5.004_02: Complex/Trig: update

The following patches do not fix actual grave errors but they do:

- make the code more robust (more discontinuities catched)
  (e.g. atan(-i), atanh(-1))
- make the results agree on signs and/or conjugate forms with the
  results MATLAB gives: the results were already correct thanks to
  the periodicity of trig funcs but now they are also consistent.
  (e.g. acos(x) did have an unnecessary discontinuity at x = 0)
- for some pure real arguments short-circuit the calculation
  to avoid rounding errors (which make epsilons appear where
  clear zeros should reign)

Tested on NetBSD 1.2G i686, Linux 2.0.25 i686, Digital UNIX 4.0 EV56.

p5p-msgid: 199708081842.VAA31214@alpha.hut.fi

1 files changed, 228 insertions, 4 deletions

diff --git a/t/lib/complex.t b/t/lib/complex.t
index 80a56254ba..c05f40f2d3 100755
--- a/t/lib/complex.t
+++ b/t/lib/complex.t
@@ -62,6 +62,21 @@ sub test_dbz {
     }
 }
 
+# test the logofzeros
+
+sub test_loz {
+    for my $op (@_) {
+	$test++;
+
+#	push(@script, qq(print "# '$op'\n";));
+	push(@script, qq(eval '$op';));
+	push(@script, qq(print 'not ' unless (\$@ =~ /Logarithm of zero/);));
+	push(@script, qq(print "ok $test\n";));
+    }
+}
+
+my $minusi = cplx(0, -1);
+
 test_dbz(
 	 'i/0',
 #	 'tan(pi/2)',	# may succeed thanks to floating point inaccuracies
@@ -69,9 +84,11 @@ test_dbz(
 	 'csc(0)',
 	 'cot(0)',
 	 'atan(i)',
+	 'atan($minusi)',
 	 'asec(0)',
 	 'acsc(0)',
 	 'acot(i)',
+	 'acot($minusi)',
 #	 'tanh(pi/2)',	# may succeed thanks to floating point inaccuracies
 #	 'sech(pi/2)',	# may succeed thanks to floating point inaccuracies
 	 'csch(0)',
@@ -79,7 +96,12 @@ test_dbz(
 	 'atanh(1)',
 	 'asech(0)',
 	 'acsch(0)',
-	 'acoth(1)'
+	 'acoth(1)',
+	);
+
+test_loz(
+	 'atanh(-1)',
+	 'acoth(-1)',
 	);
 
 # test the 0**0
@@ -342,7 +364,7 @@ __END__
 |'z - ~z':'2*i*Im(z)'
 |'z * ~z':'abs(z) * abs(z)'
 
-{ (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
+{ (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] }
 
 |'(root(z, 4))[1] ** 4':'z'
 |'(root(z, 5))[3] ** 5':'z'
@@ -350,8 +372,8 @@ __END__
 |'abs(z)':'r'
 |'acot(z)':'acotan(z)'
 |'acsc(z)':'acosec(z)'
-|'acsc(z)':'asin(1 / z)'
-|'asec(z)':'acos(1 / z)'
+|'abs(acsc(z))':'abs(asin(1 / z))'
+|'abs(asec(z))':'abs(acos(1 / z))'
 |'cbrt(z)':'cbrt(r) * exp(i * t/3)'
 |'cos(acos(z))':'z'
 |'cos(z) ** 2 + sin(z) ** 2':1
@@ -409,144 +431,346 @@ __END__
 |'atanh(tanh(z))':'z'
 
 &sin
+(-2.0,0):(  -0.90929742682568,  0               )
+(-1.0,0):(  -0.84147098480790,  0               )
+(-0.5,0):(  -0.47942553860420,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.47942553860420,  0               )
+( 1.0,0):(   0.84147098480790,  0               )
+( 2.0,0):(   0.90929742682568,  0               )
+
+&sin
 ( 2, 3):(  9.15449914691143, -4.16890695996656)
 (-2, 3):( -9.15449914691143, -4.16890695996656)
 (-2,-3):( -9.15449914691143,  4.16890695996656)
 ( 2,-3):(  9.15449914691143,  4.16890695996656)
 
