Be more robust on "extreme" (large absolute value)

arguments.  Originally reported by Daniel Connelly
as a problem with asinh() on large negative arguments,
asinh() used to bail out because an argument to log()
ended up being zero.  Ilya Zakharevich proposed using
Taylor's series in such cases, which for such large
arguments is a very good approximation.

p4raw-id: //depot/cfgperl@5951

1 files changed, 179 insertions, 147 deletions

diff --git a/lib/Math/Complex.pm b/lib/Math/Complex.pm
index 1a47f4af5e..b13ab77f7a 100644
--- a/lib/Math/Complex.pm
+++ b/lib/Math/Complex.pm
@@ -13,7 +13,7 @@ use strict;
 
 our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS);
 
-my ( $i, $ip2, %logn );
+my ( $i, %logn );
 
 $VERSION = sprintf("%s", q$Id: Complex.pm,v 1.26 1998/11/01 00:00:00 dsl Exp $ =~ /(\d+\.\d+)/);
 
@@ -49,6 +49,7 @@ use overload
 	'*'	=> \&multiply,
 	'/'	=> \&divide,
 	'**'	=> \&power,
+	'=='	=> \&numeq,
 	'<=>'	=> \&spaceship,
 	'neg'	=> \&negate,
 	'~'	=> \&conjugate,
@@ -71,6 +72,9 @@ my %DISPLAY_FORMAT = ('style' => 'cartesian',
 		      'polar_pretty_print' => 1);
 my $eps            = 1e-14;		# Epsilon
 
+my $Inf		   = CORE::exp(CORE::exp(30));
+$Inf = "Inf" if !defined $Inf || !$Inf > 0;
+
 #
 # Object attributes (internal):
 #	cartesian	[real, imaginary] -- cartesian form
@@ -228,6 +232,13 @@ sub i () {
 }
 
 #
+# ip2
+#
+# Half of i.
+#
+sub ip2 () { i / 2 }
+
+#
 # Attribute access/set routines
 #
 
@@ -262,7 +273,8 @@ sub update_polar {
 	my ($x, $y) = @{$self->{'cartesian'}};
 	$self->{p_dirty} = 0;
 	return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
-	return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), CORE::atan2($y, $x)];
+	return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
+				   CORE::atan2($y, $x)];
 }
 
 #
@@ -342,7 +354,7 @@ sub _divbyzero {
 
     if (defined $_[1]) {
 	$mess .= "(Because in the definition of $_[0], the divisor ";
-	$mess .= "$_[1] " unless ($_[1] eq '0');
+	$mess .= "$_[1] " unless ("$_[1]" eq '0');
 	$mess .= "is 0)\n";
     }
 
@@ -416,8 +428,8 @@ sub power {
 	    return 1 if $z2 == 0 || $z1 == 1;
 	    return 0 if $z1 == 0 && Re($z2) > 0;
 	}
-	my $w = $inverted ? CORE::exp($z1 * CORE::log($z2))
-	                  : CORE::exp($z2 * CORE::log($z1));
+	my $w = $inverted ? &exp($z1 * &log($z2))
+	                  : &exp($z2 * &log($z1));
 	# If both arguments cartesian, return cartesian, else polar.
 	return $z1->{c_dirty} == 0 &&
 	       (not ref $z2 or $z2->{c_dirty} == 0) ?
@@ -440,6 +452,19 @@ sub spaceship {
 }
 
