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 my $VERSION = sprintf("%s", q$Id: complex.t,v 1.9 1998/11/01 00:00:00 dsl 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,$z2);' .
29 if ($^O eq 'unicos') { # For some reason root() produces very inaccurate
30 $eps = 1e-11; # results in Cray UNICOS, and occasionally also
31 } # cos(), sin(), cosh(), sinh(). The division
32 # of doubles is the current suspect.
36 next if $_ eq '' || /^\#/;
38 $test_set = 0; # Assume not a test over a set of values
44 set($1, \@set, \@val);
48 $test_set = 1; # Requests we loop over the set...
50 my @args = split(/:/);
53 for ($i = 0; $i < @set; $i++) {
56 # textual value as found in set definition
58 test($zvalue, $target, @args);
61 test($op, undef, @args);
71 push(@script, <<'EOT');
76 print 'not ' unless Re($z) == 2 and Im($z) == 3;
78 push(@script, qq(print "ok $test\\n"}\n));
81 push(@script, <<'EOT');
85 print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and
86 (arg($z) - pi / 4 ) < $eps and
87 (Re($z) - 3 ) < $eps and
90 push(@script, qq(print "ok $test\\n"}\n));
93 push(@script, <<'EOT');
97 print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and
98 (abs($z) - sqrt(2) ) < $eps and
99 (Re($z) + 1 ) < $eps and
100 (Im($z) + 1 ) < $eps;
102 push(@script, qq(print "ok $test\\n"}\n));
109 my $pi = cplx(pi, 0);
110 my $pii = cplx(0, pi);
111 my $pip2 = cplx(pi/2, 0);
112 my $zero = cplx(0, 0);
115 push(@script, $constants);
118 # test the divbyzeros
124 push(@script, <<EOT);
126 print 'not ' unless (\$@ =~ /Division by zero/);
128 push(@script, qq(print "ok $test\\n";\n));
132 # test the logofzeros
138 push(@script, <<EOT);
140 print 'not ' unless (\$@ =~ /Logarithm of zero/);
142 push(@script, qq(print "ok $test\\n";\n));
150 # 'acoth(-1)', # Log of zero.
159 # 'atanh(-1)', # Log of zero.
182 push(@script, <<EOT);
184 print 'not ' unless (\$@ =~ /root must be/);
186 push(@script, qq(print "ok $test\\n";\n));
190 test_broot(qw(-3 -2.1 0 0.99));
193 eval join '', @script;
199 push(@script, qq(print "# $op=\n";));
203 my ($op, $z, @args) = @_;
207 $baop = 1 if ($op =~ s/;=$//);
208 for ($i = 0; $i < @args; $i++) {
209 $val = value($args[$i]);
210 push @script, "\$z$i = $val;\n";
213 $args = "'$op'"; # Really the value
214 $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0";
215 push @script, "\$res = $try; ";
216 push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n";
221 $args = "'$args[0]'";
223 $try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1";
224 $args = "'$args[0]', '$args[1]'";
226 push @script, "\$res = $try; ";
227 push @script, "check($test, '$try', \$res, \$z$#args, $args);\n";
228 if (@args > 2 and $baop) { # binary assignment ops
230 # check the op= works
233 my \$za = cplx(ref \$z0 ? \@{\$z0->cartesian} : (\$z0, 0));
