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 $VERSION = sprintf("%s", q$Id: complex.t,v 1.8 1998/02/05 16:03:39 jhi Exp $ =~ /(\d+\.d+)/);
19 my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val);
24 'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10, $z0,$z1,$z2);' .
31 next if $_ eq '' || /^\#/;
33 $test_set = 0; # Assume not a test over a set of values
39 set($1, \@set, \@val);
43 $test_set = 1; # Requests we loop over the set...
45 my @args = split(/:/);
48 for ($i = 0; $i < @set; $i++) {
51 # textual value as found in set definition
53 test($zvalue, $target, @args);
56 test($op, undef, @args);
66 push(@script, <<'EOT');
71 print 'not ' unless Re($z) == 2 and Im($z) == 3;
73 push(@script, qq(print "ok $test\\n"}\n));
76 push(@script, <<'EOT');
80 print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and
81 (arg($z) - pi / 4 ) < $eps and
82 (Re($z) - 3 ) < $eps and
85 push(@script, qq(print "ok $test\\n"}\n));
88 push(@script, <<'EOT');
92 print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and
93 (abs($z) - sqrt(2) ) < $eps and
94 (Re($z) + 1 ) < $eps and
97 push(@script, qq(print "ok $test\\n"}\n));
104 my $pi = cplx(pi, 0);
105 my $pii = cplx(0, pi);
106 my $pip2 = cplx(pi/2, 0);
107 my $zero = cplx(0, 0);
110 push(@script, $constants);
113 # test the divbyzeros
119 push(@script, <<EOT);
121 print 'not ' unless (\$@ =~ /Division by zero/);
123 push(@script, qq(print "ok $test\\n";\n));
127 # test the logofzeros
133 push(@script, <<EOT);
135 print 'not ' unless (\$@ =~ /Logarithm of zero/);
137 push(@script, qq(print "ok $test\\n";\n));
145 # 'acoth(-1)', # Log of zero.
154 # 'atanh(-1)', # Log of zero.
176 push(@script, <<'EOT');
177 eval 'cplx(0)**cplx(0)';
178 print 'not ' unless ($@ =~ /zero raised to the zeroth/);
180 push(@script, qq(print "ok $test\\n";\n));
191 push(@script, <<EOT);
193 print 'not ' unless (\$@ =~ /root must be/);
195 push(@script, qq(print "ok $test\\n";\n));
199 test_broot(qw(-3 -2.1 0 0.99));
202 eval join '', @script;
208 push(@script, qq(print "# $op=\n";));
212 my ($op, $z, @args) = @_;
216 $baop = 1 if ($op =~ s/;=$//);
217 for ($i = 0; $i < @args; $i++) {
218 $val = value($args[$i]);
219 push @script, "\$z$i = $val;\n";
222 $args = "'$op'"; # Really the value
223 $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
224 push @script, "\$res = $try; ";
225 push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n";
230 $args = "'$args[0]'";
232 $try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1";
233 $args = "'$args[0]', '$args[1]'";
235 push @script, "\$res = $try; ";
236 push @script, "check($test, '$try', \$res, \$z$#args, $args);\n";
237 if (@args > 2 and $baop) { # binary assignment ops
239 # check the op= works