 #
+# (numeq)
+#
+# Computes z1 == z2.
+#
+# (Required in addition to spaceship() because of NaNs.)
+sub numeq {
+	my ($z1, $z2, $inverted) = @_;
+	my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
+	my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+	return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
+}
+
+#
 # (negate)
 #
 # Computes -z.
@@ -477,7 +502,13 @@ sub conjugate {
 #
 sub abs {
 	my ($z, $rho) = @_;
-	return $z unless ref $z;
+	unless (ref $z) {
+	    if (@_ == 2) {
+		$_[0] = $_[1];
+	    } else {
+		return CORE::abs($z);
+	    }
+	}
 	if (defined $rho) {
 	    $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
 	    $z->{p_dirty} = 0;
@@ -533,7 +564,8 @@ sub arg {
 sub sqrt {
 	my ($z) = @_;
 	my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
-	return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) if $im == 0;
+	return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
+	    if $im == 0;
 	my ($r, $t) = @{$z->polar};
 	return (ref $z)->emake(CORE::sqrt($r), $t/2);
 }
@@ -547,9 +579,12 @@ sub sqrt {
 #
 sub cbrt {
 	my ($z) = @_;
-	return $z < 0 ? -CORE::exp(CORE::log(-$z)/3) : ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
+	return $z < 0 ?
+	    -CORE::exp(CORE::log(-$z)/3) :
+		($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
 	    unless ref $z;
 	my ($r, $t) = @{$z->polar};
+	return 0 if $r == 0;
 	return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
 }
 
@@ -559,7 +594,7 @@ sub cbrt {
 # Die on bad root.
 #
 sub _rootbad {
-    my $mess = "Root $_[0] not defined, root must be positive integer.\n";
+    my $mess = "Root $_[0] illegal, root rank must be positive integer.\n";
 
     my @up = caller(1);
 
@@ -581,7 +616,8 @@ sub _rootbad {
 sub root {
 	my ($z, $n) = @_;
 	_rootbad($n) if ($n < 1 or int($n) != $n);
-	my ($r, $t) = ref $z ? @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
+	my ($r, $t) = ref $z ?
+	    @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
 	my @root;
 	my $k;
 	my $theta_inc = pit2 / $n;
@@ -725,7 +761,7 @@ sub logn {
 	$z = cplx($z, 0) unless ref $z;
 	my $logn = $logn{$n};
 	$logn = $logn{$n} = CORE::log($n) unless defined $logn;	# Cache log(n)
-	return CORE::log($z) / $logn;
+	return &log($z) / $logn;
 }
 
 #
@@ -735,11 +771,14 @@ sub logn {
 #
 sub cos {
 	my ($z) = @_;
+	return CORE::cos($z) unless ref $z;
 	my ($x, $y) = @{$z->cartesian};
 	my $ey = CORE::exp($y);
-	my $ey_1 = 1 / $ey;
-	return (ref $z)->make(CORE::cos($x) * ($ey + $ey_1)/2,
-			      CORE::sin($x) * ($ey_1 - $ey)/2);
+	my $sx = CORE::sin($x);
+	my $cx = CORE::cos($x);
+	my $ey_1 = $ey ? 1 / $ey : $Inf;
+	return (ref $z)->make($cx * ($ey + $ey_1)/2,
+			      $sx * ($ey_1 - $ey)/2);
 }
 
 #
@@ -749,11 +788,14 @@ sub cos {
 #
 sub sin {
 	my ($z) = @_;
+	return CORE::sin($z) unless ref $z;
 	my ($x, $y) = @{$z->cartesian};
 	my $ey = CORE::exp($y);
-	my $ey_1 = 1 / $ey;
-	return (ref $z)->make(CORE::sin($x) * ($ey + $ey_1)/2,
-			      CORE::cos($x) * ($ey - $ey_1)/2);
+	my $sx = CORE::sin($x);
+	my $cx = CORE::cos($x);
+	my $ey_1 = $ey ? 1 / $ey : $Inf;
+	return (ref $z)->make($sx * ($ey + $ey_1)/2,
+			      $cx * ($ey - $ey_1)/2);
 }
 
 #
@@ -763,9 +805,9 @@ sub sin {
 #
 sub tan {
 	my ($z) = @_;
-	my $cz = CORE::cos($z);
-	_divbyzero "tan($z)", "cos($z)" if (CORE::abs($cz) < $eps);
-	return CORE::sin($z) / $cz;
+	my $cz = &cos($z);
+	_divbyzero "tan($z)", "cos($z)" if $cz == 0;
+	return &sin($z) / $cz;
 }
 