235 my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->cartesian} : (\$z1, 0);
237 my \$zb = cplx(\$z1r, \$z1i);
240 my (\$zbr, \$zbi) = \@{\$zb->cartesian};
242 check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args);
245 # check that the rhs has not changed
246 push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i););
247 push @script, qq(print "ok $test\\n";\n);
254 my ($set, $setref, $valref) = @_;
257 my @set = split(/;\s*/, $set);
260 for ($i = 0; $i < @set; $i++) {
261 push(@{$valref}, $set[$i]);
262 my $val = value($set[$i]);
263 push @script, "\$s$i = $val;\n";
264 push @{$setref}, "\$s$i";
270 if (/^\s*\((.*),(.*)\)/) {
271 return "cplx($1,$2)";
273 elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) {
276 elsif (/^\s*\[(.*),(.*)\]/) {
277 return "cplxe($1,$2)";
279 elsif (/^\s*'(.*)'/) {
281 $ex =~ s/\bz\b/$target/g;
282 $ex =~ s/\br\b/abs($target)/g;
283 $ex =~ s/\bt\b/arg($target)/g;
284 $ex =~ s/\ba\b/Re($target)/g;
285 $ex =~ s/\bb\b/Im($target)/g;
288 elsif (/^\s*"(.*)"/) {
295 my ($test, $try, $got, $expected, @z) = @_;
299 if ("$got" eq "$expected"
301 ($expected =~ /^-?\d/ && $got == $expected)
303 (abs($got - $expected) < $eps)
307 print "not ok $test\n";
308 my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]";
309 print "# '$try' expected: '$expected' got: '$got' for $args\n";
315 return ($z1 + i*$z2) * ($z1 - i*$z2);
320 return ($z1 + $z2) * ($z1 - $z2);
350 [2,2*pi/3]:(1,0):[2,2*pi/3]
362 [4,pi]:[2,pi/2]:[2,pi/2]
363 [2,pi/2]:[4,pi]:[0.5,-(pi)/2]
368 (2,3):(4,0):(-119,-120)
430 [6, -2*pi/3]:"[6,-2pi/3]"
431 [0.5, -9*pi/11]:"[0.5,-9pi/11]"
433 { (4,3); [3,2]; (-3,4); (0,2); [2,1] }
436 |'z - ~z':'2*i*Im(z)'
437 |'z * ~z':'abs(z) * abs(z)'
439 { (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
441 |'(root(z, 4))[1] ** 4':'z'
442 |'(root(z, 5))[3] ** 5':'z'
443 |'(root(z, 8))[7] ** 8':'z'
445 |'acot(z)':'acotan(z)'
446 |'acsc(z)':'acosec(z)'
447 |'acsc(z)':'asin(1 / z)'
448 |'asec(z)':'acos(1 / z)'
449 |'cbrt(z)':'cbrt(r) * exp(i * t/3)'
451 |'addsq(cos(z), sin(z))':1
452 |'cos(z)':'cosh(i*z)'
453 |'subsq(cosh(z), sinh(z))':1
455 |'cot(z)':'1 / tan(z)'
458 |'csc(z)':'1 / sin(z)'
461 |'exp(z)':'exp(a) * exp(i * b)'
464 |'log(z)':'log(r) + i*t'
465 |'log10(z)':'log(z) / log(10)'
466 |'logn(z, 2)':'log(z) / log(2)'
467 |'logn(z, 3)':'log(z) / log(3)'
469 |'sec(z)':'1 / cos(z)'
471 |'sin(i * z)':'i * sinh(z)'
472 |'sqrt(z) * sqrt(z)':'z'
473 |'sqrt(z)':'sqrt(r) * exp(i * t/2)'
475 |'z**z':'exp(z * log(z))'
477 { (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) }
479 |'cosh(acosh(z))':'z'
480 |'coth(acoth(z))':'z'
481 |'coth(z)':'1 / tanh(z)'
482 |'coth(z)':'cotanh(z)'
483 |'csch(acsch(z))':'z'
484 |'csch(z)':'1 / sinh(z)'
485 |'csch(z)':'cosech(z)'