242 my \$za = cplx(ref \$z0 ? \@{\$z0->cartesian} : (\$z0, 0));
244 my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->cartesian} : (\$z1, 0);
246 my \$zb = cplx(\$z1r, \$z1i);
249 my (\$zbr, \$zbi) = \@{\$zb->cartesian};
251 check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args);
254 # check that the rhs has not changed
255 push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
256 push @script, qq(print "ok $test\\n";\n);
263 my ($set, $setref, $valref) = @_;
266 my @set = split(/;\s*/, $set);
269 for ($i = 0; $i < @set; $i++) {
270 push(@{$valref}, $set[$i]);
271 my $val = value($set[$i]);
272 push @script, "\$s$i = $val;\n";
273 push @{$setref}, "\$s$i";
279 if (/^\s*\((.*),(.*)\)/) {
280 return "cplx($1,$2)";
282 elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
285 elsif (/^\s*\[(.*),(.*)\]/) {
286 return "cplxe($1,$2)";
288 elsif (/^\s*'(.*)'/) {
290 $ex =~ s/\bz\b/$target/g;
291 $ex =~ s/\br\b/abs($target)/g;
292 $ex =~ s/\bt\b/arg($target)/g;
293 $ex =~ s/\ba\b/Re($target)/g;
294 $ex =~ s/\bb\b/Im($target)/g;
297 elsif (/^\s*"(.*)"/) {
304 my ($test, $try, $got, $expected, @z) = @_;
308 if ("$got" eq "$expected"
310 ($expected =~ /^-?\d/ && $got == $expected)
312 (abs($got - $expected) < $eps)
316 print "not ok $test\n";
317 my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]";
318 print "# '$try' expected: '$expected' got: '$got' for $args\n";
324 return ($z1 + i*$z2) * ($z1 - i*$z2);
329 return ($z1 + $z2) * ($z1 - $z2);
359 [2,2*pi/3]:(1,0):[2,2*pi/3]
371 [4,pi]:[2,pi/2]:[2,pi/2]
372 [2,pi/2]:[4,pi]:[0.5,-(pi)/2]
377 (2,3):(4,0):(-119,-120)
438 [6, -2*pi/3]:"[6,-2pi/3]"
439 [0.5, -9*pi/11]:"[0.5,-9pi/11]"
441 { (4,3); [3,2]; (-3,4); (0,2); [2,1] }
444 |'z - ~z':'2*i*Im(z)'
445 |'z * ~z':'abs(z) * abs(z)'
447 { (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
449 |'(root(z, 4))[1] ** 4':'z'
450 |'(root(z, 5))[3] ** 5':'z'
451 |'(root(z, 8))[7] ** 8':'z'
453 |'acot(z)':'acotan(z)'
454 |'acsc(z)':'acosec(z)'
455 |'acsc(z)':'asin(1 / z)'
456 |'asec(z)':'acos(1 / z)'
457 |'cbrt(z)':'cbrt(r) * exp(i * t/3)'
459 |'addsq(cos(z), sin(z))':1
460 |'cos(z)':'cosh(i*z)'
461 |'subsq(cosh(z), sinh(z))':1
463 |'cot(z)':'1 / tan(z)'
466 |'csc(z)':'1 / sin(z)'
469 |'exp(z)':'exp(a) * exp(i * b)'
472 |'log(z)':'log(r) + i*t'
473 |'log10(z)':'log(z) / log(10)'
474 |'logn(z, 2)':'log(z) / log(2)'
475 |'logn(z, 3)':'log(z) / log(3)'
477 |'sec(z)':'1 / cos(z)'
479 |'sin(i * z)':'i * sinh(z)'
480 |'sqrt(z) * sqrt(z)':'z'
481 |'sqrt(z)':'sqrt(r) * exp(i * t/2)'
483 |'z**z':'exp(z * log(z))'
485 { (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
487 |'cosh(acosh(z))':'z'
488 |'coth(acoth(z))':'z'
489 |'coth(z)':'1 / tanh(z)'
490 |'coth(z)':'cotanh(z)'
491 |'csch(acsch(z))':'z'
492 |'csch(z)':'1 / sinh(z)'
493 |'csch(z)':'cosech(z)'
