| 1 | #!./perl |
| 2 | |
| 3 | # $RCSfile: complex.t,v $ |
| 4 | # |
| 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 |
| 9 | |
| 10 | BEGIN { |
| 11 | chdir 't' if -d 't'; |
| 12 | @INC = '../lib'; |
| 13 | } |
| 14 | |
| 15 | use Math::Complex; |
| 16 | |
| 17 | use vars qw($VERSION); |
| 18 | |
| 19 | $VERSION = 1.91; |
| 20 | |
| 21 | my ($args, $op, $target, $test, $test_set, $try, $val, $zvalue, @set, @val); |
| 22 | |
| 23 | $test = 0; |
| 24 | $| = 1; |
| 25 | my @script = ( |
| 26 | 'my ($res, $s0,$s1,$s2,$s3,$s4,$s5,$s6,$s7,$s8,$s9,$s10, $z0,$z1,$z2);' . |
| 27 | "\n\n" |
| 28 | ); |
| 29 | my $eps = 1e-13; |
| 30 | |
| 31 | if ($^O eq 'unicos') { # For some reason root() produces very inaccurate |
| 32 | $eps = 1e-10; # results in Cray UNICOS, and occasionally also |
| 33 | } # cos(), sin(), cosh(), sinh(). The division |
| 34 | # of doubles is the current suspect. |
| 35 | |
| 36 | while (<DATA>) { |
| 37 | s/^\s+//; |
| 38 | next if $_ eq '' || /^\#/; |
| 39 | chomp; |
| 40 | $test_set = 0; # Assume not a test over a set of values |
| 41 | if (/^&(.+)/) { |
| 42 | $op = $1; |
| 43 | next; |
| 44 | } |
| 45 | elsif (/^\{(.+)\}/) { |
| 46 | set($1, \@set, \@val); |
| 47 | next; |
| 48 | } |
| 49 | elsif (s/^\|//) { |
| 50 | $test_set = 1; # Requests we loop over the set... |
| 51 | } |
| 52 | my @args = split(/:/); |
| 53 | if ($test_set == 1) { |
| 54 | my $i; |
| 55 | for ($i = 0; $i < @set; $i++) { |
| 56 | # complex number |
| 57 | $target = $set[$i]; |
| 58 | # textual value as found in set definition |
| 59 | $zvalue = $val[$i]; |
| 60 | test($zvalue, $target, @args); |
| 61 | } |
| 62 | } else { |
| 63 | test($op, undef, @args); |
| 64 | } |
| 65 | } |
| 66 | |
| 67 | # |
| 68 | |
| 69 | sub test_mutators { |
| 70 | my $op; |
| 71 | |
| 72 | $test++; |
| 73 | push(@script, <<'EOT'); |
| 74 | { |
| 75 | my $z = cplx( 1, 1); |
| 76 | $z->Re(2); |
| 77 | $z->Im(3); |
| 78 | print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n"; |
| 79 | print 'not ' unless Re($z) == 2 and Im($z) == 3; |
| 80 | EOT |
| 81 | push(@script, qq(print "ok $test\\n"}\n)); |
| 82 | |
| 83 | $test++; |
| 84 | push(@script, <<'EOT'); |
| 85 | { |
| 86 | my $z = cplx( 1, 1); |
| 87 | $z->abs(3 * sqrt(2)); |
| 88 | print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n"; |
| 89 | print 'not ' unless (abs($z) - 3 * sqrt(2)) < $eps and |
| 90 | (arg($z) - pi / 4 ) < $eps and |
| 91 | (Re($z) - 3 ) < $eps and |
| 92 | (Im($z) - 3 ) < $eps; |
| 93 | EOT |
| 94 | push(@script, qq(print "ok $test\\n"}\n)); |
| 95 | |
| 96 | $test++; |
| 97 | push(@script, <<'EOT'); |
| 98 | { |
| 99 | my $z = cplx( 1, 1); |
| 100 | $z->arg(-3 / 4 * pi); |
| 101 | print "# $test Re(z) = ",$z->Re(), " Im(z) = ", $z->Im(), " z = $z\n"; |
| 102 | print 'not ' unless (arg($z) + 3 / 4 * pi) < $eps and |
| 103 | (abs($z) - sqrt(2) ) < $eps and |