486 |'sech(asech(z))':'z'
487 |'sech(z)':'1 / cosh(z)'
488 |'sinh(asinh(z))':'z'
489 |'tanh(atanh(z))':'z'
491 { (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) }
493 |'acos(cos(z)) ** 2':'z * z'
494 |'acosh(cosh(z)) ** 2':'z * z'
495 |'acoth(z)':'acotanh(z)'
496 |'acoth(z)':'atanh(1 / z)'
497 |'acsch(z)':'acosech(z)'
498 |'acsch(z)':'asinh(1 / z)'
499 |'asech(z)':'acosh(1 / z)'
501 |'asinh(sinh(z))':'z'
503 |'atanh(tanh(z))':'z'
506 (-2.0,0):( 0.69314718055995, 3.14159265358979)
507 (-1.0,0):( 0 , 3.14159265358979)
508 (-0.5,0):( -0.69314718055995, 3.14159265358979)
509 ( 0.5,0):( -0.69314718055995, 0 )
511 ( 2.0,0):( 0.69314718055995, 0 )
514 ( 2, 3):( 1.28247467873077, 0.98279372324733)
515 (-2, 3):( 1.28247467873077, 2.15879893034246)
516 (-2,-3):( 1.28247467873077, -2.15879893034246)
517 ( 2,-3):( 1.28247467873077, -0.98279372324733)
520 (-2.0,0):( -0.90929742682568, 0 )
521 (-1.0,0):( -0.84147098480790, 0 )
522 (-0.5,0):( -0.47942553860420, 0 )
524 ( 0.5,0):( 0.47942553860420, 0 )
525 ( 1.0,0):( 0.84147098480790, 0 )
526 ( 2.0,0):( 0.90929742682568, 0 )
529 ( 2, 3):( 9.15449914691143, -4.16890695996656)
530 (-2, 3):( -9.15449914691143, -4.16890695996656)
531 (-2,-3):( -9.15449914691143, 4.16890695996656)
532 ( 2,-3):( 9.15449914691143, 4.16890695996656)
535 (-2.0,0):( -0.41614683654714, 0 )
536 (-1.0,0):( 0.54030230586814, 0 )
537 (-0.5,0):( 0.87758256189037, 0 )
539 ( 0.5,0):( 0.87758256189037, 0 )
540 ( 1.0,0):( 0.54030230586814, 0 )
541 ( 2.0,0):( -0.41614683654714, 0 )
544 ( 2, 3):( -4.18962569096881, -9.10922789375534)
545 (-2, 3):( -4.18962569096881, 9.10922789375534)
546 (-2,-3):( -4.18962569096881, -9.10922789375534)
547 ( 2,-3):( -4.18962569096881, 9.10922789375534)
550 (-2.0,0):( 2.18503986326152, 0 )
551 (-1.0,0):( -1.55740772465490, 0 )
552 (-0.5,0):( -0.54630248984379, 0 )
554 ( 0.5,0):( 0.54630248984379, 0 )
555 ( 1.0,0):( 1.55740772465490, 0 )
556 ( 2.0,0):( -2.18503986326152, 0 )
559 ( 2, 3):( -0.00376402564150, 1.00323862735361)
560 (-2, 3):( 0.00376402564150, 1.00323862735361)
561 (-2,-3):( 0.00376402564150, -1.00323862735361)
562 ( 2,-3):( -0.00376402564150, -1.00323862735361)
565 (-2.0,0):( -2.40299796172238, 0 )
566 (-1.0,0):( 1.85081571768093, 0 )
567 (-0.5,0):( 1.13949392732455, 0 )
569 ( 0.5,0):( 1.13949392732455, 0 )
570 ( 1.0,0):( 1.85081571768093, 0 )
571 ( 2.0,0):( -2.40299796172238, 0 )
574 ( 2, 3):( -0.04167496441114, 0.09061113719624)
575 (-2, 3):( -0.04167496441114, -0.09061113719624)
576 (-2,-3):( -0.04167496441114, 0.09061113719624)
577 ( 2,-3):( -0.04167496441114, -0.09061113719624)
580 (-2.0,0):( -1.09975017029462, 0 )
581 (-1.0,0):( -1.18839510577812, 0 )
582 (-0.5,0):( -2.08582964293349, 0 )
583 ( 0.5,0):( 2.08582964293349, 0 )
584 ( 1.0,0):( 1.18839510577812, 0 )