494 |'sech(asech(z))':'z'
495 |'sech(z)':'1 / cosh(z)'
496 |'sinh(asinh(z))':'z'
497 |'tanh(atanh(z))':'z'
499 { (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
501 |'acos(cos(z)) ** 2':'z * z'
502 |'acosh(cosh(z)) ** 2':'z * z'
503 |'acoth(z)':'acotanh(z)'
504 |'acoth(z)':'atanh(1 / z)'
505 |'acsch(z)':'acosech(z)'
506 |'acsch(z)':'asinh(1 / z)'
507 |'asech(z)':'acosh(1 / z)'
509 |'asinh(sinh(z))':'z'
511 |'atanh(tanh(z))':'z'
514 (-2.0,0):( 0.69314718055995, 3.14159265358979)
515 (-1.0,0):( 0 , 3.14159265358979)
516 (-0.5,0):( -0.69314718055995, 3.14159265358979)
517 ( 0.5,0):( -0.69314718055995, 0 )
519 ( 2.0,0):( 0.69314718055995, 0 )
522 ( 2, 3):( 1.28247467873077, 0.98279372324733)
523 (-2, 3):( 1.28247467873077, 2.15879893034246)
524 (-2,-3):( 1.28247467873077, -2.15879893034246)
525 ( 2,-3):( 1.28247467873077, -0.98279372324733)
528 (-2.0,0):( -0.90929742682568, 0 )
529 (-1.0,0):( -0.84147098480790, 0 )
530 (-0.5,0):( -0.47942553860420, 0 )
532 ( 0.5,0):( 0.47942553860420, 0 )
533 ( 1.0,0):( 0.84147098480790, 0 )
534 ( 2.0,0):( 0.90929742682568, 0 )
537 ( 2, 3):( 9.15449914691143, -4.16890695996656)
538 (-2, 3):( -9.15449914691143, -4.16890695996656)
539 (-2,-3):( -9.15449914691143, 4.16890695996656)
540 ( 2,-3):( 9.15449914691143, 4.16890695996656)
543 (-2.0,0):( -0.41614683654714, 0 )
544 (-1.0,0):( 0.54030230586814, 0 )
545 (-0.5,0):( 0.87758256189037, 0 )
547 ( 0.5,0):( 0.87758256189037, 0 )
548 ( 1.0,0):( 0.54030230586814, 0 )
549 ( 2.0,0):( -0.41614683654714, 0 )
552 ( 2, 3):( -4.18962569096881, -9.10922789375534)
553 (-2, 3):( -4.18962569096881, 9.10922789375534)
554 (-2,-3):( -4.18962569096881, -9.10922789375534)
555 ( 2,-3):( -4.18962569096881, 9.10922789375534)
558 (-2.0,0):( 2.18503986326152, 0 )
559 (-1.0,0):( -1.55740772465490, 0 )
560 (-0.5,0):( -0.54630248984379, 0 )
562 ( 0.5,0):( 0.54630248984379, 0 )
563 ( 1.0,0):( 1.55740772465490, 0 )
564 ( 2.0,0):( -2.18503986326152, 0 )
567 ( 2, 3):( -0.00376402564150, 1.00323862735361)
568 (-2, 3):( 0.00376402564150, 1.00323862735361)
569 (-2,-3):( 0.00376402564150, -1.00323862735361)
570 ( 2,-3):( -0.00376402564150, -1.00323862735361)
573 (-2.0,0):( -2.40299796172238, 0 )
574 (-1.0,0):( 1.85081571768093, 0 )
575 (-0.5,0):( 1.13949392732455, 0 )
577 ( 0.5,0):( 1.13949392732455, 0 )
578 ( 1.0,0):( 1.85081571768093, 0 )
579 ( 2.0,0):( -2.40299796172238, 0 )
582 ( 2, 3):( -0.04167496441114, 0.09061113719624)
583 (-2, 3):( -0.04167496441114, -0.09061113719624)
584 (-2,-3):( -0.04167496441114, 0.09061113719624)
585 ( 2,-3):( -0.04167496441114, -0.09061113719624)
588 (-2.0,0):( -1.09975017029462, 0 )
589 (-1.0,0):( -1.18839510577812, 0 )
590 (-0.5,0):( -2.08582964293349, 0 )
591 ( 0.5,0):( 2.08582964293349, 0 )
592 ( 1.0,0):( 1.18839510577812, 0 )