 #
@@ -775,7 +817,7 @@ sub tan {
 #
 sub sec {
 	my ($z) = @_;
-	my $cz = CORE::cos($z);
+	my $cz = &cos($z);
 	_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
 	return 1 / $cz;
 }
@@ -787,7 +829,7 @@ sub sec {
 #
 sub csc {
 	my ($z) = @_;
-	my $sz = CORE::sin($z);
+	my $sz = &sin($z);
 	_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
 	return 1 / $sz;
 }
@@ -806,9 +848,9 @@ sub cosec { Math::Complex::csc(@_) }
 #
 sub cot {
 	my ($z) = @_;
-	my $sz = CORE::sin($z);
+	my $sz = &sin($z);
 	_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
-	return CORE::cos($z) / $sz;
+	return &cos($z) / $sz;
 }
 
 #
@@ -825,8 +867,10 @@ sub cotan { Math::Complex::cot(@_) }
 #
 sub acos {
 	my $z = $_[0];
-	return CORE::atan2(CORE::sqrt(1-$z*$z), $z) if (! ref $z) && CORE::abs($z) <= 1;
+	return CORE::atan2(CORE::sqrt(1-$z*$z), $z)
+	    if (! ref $z) && CORE::abs($z) <= 1;
 	my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+	return 0 if $x == 1 && $y == 0;
 	my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
 	my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
 	my $alpha = ($t1 + $t2)/2;
@@ -847,8 +891,10 @@ sub acos {
 #
 sub asin {
 	my $z = $_[0];
-	return CORE::atan2($z, CORE::sqrt(1-$z*$z)) if (! ref $z) && CORE::abs($z) <= 1;
+	return CORE::atan2($z, CORE::sqrt(1-$z*$z))
+	    if (! ref $z) && CORE::abs($z) <= 1;
 	my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+	return 0 if $x == 0 && $y == 0;
 	my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
 	my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
 	my $alpha = ($t1 + $t2)/2;
@@ -870,11 +916,12 @@ sub asin {
 sub atan {
 	my ($z) = @_;
 	return CORE::atan2($z, 1) unless ref $z;
+	my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+	return 0 if $x == 0 && $y == 0;
 	_divbyzero "atan(i)"  if ( $z == i);
-	_divbyzero "atan(-i)" if (-$z == i);
-	my $log = CORE::log((i + $z) / (i - $z));
-	$ip2 = 0.5 * i unless defined $ip2;
-	return $ip2 * $log;
+	_logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
+	my $log = &log((i + $z) / (i - $z));
+	return ip2 * $log;
 }
 
 #
@@ -913,10 +960,11 @@ sub acosec { Math::Complex::acsc(@_) }
 #
 sub acot {
 	my ($z) = @_;
-	_divbyzero "acot(0)"  if (CORE::abs($z)     < $eps);
-	return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) unless ref $z;
-	_divbyzero "acot(i)"  if (CORE::abs($z - i) < $eps);
-	_logofzero "acot(-i)" if (CORE::abs($z + i) < $eps);
+	_divbyzero "acot(0)"  if $z == 0;
+	return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
+	    unless ref $z;
+	_divbyzero "acot(i)"  if ($z - i == 0);
+	_logofzero "acot(-i)" if ($z + i == 0);
 	return atan(1 / $z);
 }
 