| 104 | (Re($z) + 1 ) < $eps and |
| 105 | (Im($z) + 1 ) < $eps; |
| 106 | EOT |
| 107 | push(@script, qq(print "ok $test\\n"}\n)); |
| 108 | } |
| 109 | |
| 110 | test_mutators(); |
| 111 | |
| 112 | my $constants = ' |
| 113 | my $i = cplx(0, 1); |
| 114 | my $pi = cplx(pi, 0); |
| 115 | my $pii = cplx(0, pi); |
| 116 | my $pip2 = cplx(pi/2, 0); |
| 117 | my $zero = cplx(0, 0); |
| 118 | '; |
| 119 | |
| 120 | push(@script, $constants); |
| 121 | |
| 122 | |
| 123 | # test the divbyzeros |
| 124 | |
| 125 | sub test_dbz { |
| 126 | for my $op (@_) { |
| 127 | $test++; |
| 128 | push(@script, <<EOT); |
| 129 | eval '$op'; |
| 130 | (\$bad) = (\$@ =~ /(.+)/); |
| 131 | print "# $test op = $op divbyzero? \$bad...\n"; |
| 132 | print 'not ' unless (\$@ =~ /Division by zero/); |
| 133 | EOT |
| 134 | push(@script, qq(print "ok $test\\n";\n)); |
| 135 | } |
| 136 | } |
| 137 | |
| 138 | # test the logofzeros |
| 139 | |
| 140 | sub test_loz { |
| 141 | for my $op (@_) { |
| 142 | $test++; |
| 143 | push(@script, <<EOT); |
| 144 | eval '$op'; |
| 145 | (\$bad) = (\$@ =~ /(.+)/); |
| 146 | print "# $test op = $op logofzero? \$bad...\n"; |
| 147 | print 'not ' unless (\$@ =~ /Logarithm of zero/); |
| 148 | EOT |
| 149 | push(@script, qq(print "ok $test\\n";\n)); |
| 150 | } |
| 151 | } |
| 152 | |
| 153 | test_dbz( |
| 154 | 'i/0', |
| 155 | 'acot(0)', |
| 156 | 'acot(+$i)', |
| 157 | # 'acoth(-1)', # Log of zero. |
| 158 | 'acoth(0)', |
| 159 | 'acoth(+1)', |
| 160 | 'acsc(0)', |
| 161 | 'acsch(0)', |
| 162 | 'asec(0)', |
| 163 | 'asech(0)', |
| 164 | 'atan($i)', |
| 165 | # 'atanh(-1)', # Log of zero. |
| 166 | 'atanh(+1)', |
| 167 | 'cot(0)', |
| 168 | 'coth(0)', |
| 169 | 'csc(0)', |
| 170 | 'csch(0)', |
| 171 | ); |
| 172 | |
| 173 | test_loz( |
| 174 | 'log($zero)', |
| 175 | 'atan(-$i)', |
| 176 | 'acot(-$i)', |
| 177 | 'atanh(-1)', |
| 178 | 'acoth(-1)', |
| 179 | ); |
| 180 | |
| 181 | # test the bad roots |
| 182 | |
| 183 | sub test_broot { |
| 184 | for my $op (@_) { |
| 185 | $test++; |
| 186 | push(@script, <<EOT); |
| 187 | eval 'root(2, $op)'; |
| 188 | (\$bad) = (\$@ =~ /(.+)/); |
| 189 | print "# $test op = $op badroot? \$bad...\n"; |
| 190 | print 'not ' unless (\$@ =~ /root rank must be/); |
| 191 | EOT |
| 192 | push(@script, qq(print "ok $test\\n";\n)); |
| 193 | } |
| 194 | } |
| 195 | |
| 196 | test_broot(qw(-3 -2.1 0 0.99)); |
| 197 | |
| 198 | sub test_display_format { |
| 199 | $test++; |
| 200 | push @script, <<EOS; |
| 201 | print "# package display_format cartesian?\n"; |
| 202 | print "not " unless Math::Complex->display_format eq 'cartesian'; |
| 203 | print "ok $test\n"; |
| 204 | EOS |
| 205 | |
| 206 | push @script, <<EOS; |
| 207 | my \$j = (root(1,3))[1]; |
| 208 | |
| 209 | \$j->display_format('polar'); |
| 210 | EOS |
| 211 | |
| 212 | $test++; |
| 213 | push @script, <<EOS; |