593 ( 2.0,0):( 1.09975017029462, 0 )
596 ( 2, 3):( 0.09047320975321, 0.04120098628857)
597 (-2, 3):( -0.09047320975321, 0.04120098628857)
598 (-2,-3):( -0.09047320975321, -0.04120098628857)
599 ( 2,-3):( 0.09047320975321, -0.04120098628857)
602 (-2.0,0):( 0.45765755436029, 0 )
603 (-1.0,0):( -0.64209261593433, 0 )
604 (-0.5,0):( -1.83048772171245, 0 )
605 ( 0.5,0):( 1.83048772171245, 0 )
606 ( 1.0,0):( 0.64209261593433, 0 )
607 ( 2.0,0):( -0.45765755436029, 0 )
610 ( 2, 3):( -0.00373971037634, -0.99675779656936)
611 (-2, 3):( 0.00373971037634, -0.99675779656936)
612 (-2,-3):( 0.00373971037634, 0.99675779656936)
613 ( 2,-3):( -0.00373971037634, 0.99675779656936)
616 (-2.0,0):( -1.57079632679490, 1.31695789692482)
617 (-1.0,0):( -1.57079632679490, 0 )
618 (-0.5,0):( -0.52359877559830, 0 )
620 ( 0.5,0):( 0.52359877559830, 0 )
621 ( 1.0,0):( 1.57079632679490, 0 )
622 ( 2.0,0):( 1.57079632679490, -1.31695789692482)
625 ( 2, 3):( 0.57065278432110, 1.98338702991654)
626 (-2, 3):( -0.57065278432110, 1.98338702991654)
627 (-2,-3):( -0.57065278432110, -1.98338702991654)
628 ( 2,-3):( 0.57065278432110, -1.98338702991654)
631 (-2.0,0):( 3.14159265358979, -1.31695789692482)
632 (-1.0,0):( 3.14159265358979, 0 )
633 (-0.5,0):( 2.09439510239320, 0 )
634 ( 0.0,0):( 1.57079632679490, 0 )
635 ( 0.5,0):( 1.04719755119660, 0 )
637 ( 2.0,0):( 0 , 1.31695789692482)
640 ( 2, 3):( 1.00014354247380, -1.98338702991654)
641 (-2, 3):( 2.14144911111600, -1.98338702991654)
642 (-2,-3):( 2.14144911111600, 1.98338702991654)
643 ( 2,-3):( 1.00014354247380, 1.98338702991654)
646 (-2.0,0):( -1.10714871779409, 0 )
647 (-1.0,0):( -0.78539816339745, 0 )
648 (-0.5,0):( -0.46364760900081, 0 )
650 ( 0.5,0):( 0.46364760900081, 0 )
651 ( 1.0,0):( 0.78539816339745, 0 )
652 ( 2.0,0):( 1.10714871779409, 0 )
655 ( 2, 3):( 1.40992104959658, 0.22907268296854)
656 (-2, 3):( -1.40992104959658, 0.22907268296854)
657 (-2,-3):( -1.40992104959658, -0.22907268296854)
658 ( 2,-3):( 1.40992104959658, -0.22907268296854)
661 (-2.0,0):( 2.09439510239320, 0 )
662 (-1.0,0):( 3.14159265358979, 0 )
663 (-0.5,0):( 3.14159265358979, -1.31695789692482)
664 ( 0.5,0):( 0 , 1.31695789692482)
666 ( 2.0,0):( 1.04719755119660, 0 )
669 ( 2, 3):( 1.42041072246703, 0.23133469857397)
670 (-2, 3):( 1.72118193112276, 0.23133469857397)
671 (-2,-3):( 1.72118193112276, -0.23133469857397)
672 ( 2,-3):( 1.42041072246703, -0.23133469857397)
675 (-2.0,0):( -0.52359877559830, 0 )
676 (-1.0,0):( -1.57079632679490, 0 )
677 (-0.5,0):( -1.57079632679490, 1.31695789692482)
678 ( 0.5,0):( 1.57079632679490, -1.31695789692482)
679 ( 1.0,0):( 1.57079632679490, 0 )
680 ( 2.0,0):( 0.52359877559830, 0 )
683 ( 2, 3):( 0.15038560432786, -0.23133469857397)
684 (-2, 3):( -0.15038560432786, -0.23133469857397)