585 ( 2.0,0):( 1.09975017029462, 0 )
588 ( 2, 3):( 0.09047320975321, 0.04120098628857)
589 (-2, 3):( -0.09047320975321, 0.04120098628857)
590 (-2,-3):( -0.09047320975321, -0.04120098628857)
591 ( 2,-3):( 0.09047320975321, -0.04120098628857)
594 (-2.0,0):( 0.45765755436029, 0 )
595 (-1.0,0):( -0.64209261593433, 0 )
596 (-0.5,0):( -1.83048772171245, 0 )
597 ( 0.5,0):( 1.83048772171245, 0 )
598 ( 1.0,0):( 0.64209261593433, 0 )
599 ( 2.0,0):( -0.45765755436029, 0 )
602 ( 2, 3):( -0.00373971037634, -0.99675779656936)
603 (-2, 3):( 0.00373971037634, -0.99675779656936)
604 (-2,-3):( 0.00373971037634, 0.99675779656936)
605 ( 2,-3):( -0.00373971037634, 0.99675779656936)
608 (-2.0,0):( -1.57079632679490, 1.31695789692482)
609 (-1.0,0):( -1.57079632679490, 0 )
610 (-0.5,0):( -0.52359877559830, 0 )
612 ( 0.5,0):( 0.52359877559830, 0 )
613 ( 1.0,0):( 1.57079632679490, 0 )
614 ( 2.0,0):( 1.57079632679490, -1.31695789692482)
617 ( 2, 3):( 0.57065278432110, 1.98338702991654)
618 (-2, 3):( -0.57065278432110, 1.98338702991654)
619 (-2,-3):( -0.57065278432110, -1.98338702991654)
620 ( 2,-3):( 0.57065278432110, -1.98338702991654)
623 (-2.0,0):( 3.14159265358979, -1.31695789692482)
624 (-1.0,0):( 3.14159265358979, 0 )
625 (-0.5,0):( 2.09439510239320, 0 )
626 ( 0.0,0):( 1.57079632679490, 0 )
627 ( 0.5,0):( 1.04719755119660, 0 )
629 ( 2.0,0):( 0 , 1.31695789692482)
632 ( 2, 3):( 1.00014354247380, -1.98338702991654)
633 (-2, 3):( 2.14144911111600, -1.98338702991654)
634 (-2,-3):( 2.14144911111600, 1.98338702991654)
635 ( 2,-3):( 1.00014354247380, 1.98338702991654)
638 (-2.0,0):( -1.10714871779409, 0 )
639 (-1.0,0):( -0.78539816339745, 0 )
640 (-0.5,0):( -0.46364760900081, 0 )
642 ( 0.5,0):( 0.46364760900081, 0 )
643 ( 1.0,0):( 0.78539816339745, 0 )
644 ( 2.0,0):( 1.10714871779409, 0 )
647 ( 2, 3):( 1.40992104959658, 0.22907268296854)
648 (-2, 3):( -1.40992104959658, 0.22907268296854)
649 (-2,-3):( -1.40992104959658, -0.22907268296854)
650 ( 2,-3):( 1.40992104959658, -0.22907268296854)
653 (-2.0,0):( 2.09439510239320, 0 )
654 (-1.0,0):( 3.14159265358979, 0 )
655 (-0.5,0):( 3.14159265358979, -1.31695789692482)
656 ( 0.5,0):( 0 , 1.31695789692482)
658 ( 2.0,0):( 1.04719755119660, 0 )
661 ( 2, 3):( 1.42041072246703, 0.23133469857397)
662 (-2, 3):( 1.72118193112276, 0.23133469857397)
663 (-2,-3):( 1.72118193112276, -0.23133469857397)
664 ( 2,-3):( 1.42041072246703, -0.23133469857397)
667 (-2.0,0):( -0.52359877559830, 0 )
668 (-1.0,0):( -1.57079632679490, 0 )
669 (-0.5,0):( -1.57079632679490, 1.31695789692482)
670 ( 0.5,0):( 1.57079632679490, -1.31695789692482)
671 ( 1.0,0):( 1.57079632679490, 0 )
672 ( 2.0,0):( 0.52359877559830, 0 )
675 ( 2, 3):( 0.15038560432786, -0.23133469857397)
676 (-2, 3):( -0.15038560432786, -0.23133469857397)