@@ -937,11 +985,13 @@ sub cosh {
 	my $ex;
 	unless (ref $z) {
 	    $ex = CORE::exp($z);
-	    return ($ex + 1/$ex)/2;
+	    return $ex ? ($ex + 1/$ex)/2 : $Inf;
 	}
 	my ($x, $y) = @{$z->cartesian};
+	my $cy = CORE::cos($y);
+	my $sy = CORE::cos($y);
 	$ex = CORE::exp($x);
-	my $ex_1 = 1 / $ex;
+	my $ex_1 = $ex ? 1 / $ex : $Inf;
 	return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
 			      CORE::sin($y) * ($ex - $ex_1)/2);
 }
@@ -955,14 +1005,17 @@ sub sinh {
 	my ($z) = @_;
 	my $ex;
 	unless (ref $z) {
+	    return 0 if $z == 0;
 	    $ex = CORE::exp($z);
-	    return ($ex - 1/$ex)/2;
+	    return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
 	}
 	my ($x, $y) = @{$z->cartesian};
+	my $cy = CORE::cos($y);
+	my $sy = CORE::sin($y);
 	$ex = CORE::exp($x);
-	my $ex_1 = 1 / $ex;
-	return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
-			      CORE::sin($y) * ($ex + $ex_1)/2);
+	my $ex_1 = $ex ? 1 / $ex : $Inf;
+	return (ref $z)->make($cy * ($ex - $ex_1)/2,
+			      $sy * ($ex + $ex_1)/2);
 }
 
 #
@@ -1016,7 +1069,7 @@ sub cosech { Math::Complex::csch(@_) }
 sub coth {
 	my ($z) = @_;
 	my $sz = sinh($z);
-	_divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
+	_divbyzero "coth($z)", "sinh($z)" if $sz == 0;
 	return cosh($z) / $sz;
 }
 
@@ -1035,25 +1088,37 @@ sub cotanh { Math::Complex::coth(@_) }
 sub acosh {
 	my ($z) = @_;
 	unless (ref $z) {
-	    return CORE::log($z + CORE::sqrt($z*$z-1)) if $z >= 1;
 	    $z = cplx($z, 0);
 	}
 	my ($re, $im) = @{$z->cartesian};
 	if ($im == 0) {
-	    return cplx(CORE::log($re + CORE::sqrt($re*$re - 1)), 0) if $re >= 1;
-	    return cplx(0, CORE::atan2(CORE::sqrt(1-$re*$re), $re)) if CORE::abs($re) <= 1;
+	    return CORE::log($re + CORE::sqrt($re*$re - 1))
+		if $re >= 1;
+	    return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
+		if CORE::abs($re) < 1;
 	}
-	return CORE::log($z + CORE::sqrt($z*$z - 1));
+	my $s = &sqrt($z*$z - 1);
+	my $t = $z + $s;
+	$t = 1/(2*$s) if $t == 0 || $t && &abs(cosh(&log($t)) - $z) > $eps;
+	return &log($t);
 }
 
 #
 # asinh
 #
-# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1))
+# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
 #
 sub asinh {
 	my ($z) = @_;
-	return CORE::log($z + CORE::sqrt($z*$z + 1));
+	unless (ref $z) {
+	    my $t = $z + CORE::sqrt($z*$z + 1);
+	    return CORE::log($t) if $t;
+	}
+	my $s = &sqrt($z*$z + 1);
+	my $t = $z + $s;
+	# Try Taylor series if looking bad.
+	$t = 1/(2*$s) if $t == 0 || $t && &abs(sinh(&log($t)) - $z) > $eps;
+	return &log($t);
 }
 
 #
@@ -1067,9 +1132,9 @@ sub atanh {
 	    return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
 	    $z = cplx($z, 0);
 	}
-	_divbyzero 'atanh(1)',  "1 - $z" if ($z ==  1);
-	_logofzero 'atanh(-1)'           if ($z == -1);
-	return 0.5 * CORE::log((1 + $z) / (1 - $z));
+	_divbyzero 'atanh(1)',  "1 - $z" if (1 - $z == 0);
+	_logofzero 'atanh(-1)'           if (1 + $z == 0);
+	return 0.5 * &log((1 + $z) / (1 - $z));
 }
 