 &cos
+(-2.0,0):(  -0.41614683654714,  0               )
+(-1.0,0):(   0.54030230586814,  0               )
+(-0.5,0):(   0.87758256189037,  0               )
+( 0.0,0):(   1               ,  0               )
+( 0.5,0):(   0.87758256189037,  0               )
+( 1.0,0):(   0.54030230586814,  0               )
+( 2.0,0):(  -0.41614683654714,  0               )
+
+&cos
 ( 2, 3):( -4.18962569096881, -9.10922789375534)
 (-2, 3):( -4.18962569096881,  9.10922789375534)
 (-2,-3):( -4.18962569096881, -9.10922789375534)
 ( 2,-3):( -4.18962569096881,  9.10922789375534)
 
 &tan
+(-2.0,0):(   2.18503986326152,  0               )
+(-1.0,0):(  -1.55740772465490,  0               )
+(-0.5,0):(  -0.54630248984379,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.54630248984379,  0               )
+( 1.0,0):(   1.55740772465490,  0               )
+( 2.0,0):(  -2.18503986326152,  0               )
+
+&tan
 ( 2, 3):( -0.00376402564150,  1.00323862735361)
 (-2, 3):(  0.00376402564150,  1.00323862735361)
 (-2,-3):(  0.00376402564150, -1.00323862735361)
 ( 2,-3):( -0.00376402564150, -1.00323862735361)
 
 &sec
+(-2.0,0):(  -2.40299796172238,  0               )
+(-1.0,0):(   1.85081571768093,  0               )
+(-0.5,0):(   1.13949392732455,  0               )
+( 0.0,0):(   1               ,  0               )
+( 0.5,0):(   1.13949392732455,  0               )
+( 1.0,0):(   1.85081571768093,  0               )
+( 2.0,0):(  -2.40299796172238,  0               )
+
+&sec
 ( 2, 3):( -0.04167496441114,  0.09061113719624)
 (-2, 3):( -0.04167496441114, -0.09061113719624)
 (-2,-3):( -0.04167496441114,  0.09061113719624)
 ( 2,-3):( -0.04167496441114, -0.09061113719624)
 
 &csc
+(-2.0,0):(  -1.09975017029462,  0               )
+(-1.0,0):(  -1.18839510577812,  0               )
+(-0.5,0):(  -2.08582964293349,  0               )
+( 0.5,0):(   2.08582964293349,  0               )
+( 1.0,0):(   1.18839510577812,  0               )
+( 2.0,0):(   1.09975017029462,  0               )
+
+&csc
 ( 2, 3):(  0.09047320975321,  0.04120098628857)
 (-2, 3):( -0.09047320975321,  0.04120098628857)
 (-2,-3):( -0.09047320975321, -0.04120098628857)
 ( 2,-3):(  0.09047320975321, -0.04120098628857)
 
 &cot
+(-2.0,0):(   0.45765755436029,  0               )
+(-1.0,0):(  -0.64209261593433,  0               )
+(-0.5,0):(  -1.83048772171245,  0               )
+( 0.5,0):(   1.83048772171245,  0               )
+( 1.0,0):(   0.64209261593433,  0               )
+( 2.0,0):(  -0.45765755436029,  0               )
+
+&cot
 ( 2, 3):( -0.00373971037634, -0.99675779656936)
 (-2, 3):(  0.00373971037634, -0.99675779656936)
 (-2,-3):(  0.00373971037634,  0.99675779656936)
 ( 2,-3):( -0.00373971037634,  0.99675779656936)
 