| 214 | print "# j display_format polar?\n"; |
| 215 | print "not " unless \$j->display_format eq 'polar'; |
| 216 | print "ok $test\n"; |
| 217 | EOS |
| 218 | |
| 219 | $test++; |
| 220 | push @script, <<EOS; |
| 221 | print "# j = \$j\n"; |
| 222 | print "not " unless "\$j" eq "[1,2pi/3]"; |
| 223 | print "ok $test\n"; |
| 224 | |
| 225 | my %display_format; |
| 226 | |
| 227 | %display_format = \$j->display_format; |
| 228 | EOS |
| 229 | |
| 230 | $test++; |
| 231 | push @script, <<EOS; |
| 232 | print "# display_format{style} polar?\n"; |
| 233 | print "not " unless \$display_format{style} eq 'polar'; |
| 234 | print "ok $test\n"; |
| 235 | EOS |
| 236 | |
| 237 | $test++; |
| 238 | push @script, <<EOS; |
| 239 | print "# keys %display_format == 2?\n"; |
| 240 | print "not " unless keys %display_format == 2; |
| 241 | print "ok $test\n"; |
| 242 | |
| 243 | \$j->display_format('style' => 'cartesian', 'format' => '%.5f'); |
| 244 | EOS |
| 245 | |
| 246 | $test++; |
| 247 | push @script, <<EOS; |
| 248 | print "# j = \$j\n"; |
| 249 | print "not " unless "\$j" eq "-0.50000+0.86603i"; |
| 250 | print "ok $test\n"; |
| 251 | |
| 252 | %display_format = \$j->display_format; |
| 253 | EOS |
| 254 | |
| 255 | $test++; |
| 256 | push @script, <<EOS; |
| 257 | print "# display_format{format} %.5f?\n"; |
| 258 | print "not " unless \$display_format{format} eq '%.5f'; |
| 259 | print "ok $test\n"; |
| 260 | EOS |
| 261 | |
| 262 | $test++; |
| 263 | push @script, <<EOS; |
| 264 | print "# keys %display_format == 3?\n"; |
| 265 | print "not " unless keys %display_format == 3; |
| 266 | print "ok $test\n"; |
| 267 | |
| 268 | \$j->display_format('format' => undef); |
| 269 | EOS |
| 270 | |
| 271 | $test++; |
| 272 | push @script, <<EOS; |
| 273 | print "# j = \$j\n"; |
| 274 | print "not " unless "\$j" =~ /^-0(?:\\.5(?:0000\\d+)?|\\.49999\\d+)\\+0.86602540\\d+i\$/; |
| 275 | print "ok $test\n"; |
| 276 | |
| 277 | \$j->display_format('style' => 'polar', 'polar_pretty_print' => 0); |
| 278 | EOS |
| 279 | |
| 280 | $test++; |
| 281 | push @script, <<EOS; |
| 282 | print "# j = \$j\n"; |
| 283 | print "not " unless "\$j" =~ /^\\[1,2\\.09439510\\d+\\]\$/; |
| 284 | print "ok $test\n"; |
| 285 | |
| 286 | \$j->display_format('style' => 'cartesian', 'format' => '(%.5g)'); |
| 287 | EOS |
| 288 | |
| 289 | $test++; |
| 290 | push @script, <<EOS; |
| 291 | print "# j = \$j\n"; |
| 292 | print "not " unless "\$j" eq "(-0.5)+(0.86603)i"; |
| 293 | print "ok $test\n"; |
| 294 | EOS |
| 295 | |
| 296 | $test++; |
| 297 | push @script, <<EOS; |
| 298 | print "# j display_format cartesian?\n"; |
| 299 | print "not " unless \$j->display_format eq 'cartesian'; |
| 300 | print "ok $test\n"; |
| 301 | EOS |
| 302 | } |
| 303 | |
| 304 | test_display_format(); |
| 305 | |
| 306 | print "1..$test\n"; |
| 307 | eval join '', @script; |