685 (-2,-3):( -0.15038560432786, 0.23133469857397)
686 ( 2,-3):( 0.15038560432786, 0.23133469857397)
689 (-2.0,0):( -0.46364760900081, 0 )
690 (-1.0,0):( -0.78539816339745, 0 )
691 (-0.5,0):( -1.10714871779409, 0 )
692 ( 0.5,0):( 1.10714871779409, 0 )
693 ( 1.0,0):( 0.78539816339745, 0 )
694 ( 2.0,0):( 0.46364760900081, 0 )
697 ( 2, 3):( 0.16087527719832, -0.22907268296854)
698 (-2, 3):( -0.16087527719832, -0.22907268296854)
699 (-2,-3):( -0.16087527719832, 0.22907268296854)
700 ( 2,-3):( 0.16087527719832, 0.22907268296854)
703 (-2.0,0):( -3.62686040784702, 0 )
704 (-1.0,0):( -1.17520119364380, 0 )
705 (-0.5,0):( -0.52109530549375, 0 )
707 ( 0.5,0):( 0.52109530549375, 0 )
708 ( 1.0,0):( 1.17520119364380, 0 )
709 ( 2.0,0):( 3.62686040784702, 0 )
712 ( 2, 3):( -3.59056458998578, 0.53092108624852)
713 (-2, 3):( 3.59056458998578, 0.53092108624852)
714 (-2,-3):( 3.59056458998578, -0.53092108624852)
715 ( 2,-3):( -3.59056458998578, -0.53092108624852)
718 (-2.0,0):( 3.76219569108363, 0 )
719 (-1.0,0):( 1.54308063481524, 0 )
720 (-0.5,0):( 1.12762596520638, 0 )
722 ( 0.5,0):( 1.12762596520638, 0 )
723 ( 1.0,0):( 1.54308063481524, 0 )
724 ( 2.0,0):( 3.76219569108363, 0 )
727 ( 2, 3):( -3.72454550491532, 0.51182256998738)
728 (-2, 3):( -3.72454550491532, -0.51182256998738)
729 (-2,-3):( -3.72454550491532, 0.51182256998738)
730 ( 2,-3):( -3.72454550491532, -0.51182256998738)
733 (-2.0,0):( -0.96402758007582, 0 )
734 (-1.0,0):( -0.76159415595576, 0 )
735 (-0.5,0):( -0.46211715726001, 0 )
737 ( 0.5,0):( 0.46211715726001, 0 )
738 ( 1.0,0):( 0.76159415595576, 0 )
739 ( 2.0,0):( 0.96402758007582, 0 )
742 ( 2, 3):( 0.96538587902213, -0.00988437503832)
743 (-2, 3):( -0.96538587902213, -0.00988437503832)
744 (-2,-3):( -0.96538587902213, 0.00988437503832)
745 ( 2,-3):( 0.96538587902213, 0.00988437503832)
748 (-2.0,0):( 0.26580222883408, 0 )
749 (-1.0,0):( 0.64805427366389, 0 )
750 (-0.5,0):( 0.88681888397007, 0 )
752 ( 0.5,0):( 0.88681888397007, 0 )
753 ( 1.0,0):( 0.64805427366389, 0 )
754 ( 2.0,0):( 0.26580222883408, 0 )
757 ( 2, 3):( -0.26351297515839, -0.03621163655877)
758 (-2, 3):( -0.26351297515839, 0.03621163655877)
759 (-2,-3):( -0.26351297515839, -0.03621163655877)
760 ( 2,-3):( -0.26351297515839, 0.03621163655877)
763 (-2.0,0):( -0.27572056477178, 0 )
764 (-1.0,0):( -0.85091812823932, 0 )
765 (-0.5,0):( -1.91903475133494, 0 )
766 ( 0.5,0):( 1.91903475133494, 0 )
767 ( 1.0,0):( 0.85091812823932, 0 )
768 ( 2.0,0):( 0.27572056477178, 0 )
771 ( 2, 3):( -0.27254866146294, -0.04030057885689)
772 (-2, 3):( 0.27254866146294, -0.04030057885689)
773 (-2,-3):( 0.27254866146294, 0.04030057885689)
774 ( 2,-3):( -0.27254866146294, 0.04030057885689)
777 (-2.0,0):( -1.03731472072755, 0 )
778 (-1.0,0):( -1.31303528549933, 0 )
779 (-0.5,0):( -2.16395341373865, 0 )