 &asin
+(-2.0,0):(  -1.57079632679490,  1.31695789692482)
+(-1.0,0):(  -1.57079632679490,  0               )
+(-0.5,0):(  -0.52359877559830,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.52359877559830,  0               )
+( 1.0,0):(   1.57079632679490,  0               )
+( 2.0,0):(   1.57079632679490, -1.31695789692482)
+
+&asin
 ( 2, 3):(  0.57065278432110,  1.98338702991654)
 (-2, 3):( -0.57065278432110,  1.98338702991654)
 (-2,-3):( -0.57065278432110, -1.98338702991654)
 ( 2,-3):(  0.57065278432110, -1.98338702991654)
 
 &acos
+(-2.0,0):(   3.14159265358979, -1.31695789692482)
+(-1.0,0):(   3.14159265358979,  0               )
+(-0.5,0):(   2.09439510239320,  0               )
+( 0.0,0):(   1.57079632679490,  0               )
+( 0.5,0):(   1.04719755119660,  0               )
+( 1.0,0):(   0               ,  0               )
+( 2.0,0):(   0               ,  1.31695789692482)
+
+&acos
 ( 2, 3):(  1.00014354247380, -1.98338702991654)
 (-2, 3):(  2.14144911111600, -1.98338702991654)
 (-2,-3):(  2.14144911111600,  1.98338702991654)
 ( 2,-3):(  1.00014354247380,  1.98338702991654)
 
 &atan
+(-2.0,0):(  -1.10714871779409,  0               )
+(-1.0,0):(  -0.78539816339745,  0               )
+(-0.5,0):(  -0.46364760900081,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.46364760900081,  0               )
+( 1.0,0):(   0.78539816339745,  0               )
+( 2.0,0):(   1.10714871779409,  0               )
+
+&atan
 ( 2, 3):(  1.40992104959658,  0.22907268296854)
 (-2, 3):( -1.40992104959658,  0.22907268296854)
 (-2,-3):( -1.40992104959658, -0.22907268296854)
 ( 2,-3):(  1.40992104959658, -0.22907268296854)
 
 &asec
+(-2.0,0):(   2.09439510239320,  0               )
+(-1.0,0):(   3.14159265358979,  0               )
+(-0.5,0):(   3.14159265358979, -1.31695789692482)
+( 0.5,0):(   0               ,  1.31695789692482)
+( 1.0,0):(   0               ,  0               )
+( 2.0,0):(   1.04719755119660,  0               )
+
+&asec
 ( 2, 3):(  1.42041072246703,  0.23133469857397)
 (-2, 3):(  1.72118193112276,  0.23133469857397)
 (-2,-3):(  1.72118193112276, -0.23133469857397)
 ( 2,-3):(  1.42041072246703, -0.23133469857397)
 
 &acsc
+(-2.0,0):(  -0.52359877559830,  0               )
+(-1.0,0):(  -1.57079632679490,  0               )
+(-0.5,0):(  -1.57079632679490,  1.31695789692482)
+( 0.5,0):(   1.57079632679490, -1.31695789692482)
+( 1.0,0):(   1.57079632679490,  0               )
+( 2.0,0):(   0.52359877559830,  0               )
+
+&acsc
 ( 2, 3):(  0.15038560432786, -0.23133469857397)
 (-2, 3):( -0.15038560432786, -0.23133469857397)
 (-2,-3):( -0.15038560432786,  0.23133469857397)
 ( 2,-3):(  0.15038560432786,  0.23133469857397)
 
 &acot
+(-2.0,0):(  -0.46364760900081,  0               )
+(-1.0,0):(  -0.78539816339745,  0               )
+(-0.5,0):(  -1.10714871779409,  0               )
+( 0.5,0):(   1.10714871779409,  0               )
+( 1.0,0):(   0.78539816339745,  0               )
+( 2.0,0):(   0.46364760900081,  0               )
+
+&acot
 ( 2, 3):(  0.16087527719832, -0.22907268296854)
 (-2, 3):( -0.16087527719832, -0.22907268296854)
 (-2,-3):( -0.16087527719832,  0.22907268296854)
 ( 2,-3):(  0.16087527719832,  0.22907268296854)
 