| 308 | die $@ if $@; |
| 309 | |
| 310 | sub abop { |
| 311 | my ($op) = @_; |
| 312 | |
| 313 | push(@script, qq(print "# $op=\n";)); |
| 314 | } |
| 315 | |
| 316 | sub test { |
| 317 | my ($op, $z, @args) = @_; |
| 318 | my ($baop) = 0; |
| 319 | $test++; |
| 320 | my $i; |
| 321 | $baop = 1 if ($op =~ s/;=$//); |
| 322 | for ($i = 0; $i < @args; $i++) { |
| 323 | $val = value($args[$i]); |
| 324 | push @script, "\$z$i = $val;\n"; |
| 325 | } |
| 326 | if (defined $z) { |
| 327 | $args = "'$op'"; # Really the value |
| 328 | $try = "abs(\$z0 - \$z1) <= $eps ? \$z1 : \$z0"; |
| 329 | push @script, "\$res = $try; "; |
| 330 | push @script, "check($test, $args[0], \$res, \$z$#args, $args);\n"; |
| 331 | } else { |
| 332 | my ($try, $args); |
| 333 | if (@args == 2) { |
| 334 | $try = "$op \$z0"; |
| 335 | $args = "'$args[0]'"; |
| 336 | } else { |
| 337 | $try = ($op =~ /^\w/) ? "$op(\$z0, \$z1)" : "\$z0 $op \$z1"; |
| 338 | $args = "'$args[0]', '$args[1]'"; |
| 339 | } |
| 340 | push @script, "\$res = $try; "; |
| 341 | push @script, "check($test, '$try', \$res, \$z$#args, $args);\n"; |
| 342 | if (@args > 2 and $baop) { # binary assignment ops |
| 343 | $test++; |
| 344 | # check the op= works |
| 345 | push @script, <<EOB; |
| 346 | { |
| 347 | my \$za = cplx(ref \$z0 ? \@{\$z0->cartesian} : (\$z0, 0)); |
| 348 | |
| 349 | my (\$z1r, \$z1i) = ref \$z1 ? \@{\$z1->cartesian} : (\$z1, 0); |
| 350 | |
| 351 | my \$zb = cplx(\$z1r, \$z1i); |
| 352 | |
| 353 | \$za $op= \$zb; |
| 354 | my (\$zbr, \$zbi) = \@{\$zb->cartesian}; |
| 355 | |
| 356 | check($test, '\$z0 $op= \$z1', \$za, \$z$#args, $args); |
| 357 | EOB |
| 358 | $test++; |
| 359 | # check that the rhs has not changed |
| 360 | push @script, qq(print "not " unless (\$zbr == \$z1r and \$zbi == \$z1i);); |
| 361 | push @script, qq(print "ok $test\\n";\n); |
| 362 | push @script, "}\n"; |
| 363 | } |
| 364 | } |
| 365 | } |
| 366 | |
| 367 | sub set { |
| 368 | my ($set, $setref, $valref) = @_; |
| 369 | @{$setref} = (); |
| 370 | @{$valref} = (); |
| 371 | my @set = split(/;\s*/, $set); |
| 372 | my @res; |
| 373 | my $i; |
| 374 | for ($i = 0; $i < @set; $i++) { |
| 375 | push(@{$valref}, $set[$i]); |
| 376 | my $val = value($set[$i]); |
| 377 | push @script, "\$s$i = $val;\n"; |
| 378 | push @{$setref}, "\$s$i"; |
| 379 | } |
| 380 | } |
| 381 | |
| 382 | sub value { |
| 383 | local ($_) = @_; |
| 384 | if (/^\s*\((.*),(.*)\)/) { |
| 385 | return "cplx($1,$2)"; |
| 386 | } |
| 387 | elsif (/^\s*([\-\+]?(?:\d+(\.\d+)?|\.\d+)(?:[e[\-\+]\d+])?)/) { |
| 388 | return "cplx($1,0)"; |
| 389 | } |
| 390 | elsif (/^\s*\[(.*),(.*)\]/) { |
| 391 | return "cplxe($1,$2)"; |
| 392 | } |
| 393 | elsif (/^\s*'(.*)'/) { |
| 394 | my $ex = $1; |
| 395 | $ex =~ s/\bz\b/$target/g; |
| 396 | $ex =~ s/\br\b/abs($target)/g; |
| 397 | $ex =~ s/\bt\b/arg($target)/g; |
