)
1073 (-2.0,0):( -1.44363547517881, 0 )
1074 (-1.0,0):( -0.88137358701954, 0 )
1075 (-0.5,0):( -0.48121182505960, 0 )
1077 ( 0.5,0):( 0.48121182505960, 0 )
1078 ( 1.0,0):( 0.88137358701954, 0 )
1079 ( 2.0,0):( 1.44363547517881, 0 )
1082 ( 2, 3):( 1.96863792579310, 0.96465850440760)
1083 (-2, 3):( -1.96863792579310, 0.96465850440761)
1084 (-2,-3):( -1.96863792579310, -0.96465850440761)
1085 ( 2,-3):( 1.96863792579310, -0.96465850440760)
1088 (-2.0,0):( 1.31695789692482, 3.14159265358979)
1089 (-1.0,0):( 0, 3.14159265358979)
1090 (-0.5,0):( 0, 2.09439510239320)
1091 ( 0.0,0):( 0, 1.57079632679490)
1092 ( 0.5,0):( 0, 1.04719755119660)
1094 ( 2.0,0):( 1.31695789692482, 0 )
1097 ( 2, 3):( 1.98338702991654, 1.00014354247380)
1098 (-2, 3):( 1.98338702991653, 2.14144911111600)
1099 (-2,-3):( 1.98338702991653, -2.14144911111600)
1100 ( 2,-3):( 1.98338702991654, -1.00014354247380)
1103 (-2.0,0):( -0.54930614433405, 1.57079632679490)
1104 (-0.5,0):( -0.54930614433405, 0 )
1106 ( 0.5,0):( 0.54930614433405, 0 )
1107 ( 2.0,0):( 0.54930614433405, 1.57079632679490)
1110 ( 2, 3):( 0.14694666622553, 1.33897252229449)
1111 (-2, 3):( -0.14694666622553, 1.33897252229449)
1112 (-2,-3):( -0.14694666622553, -1.33897252229449)
1113 ( 2,-3):( 0.14694666622553, -1.33897252229449)
1116 (-2.0,0):( 0 , 2.09439510239320)
1117 (-1.0,0):( 0 , 3.14159265358979)
1118 (-0.5,0):( 1.31695789692482, 3.14159265358979)
1119 ( 0.5,0):( 1.31695789692482, 0 )
1121 ( 2.0,0):( 0 , 1.04719755119660)
1124 ( 2, 3):( 0.23133469857397, -1.42041072246703)
1125 (-2, 3):( 0.23133469857397, -1.72118193112276)
1126 (-2,-3):( 0.23133469857397, 1.72118193112276)
1127 ( 2,-3):( 0.23133469857397, 1.42041072246703)
1130 (-2.0,0):( -0.48121182505960, 0 )
1131 (-1.0,0):( -0.88137358701954, 0 )
1132 (-0.5,0):( -1.44363547517881, 0 )
1133 ( 0.5,0):( 1.44363547517881, 0 )
1134 ( 1.0,0):( 0.88137358701954, 0 )
1135 ( 2.0,0):( 0.48121182505960, 0 )
1138 ( 2, 3):( 0.15735549884499, -0.22996290237721)
1139 (-2, 3):( -0.15735549884499, -0.22996290237721)
1140 (-2,-3):( -0.15735549884499, 0.22996290237721)
1141 ( 2,-3):( 0.15735549884499, 0.22996290237721)
1144 (-2.0,0):( -0.54930614433405, 0 )
1145 (-0.5,0):( -0.54930614433405, 1.57079632679490)
1146 ( 0.5,0):( 0.54930614433405, 1.57079632679490)
1147 ( 2.0,0):( 0.54930614433405, 0 )
1150 ( 2, 3):( 0.14694666622553, -0.23182380450040)
1151 (-2, 3):( -0.14694666622553, -0.23182380450040)
1152 (-2,-3):( -0.14694666622553, 0.23182380450040)
1153 ( 2,-3):( 0.14694666622553, 0.23182380450040)