677 (-2,-3):( -0.15038560432786, 0.23133469857397)
678 ( 2,-3):( 0.15038560432786, 0.23133469857397)
681 (-2.0,0):( -0.46364760900081, 0 )
682 (-1.0,0):( -0.78539816339745, 0 )
683 (-0.5,0):( -1.10714871779409, 0 )
684 ( 0.5,0):( 1.10714871779409, 0 )
685 ( 1.0,0):( 0.78539816339745, 0 )
686 ( 2.0,0):( 0.46364760900081, 0 )
689 ( 2, 3):( 0.16087527719832, -0.22907268296854)
690 (-2, 3):( -0.16087527719832, -0.22907268296854)
691 (-2,-3):( -0.16087527719832, 0.22907268296854)
692 ( 2,-3):( 0.16087527719832, 0.22907268296854)
695 (-2.0,0):( -3.62686040784702, 0 )
696 (-1.0,0):( -1.17520119364380, 0 )
697 (-0.5,0):( -0.52109530549375, 0 )
699 ( 0.5,0):( 0.52109530549375, 0 )
700 ( 1.0,0):( 1.17520119364380, 0 )
701 ( 2.0,0):( 3.62686040784702, 0 )
704 ( 2, 3):( -3.59056458998578, 0.53092108624852)
705 (-2, 3):( 3.59056458998578, 0.53092108624852)
706 (-2,-3):( 3.59056458998578, -0.53092108624852)
707 ( 2,-3):( -3.59056458998578, -0.53092108624852)
710 (-2.0,0):( 3.76219569108363, 0 )
711 (-1.0,0):( 1.54308063481524, 0 )
712 (-0.5,0):( 1.12762596520638, 0 )
714 ( 0.5,0):( 1.12762596520638, 0 )
715 ( 1.0,0):( 1.54308063481524, 0 )
716 ( 2.0,0):( 3.76219569108363, 0 )
719 ( 2, 3):( -3.72454550491532, 0.51182256998738)
720 (-2, 3):( -3.72454550491532, -0.51182256998738)
721 (-2,-3):( -3.72454550491532, 0.51182256998738)
722 ( 2,-3):( -3.72454550491532, -0.51182256998738)
725 (-2.0,0):( -0.96402758007582, 0 )
726 (-1.0,0):( -0.76159415595576, 0 )
727 (-0.5,0):( -0.46211715726001, 0 )
729 ( 0.5,0):( 0.46211715726001, 0 )
730 ( 1.0,0):( 0.76159415595576, 0 )
731 ( 2.0,0):( 0.96402758007582, 0 )
734 ( 2, 3):( 0.96538587902213, -0.00988437503832)
735 (-2, 3):( -0.96538587902213, -0.00988437503832)
736 (-2,-3):( -0.96538587902213, 0.00988437503832)
737 ( 2,-3):( 0.96538587902213, 0.00988437503832)
740 (-2.0,0):( 0.26580222883408, 0 )
741 (-1.0,0):( 0.64805427366389, 0 )
742 (-0.5,0):( 0.88681888397007, 0 )
744 ( 0.5,0):( 0.88681888397007, 0 )
745 ( 1.0,0):( 0.64805427366389, 0 )
746 ( 2.0,0):( 0.26580222883408, 0 )
749 ( 2, 3):( -0.26351297515839, -0.03621163655877)
750 (-2, 3):( -0.26351297515839, 0.03621163655877)
751 (-2,-3):( -0.26351297515839, -0.03621163655877)
752 ( 2,-3):( -0.26351297515839, 0.03621163655877)
755 (-2.0,0):( -0.27572056477178, 0 )
756 (-1.0,0):( -0.85091812823932, 0 )
757 (-0.5,0):( -1.91903475133494, 0 )
758 ( 0.5,0):( 1.91903475133494, 0 )
759 ( 1.0,0):( 0.85091812823932, 0 )
760 ( 2.0,0):( 0.27572056477178, 0 )
763 ( 2, 3):( -0.27254866146294, -0.04030057885689)
764 (-2, 3):( 0.27254866146294, -0.04030057885689)
765 (-2,-3):( 0.27254866146294, 0.04030057885689)
766 ( 2,-3):( -0.27254866146294, 0.04030057885689)
769 (-2.0,0):( -1.03731472072755, 0 )
770 (-1.0,0):( -1.31303528549933, 0 )
771 (-0.5,0):( -2.16395341373865, 0 )