 #
@@ -1079,7 +1144,7 @@ sub atanh {
 #
 sub asech {
 	my ($z) = @_;
-	_divbyzero 'asech(0)', $z if ($z == 0);
+	_divbyzero 'asech(0)', "$z" if ($z == 0);
 	return acosh(1 / $z);
 }
 
@@ -1108,14 +1173,14 @@ sub acosech { Math::Complex::acsch(@_) }
 #
 sub acoth {
 	my ($z) = @_;
-	_divbyzero 'acoth(0)'            if (CORE::abs($z)     < $eps);
+	_divbyzero 'acoth(0)'            if ($z == 0);
 	unless (ref $z) {
 	    return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
 	    $z = cplx($z, 0);
 	}
-	_divbyzero 'acoth(1)',  "$z - 1" if (CORE::abs($z - 1) < $eps);
-	_logofzero 'acoth(-1)', "1 / $z" if (CORE::abs($z + 1) < $eps);
-	return CORE::log((1 + $z) / ($z - 1)) / 2;
+	_divbyzero 'acoth(1)',  "$z - 1" if ($z - 1 == 0);
+	_logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
+	return &log((1 + $z) / ($z - 1)) / 2;
 }
 
 #
@@ -1141,8 +1206,8 @@ sub atan2 {
 	    ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
 	}
 	if ($im2 == 0) {
-	    return cplx(CORE::atan2($re1, $re2), 0) if $im1 == 0;
-	    return cplx(($im1<=>0) * pip2, 0) if $re2 == 0;
+	    return CORE::atan2($re1, $re2) if $im1 == 0;
+	    return ($im1<=>0) * pip2 if $re2 == 0;
 	}
 	my $w = atan($z1/$z2);
 	my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0);
@@ -1235,67 +1300,57 @@ sub stringify_cartesian {
 	my ($x, $y) = @{$z->cartesian};
 	my ($re, $im);
 
-	$x = int($x + ($x < 0 ? -1 : 1) * $eps)
-		if int(CORE::abs($x)) != int(CORE::abs($x) + $eps);
-	$y = int($y + ($y < 0 ? -1 : 1) * $eps)
-		if int(CORE::abs($y)) != int(CORE::abs($y) + $eps);
-
-	$re = "$x" if CORE::abs($x) >= $eps;
-
 	my %format = $z->display_format;
 	my $format = $format{format};
 
-	if ($y == 1)			       { $im = 'i' }
-	elsif ($y == -1)		       { $im = '-i' }
-	elsif (CORE::abs($y) >= $eps) {
-	    $im = (defined $format ? sprintf($format, $y) : $y) . "i";
+	if ($x) {
+	    if ($x =~ /^NaN[QS]?$/i) {
+		$re = $x;
+	    } else {
+		if ($x =~ /^-?$Inf$/oi) {
+		    $re = $x;
+		} else {
+		    $re = defined $format ? sprintf($format, $x) : $x;
+		}
+	    }
+	} else {
+	    undef $re;
+	}
+
+	if ($y) {
+	    if ($y == 1)     { $im = ""  }
+	    elsif ($y == -1) { $im = "-" }
+	    elsif ($y =~ /^(NaN[QS]?)$/i) {
+		$im = $y;
+	    } else {
+		if ($y =~ /^-?$Inf$/oi) {
+		    $im = $y;
+		} else {
+		    $im = defined $format ? sprintf($format, $y) : $y;
+		}
+	    }
+	    $im .= "i";
+	} else {
+	    undef $im;
 	}
 