 &sinh
+(-2.0,0):(  -3.62686040784702,  0               )
+(-1.0,0):(  -1.17520119364380,  0               )
+(-0.5,0):(  -0.52109530549375,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.52109530549375,  0               )
+( 1.0,0):(   1.17520119364380,  0               )
+( 2.0,0):(   3.62686040784702,  0               )
+
+&sinh
 ( 2, 3):( -3.59056458998578,  0.53092108624852)
 (-2, 3):(  3.59056458998578,  0.53092108624852)
 (-2,-3):(  3.59056458998578, -0.53092108624852)
 ( 2,-3):( -3.59056458998578, -0.53092108624852)
 
 &cosh
+(-2.0,0):(   3.76219569108363,  0               )
+(-1.0,0):(   1.54308063481524,  0               )
+(-0.5,0):(   1.12762596520638,  0               )
+( 0.0,0):(   1               ,  0               )
+( 0.5,0):(   1.12762596520638,  0               )
+( 1.0,0):(   1.54308063481524,  0               )
+( 2.0,0):(   3.76219569108363,  0               )
+
+&cosh
 ( 2, 3):( -3.72454550491532,  0.51182256998738)
 (-2, 3):( -3.72454550491532, -0.51182256998738)
 (-2,-3):( -3.72454550491532,  0.51182256998738)
 ( 2,-3):( -3.72454550491532, -0.51182256998738)
 
 &tanh
+(-2.0,0):(  -0.96402758007582,  0               )
+(-1.0,0):(  -0.76159415595576,  0               )
+(-0.5,0):(  -0.46211715726001,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.46211715726001,  0               )
+( 1.0,0):(   0.76159415595576,  0               )
+( 2.0,0):(   0.96402758007582,  0               )
+
+&tanh
 ( 2, 3):(  0.96538587902213, -0.00988437503832)
 (-2, 3):( -0.96538587902213, -0.00988437503832)
 (-2,-3):( -0.96538587902213,  0.00988437503832)
 ( 2,-3):(  0.96538587902213,  0.00988437503832)
 
 &sech
+(-2.0,0):(   0.26580222883408,  0               )
+(-1.0,0):(   0.64805427366389,  0               )
+(-0.5,0):(   0.88681888397007,  0               )
+( 0.0,0):(   1               ,  0               )
+( 0.5,0):(   0.88681888397007,  0               )
+( 1.0,0):(   0.64805427366389,  0               )
+( 2.0,0):(   0.26580222883408,  0               )
+
+&sech
 ( 2, 3):( -0.26351297515839, -0.03621163655877)
 (-2, 3):( -0.26351297515839,  0.03621163655877)
 (-2,-3):( -0.26351297515839, -0.03621163655877)
 ( 2,-3):( -0.26351297515839,  0.03621163655877)
 
 &csch
+(-2.0,0):(  -0.27572056477178,  0               )
+(-1.0,0):(  -0.85091812823932,  0               )
+(-0.5,0):(  -1.91903475133494,  0               )
+( 0.5,0):(   1.91903475133494,  0               )
+( 1.0,0):(   0.85091812823932,  0               )
+( 2.0,0):(   0.27572056477178,  0               )
+
+&csch
 ( 2, 3):( -0.27254866146294, -0.04030057885689)
 (-2, 3):(  0.27254866146294, -0.04030057885689)
 (-2,-3):(  0.27254866146294,  0.04030057885689)
 ( 2,-3):( -0.27254866146294,  0.04030057885689)
 
 &coth
+(-2.0,0):(  -1.03731472072755,  0               )
+(-1.0,0):(  -1.31303528549933,  0               )
+(-0.5,0):(  -2.16395341373865,  0               )
+( 0.5,0):(   2.16395341373865,  0               )
+( 1.0,0):(   1.31303528549933,  0               )
+( 2.0,0):(   1.03731472072755,  0               )
+
+&coth
 ( 2, 3):(  1.03574663776500,  0.01060478347034)
 (-2, 3):( -1.03574663776500,  0.01060478347034)
 (-2,-3):( -1.03574663776500, -0.01060478347034)
 ( 2,-3):(  1.03574663776500, -0.01060478347034)
 