772 ( 0.5,0):( 2.16395341373865, 0 )
773 ( 1.0,0):( 1.31303528549933, 0 )
774 ( 2.0,0):( 1.03731472072755, 0 )
777 ( 2, 3):( 1.03574663776500, 0.01060478347034)
778 (-2, 3):( -1.03574663776500, 0.01060478347034)
779 (-2,-3):( -1.03574663776500, -0.01060478347034)
780 ( 2,-3):( 1.03574663776500, -0.01060478347034)
783 (-2.0,0):( -1.44363547517881, 0 )
784 (-1.0,0):( -0.88137358701954, 0 )
785 (-0.5,0):( -0.48121182505960, 0 )
787 ( 0.5,0):( 0.48121182505960, 0 )
788 ( 1.0,0):( 0.88137358701954, 0 )
789 ( 2.0,0):( 1.44363547517881, 0 )
792 ( 2, 3):( 1.96863792579310, 0.96465850440760)
793 (-2, 3):( -1.96863792579310, 0.96465850440761)
794 (-2,-3):( -1.96863792579310, -0.96465850440761)
795 ( 2,-3):( 1.96863792579310, -0.96465850440760)
798 (-2.0,0):( -1.31695789692482, 3.14159265358979)
799 (-1.0,0):( 0, 3.14159265358979)
800 (-0.5,0):( 0, 2.09439510239320)
801 ( 0.0,0):( 0, 1.57079632679490)
802 ( 0.5,0):( 0, 1.04719755119660)
804 ( 2.0,0):( 1.31695789692482, 0 )
807 ( 2, 3):( 1.98338702991654, 1.00014354247380)
808 (-2, 3):( -1.98338702991653, -2.14144911111600)
809 (-2,-3):( -1.98338702991653, 2.14144911111600)
810 ( 2,-3):( 1.98338702991654, -1.00014354247380)
813 (-2.0,0):( -0.54930614433405, 1.57079632679490)
814 (-0.5,0):( -0.54930614433405, 0 )
816 ( 0.5,0):( 0.54930614433405, 0 )
817 ( 2.0,0):( 0.54930614433405, 1.57079632679490)
820 ( 2, 3):( 0.14694666622553, 1.33897252229449)
821 (-2, 3):( -0.14694666622553, 1.33897252229449)
822 (-2,-3):( -0.14694666622553, -1.33897252229449)
823 ( 2,-3):( 0.14694666622553, -1.33897252229449)
826 (-2.0,0):( 0 , 2.09439510239320)
827 (-1.0,0):( 0 , 3.14159265358979)
828 (-0.5,0):( -1.31695789692482, 3.14159265358979)
829 ( 0.5,0):( 1.31695789692482, 0 )
831 ( 2.0,0):( 0 , 1.04719755119660)
834 ( 2, 3):( 0.23133469857397, -1.42041072246703)
835 (-2, 3):( -0.23133469857397, 1.72118193112276)
836 (-2,-3):( -0.23133469857397, -1.72118193112276)
837 ( 2,-3):( 0.23133469857397, 1.42041072246703)
840 (-2.0,0):( -0.48121182505960, 0 )
841 (-1.0,0):( -0.88137358701954, 0 )
842 (-0.5,0):( -1.44363547517881, 0 )
843 ( 0.5,0):( 1.44363547517881, 0 )
844 ( 1.0,0):( 0.88137358701954, 0 )
845 ( 2.0,0):( 0.48121182505960, 0 )
848 ( 2, 3):( 0.15735549884499, -0.22996290237721)
849 (-2, 3):( -0.15735549884499, -0.22996290237721)
850 (-2,-3):( -0.15735549884499, 0.22996290237721)
851 ( 2,-3):( 0.15735549884499, 0.22996290237721)
854 (-2.0,0):( -0.54930614433405, 0 )
855 (-0.5,0):( -0.54930614433405, 1.57079632679490)
856 ( 0.5,0):( 0.54930614433405, 1.57079632679490)
857 ( 2.0,0):( 0.54930614433405, 0 )
860 ( 2, 3):( 0.14694666622553, -0.23182380450040)
861 (-2, 3):( -0.14694666622553, -0.23182380450040)
862 (-2,-3):( -0.14694666622553, 0.23182380450040)
863 ( 2,-3):( 0.14694666622553, 0.23182380450040)