-	my $str = '';
-	$str = defined $format ? sprintf($format, $re) : $re
-	    if defined $re;
+	my $str = $re;
+
 	if (defined $im) {
 	    if ($y < 0) {
 		$str .= $im;
-	    } elsif ($y > 0)  {
+	    } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i)  {
 		$str .= "+" if defined $re;
 		$str .= $im;
 	    }
+	} elsif (!defined $re) {
+	    $str = "0";
 	}
 
 	return $str;
 }
 
 
-# Helper for stringify_polar, a Greatest Common Divisor with a memory.
-
-sub _gcd {
-    my ($a, $b) = @_;
-
-    use integer;
-
-    # Loops forever if given negative inputs.
-
-    if    ($b and $a > $b) { return gcd($a % $b, $b) }
-    elsif ($a and $b > $a) { return gcd($b % $a, $a) }
-    else                   { return $a ? $a : $b     }
-}
-
-my %gcd;
-
-sub gcd {
-    my ($a, $b) = @_;
-
-    my $id = "$a $b";
-
-    unless (exists $gcd{$id}) {
-	$gcd{$id} = _gcd($a, $b);
-	$gcd{"$b $a"} = $gcd{$id};
-    }
-
-    return $gcd{$id};
-}
-
 #
 # ->stringify_polar
 #
@@ -1306,68 +1361,45 @@ sub stringify_polar {
 	my ($r, $t) = @{$z->polar};
 	my $theta;
 
-	return '[0,0]' if $r <= $eps;
-
 	my %format = $z->display_format;
+	my $format = $format{format};
 
-	my $nt = $t / pit2;
-	$nt = ($nt - int($nt)) * pit2;
-	$nt += pit2 if $nt < 0;			# Range [0, 2pi]
-
-	if (CORE::abs($nt) <= $eps)		{ $theta = 0 }
-	elsif (CORE::abs(pi-$nt) <= $eps)	{ $theta = 'pi' }
-
-	if (defined $theta) {
-		$r = int($r + ($r < 0 ? -1 : 1) * $eps)
-			if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
-		$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
-			if ($theta ne 'pi' and
-			    int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
-		return "\[$r,$theta\]";
+	if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) {
+	    $theta = $t; 
+	} elsif ($t == pi) {
+	    $theta = "pi";
+	} elsif ($r == 0 || $t == 0) {
+	    $theta = defined $format ? sprintf($format, $t) : $t;
 	}
 
+	return "[$r,$theta]" if defined $theta;
+
 	#
-	# Okay, number is not a real. Try to identify pi/n and friends...
+	# Try to identify pi/n and friends.
 	#
 
-	$nt -= pit2 if $nt > pi;
-
-	if ($format{polar_pretty_print} && CORE::abs($nt) >= deg1) {
-	    my ($n, $k, $kpi);
-
-	    for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
-		$n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
-		if (CORE::abs($kpi/$n - $nt) <= $eps) {
-		    $n = CORE::abs($n);
-		    my $gcd = gcd($k, $n);
-		    if ($gcd > 1) {
-			$k /= $gcd;
-			$n /= $gcd;
-		    }
-		    next if $n > 360;
-		    $theta = ($nt < 0 ? '-':'').
-			     ($k == 1 ? 'pi':"${k}pi");
-		    $theta .= '/'.$n if $n > 1;
+	$t -= int(CORE::abs($t) / pit2) * pit2;
+
+	if ($format{polar_pretty_print}) {
+	    my ($a, $b);
+	    for $a (2, 3, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72) {
+		$b = $t * $a / pi;
+		if (int($b) == $b) {
+		    $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
+		    $theta = "${b}pi/$a";
 		    last;
 		}
 	    }
 	}
 
-	$theta = $nt unless defined $theta;
-
-	$r = int($r + ($r < 0 ? -1 : 1) * $eps)
-		if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
-	$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
-		if ($theta !~ m(^-?\d*pi/\d+$) and
-		    int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
-
-	my $format = $format{format};
         if (defined $format) {
 	    $r     = sprintf($format, $r);
-	    $theta = sprintf($format, $theta);
+	    $theta = sprintf($format, $theta) unless defined $theta;
+	} else {
+	    $theta = $t unless defined $theta;
 	}
 
-	return "\[$r,$theta\]";
+	return "[$r,$theta]";
 }
 
 1;