 &asinh
+(-2.0,0):(  -1.44363547517881,  0               )
+(-1.0,0):(  -0.88137358701954,  0               )
+(-0.5,0):(  -0.48121182505960,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.48121182505960,  0               )
+( 1.0,0):(   0.88137358701954,  0               )
+( 2.0,0):(   1.44363547517881,  0               )
+
+&asinh
 ( 2, 3):(  1.96863792579310,  0.96465850440760)
 (-2, 3):( -1.96863792579310,  0.96465850440761)
 (-2,-3):( -1.96863792579310, -0.96465850440761)
 ( 2,-3):(  1.96863792579310, -0.96465850440760)
 
 &acosh
+(-2.0,0):(  -1.31695789692482,  3.14159265358979)
+(-1.0,0):(   0,                 3.14159265358979)
+(-0.5,0):(   0,                 2.09439510239320)
+( 0.0,0):(   0,                 1.57079632679490)
+( 0.5,0):(   0,                 1.04719755119660)
+( 1.0,0):(   0               ,  0               )
+( 2.0,0):(   1.31695789692482,  0               )
+
+&acosh
 ( 2, 3):(  1.98338702991654,  1.00014354247380)
 (-2, 3):( -1.98338702991653, -2.14144911111600)
 (-2,-3):( -1.98338702991653,  2.14144911111600)
 ( 2,-3):(  1.98338702991654, -1.00014354247380)
 
 &atanh
+(-2.0,0):(  -0.54930614433405,  1.57079632679490)
+(-0.5,0):(  -0.54930614433405,  0               )
+( 0.0,0):(   0               ,  0               )
+( 0.5,0):(   0.54930614433405,  0               )
+( 2.0,0):(   0.54930614433405,  1.57079632679490)
+
+&atanh
 ( 2, 3):(  0.14694666622553,  1.33897252229449)
 (-2, 3):( -0.14694666622553,  1.33897252229449)
 (-2,-3):( -0.14694666622553, -1.33897252229449)
 ( 2,-3):(  0.14694666622553, -1.33897252229449)
 
 &asech
+(-2.0,0):(   0               , 2.09439510239320)
+(-1.0,0):(   0               , 3.14159265358979)
+(-0.5,0):(  -1.31695789692482, 3.14159265358979)
+( 0.5,0):(   1.31695789692482, 0               )
+( 1.0,0):(   0               , 0               )
+( 2.0,0):(   0               , 1.04719755119660)
+
+&asech
 ( 2, 3):(  0.23133469857397, -1.42041072246703)
 (-2, 3):( -0.23133469857397,  1.72118193112276)
 (-2,-3):( -0.23133469857397, -1.72118193112276)
 ( 2,-3):(  0.23133469857397,  1.42041072246703)
 
 &acsch
+(-2.0,0):(  -0.48121182505960, 0               )
+(-1.0,0):(  -0.88137358701954, 0               )
+(-0.5,0):(  -1.44363547517881, 0               )
+( 0.5,0):(   1.44363547517881, 0               )
+( 1.0,0):(   0.88137358701954, 0               )
+( 2.0,0):(   0.48121182505960, 0               )
+
+&acsch
 ( 2, 3):(  0.15735549884499, -0.22996290237721)
 (-2, 3):( -0.15735549884499, -0.22996290237721)
 (-2,-3):( -0.15735549884499,  0.22996290237721)
 ( 2,-3):(  0.15735549884499,  0.22996290237721)
 
 &acoth
+(-2.0,0):(  -0.54930614433405, 0               )
+(-0.5,0):(  -0.54930614433405, 1.57079632679490)
+( 0.5,0):(   0.54930614433405, 1.57079632679490)
+( 2.0,0):(   0.54930614433405, 0               )
+
+&acoth
 ( 2, 3):(  0.14694666622553, -0.23182380450040)
 (-2, 3):( -0.14694666622553, -0.23182380450040)
 (-2,-3):( -0.14694666622553,  0.23182380450040)