780 ( 0.5,0):( 2.16395341373865, 0 )
781 ( 1.0,0):( 1.31303528549933, 0 )
782 ( 2.0,0):( 1.03731472072755, 0 )
785 ( 2, 3):( 1.03574663776500, 0.01060478347034)
786 (-2, 3):( -1.03574663776500, 0.01060478347034)
787 (-2,-3):( -1.03574663776500, -0.01060478347034)
788 ( 2,-3):( 1.03574663776500, -0.01060478347034)
791 (-2.0,0):( -1.44363547517881, 0 )
792 (-1.0,0):( -0.88137358701954, 0 )
793 (-0.5,0):( -0.48121182505960, 0 )
795 ( 0.5,0):( 0.48121182505960, 0 )
796 ( 1.0,0):( 0.88137358701954, 0 )
797 ( 2.0,0):( 1.44363547517881, 0 )
800 ( 2, 3):( 1.96863792579310, 0.96465850440760)
801 (-2, 3):( -1.96863792579310, 0.96465850440761)
802 (-2,-3):( -1.96863792579310, -0.96465850440761)
803 ( 2,-3):( 1.96863792579310, -0.96465850440760)
806 (-2.0,0):( -1.31695789692482, 3.14159265358979)
807 (-1.0,0):( 0, 3.14159265358979)
808 (-0.5,0):( 0, 2.09439510239320)
809 ( 0.0,0):( 0, 1.57079632679490)
810 ( 0.5,0):( 0, 1.04719755119660)
812 ( 2.0,0):( 1.31695789692482, 0 )
815 ( 2, 3):( 1.98338702991654, 1.00014354247380)
816 (-2, 3):( -1.98338702991653, -2.14144911111600)
817 (-2,-3):( -1.98338702991653, 2.14144911111600)
818 ( 2,-3):( 1.98338702991654, -1.00014354247380)
821 (-2.0,0):( -0.54930614433405, 1.57079632679490)
822 (-0.5,0):( -0.54930614433405, 0 )
824 ( 0.5,0):( 0.54930614433405, 0 )
825 ( 2.0,0):( 0.54930614433405, 1.57079632679490)
828 ( 2, 3):( 0.14694666622553, 1.33897252229449)
829 (-2, 3):( -0.14694666622553, 1.33897252229449)
830 (-2,-3):( -0.14694666622553, -1.33897252229449)
831 ( 2,-3):( 0.14694666622553, -1.33897252229449)
834 (-2.0,0):( 0 , 2.09439510239320)
835 (-1.0,0):( 0 , 3.14159265358979)
836 (-0.5,0):( -1.31695789692482, 3.14159265358979)
837 ( 0.5,0):( 1.31695789692482, 0 )
839 ( 2.0,0):( 0 , 1.04719755119660)
842 ( 2, 3):( 0.23133469857397, -1.42041072246703)
843 (-2, 3):( -0.23133469857397, 1.72118193112276)
844 (-2,-3):( -0.23133469857397, -1.72118193112276)
845 ( 2,-3):( 0.23133469857397, 1.42041072246703)
848 (-2.0,0):( -0.48121182505960, 0 )
849 (-1.0,0):( -0.88137358701954, 0 )
850 (-0.5,0):( -1.44363547517881, 0 )
851 ( 0.5,0):( 1.44363547517881, 0 )
852 ( 1.0,0):( 0.88137358701954, 0 )
853 ( 2.0,0):( 0.48121182505960, 0 )
856 ( 2, 3):( 0.15735549884499, -0.22996290237721)
857 (-2, 3):( -0.15735549884499, -0.22996290237721)
858 (-2,-3):( -0.15735549884499, 0.22996290237721)
859 ( 2,-3):( 0.15735549884499, 0.22996290237721)
862 (-2.0,0):( -0.54930614433405, 0 )
863 (-0.5,0):( -0.54930614433405, 1.57079632679490)
864 ( 0.5,0):( 0.54930614433405, 1.57079632679490)
865 ( 2.0,0):( 0.54930614433405, 0 )
868 ( 2, 3):( 0.14694666622553, -0.23182380450040)
869 (-2, 3):( -0.14694666622553, -0.23182380450040)
870 (-2,-3):( -0.14694666622553, 0.23182380450040)
871 ( 2,-3):( 0.14694666622553, 0.23182380450040)