| 398 | $ex =~ s/\ba\b/Re($target)/g; |
| 399 | $ex =~ s/\bb\b/Im($target)/g; |
| 400 | return $ex; |
| 401 | } |
| 402 | elsif (/^\s*"(.*)"/) { |
| 403 | return "\"$1\""; |
| 404 | } |
| 405 | return $_; |
| 406 | } |
| 407 | |
| 408 | sub check { |
| 409 | my ($test, $try, $got, $expected, @z) = @_; |
| 410 | |
| 411 | print "# @_\n"; |
| 412 | |
| 413 | if ("$got" eq "$expected" |
| 414 | || |
| 415 | ($expected =~ /^-?\d/ && $got == $expected) |
| 416 | || |
| 417 | (abs($got - $expected) < $eps) |
| 418 | ) { |
| 419 | print "ok $test\n"; |
| 420 | } else { |
| 421 | print "not ok $test\n"; |
| 422 | my $args = (@z == 1) ? "z = $z[0]" : "z0 = $z[0], z1 = $z[1]"; |
| 423 | print "# '$try' expected: '$expected' got: '$got' for $args\n"; |
| 424 | } |
| 425 | } |
| 426 | |
| 427 | sub addsq { |
| 428 | my ($z1, $z2) = @_; |
| 429 | return ($z1 + i*$z2) * ($z1 - i*$z2); |
| 430 | } |
| 431 | |
| 432 | sub subsq { |
| 433 | my ($z1, $z2) = @_; |
| 434 | return ($z1 + $z2) * ($z1 - $z2); |
| 435 | } |
| 436 | |
| 437 | __END__ |
| 438 | &+;= |
| 439 | (3,4):(3,4):(6,8) |
| 440 | (-3,4):(3,-4):(0,0) |
| 441 | (3,4):-3:(0,4) |
| 442 | 1:(4,2):(5,2) |
| 443 | [2,0]:[2,pi]:(0,0) |
| 444 | |
| 445 | &++ |
| 446 | (2,1):(3,1) |
| 447 | |
| 448 | &-;= |
| 449 | (2,3):(-2,-3) |
| 450 | [2,pi/2]:[2,-(pi)/2] |
| 451 | 2:[2,0]:(0,0) |
| 452 | [3,0]:2:(1,0) |
| 453 | 3:(4,5):(-1,-5) |
| 454 | (4,5):3:(1,5) |
| 455 | (2,1):(3,5):(-1,-4) |
| 456 | |
| 457 | &-- |
| 458 | (1,2):(0,2) |
| 459 | [2,pi]:[3,pi] |
| 460 | |
| 461 | &*;= |
| 462 | (0,1):(0,1):(-1,0) |
| 463 | (4,5):(1,0):(4,5) |
| 464 | [2,2*pi/3]:(1,0):[2,2*pi/3] |
| 465 | 2:(0,1):(0,2) |
| 466 | (0,1):3:(0,3) |
| 467 | (0,1):(4,1):(-1,4) |
| 468 | (2,1):(4,-1):(9,2) |
| 469 | |
| 470 | &/;= |
| 471 | (3,4):(3,4):(1,0) |
| 472 | (4,-5):1:(4,-5) |
| 473 | 1:(0,1):(0,-1) |
| 474 | (0,6):(0,2):(3,0) |
| 475 | (9,2):(4,-1):(2,1) |
| 476 | [4,pi]:[2,pi/2]:[2,pi/2] |
| 477 | [2,pi/2]:[4,pi]:[0.5,-(pi)/2] |
| 478 | |
| 479 | &**;= |
| 480 | (2,0):(3,0):(8,0) |
| 481 | (3,0):(2,0):(9,0) |
| 482 | (2,3):(4,0):(-119,-120) |
| 483 | (0,0):(1,0):(0,0) |
| 484 | (0,0):(2,3):(0,0) |
| 485 | (1,0):(0,0):(1,0) |
| 486 | (1,0):(1,0):(1,0) |
| 487 | (1,0):(2,3):(1,0) |
| 488 | (2,3):(0,0):(1,0) |
| 489 | (2,3):(1,0):(2,3) |
| 490 | (0,0):(0,0):(1,0) |
| 491 | |
| 492 | &Re |
| 493 | (3,4):3 |
| 494 | (-3,4):-3 |
| 495 | [1,pi/2]:0 |
| 496 | |
| 497 | &Im |
| 498 | (3,4):4 |
| 499 | (3,-4):-4 |
| 500 | [1,pi/2]:1 |
| 501 | |
| 502 | &abs |
| 503 | (3,4):5 |
| 504 | (-3,4):5 |
| 505 | |
| 506 | &arg |
| 507 | [2,0]:0 |
| 508 | [-2,0]:pi |
| 509 | |
| 510 | &~ |
| 511 | (4,5):(4,-5) |
| 512 | (-3,4):(-3,-4) |
| 513 | [2,pi/2]:[2,-(pi)/2] |
| 514 | |
| 515 | &< |
| 516 | (3,4):(1,2):0 |
| 517 | (3,4):(3,2):0 |
| 518 | (3,4):(3,8):1 |
| 519 | (4,4):(5,129):1 |
| 520 | |
| 521 | &== |
| 522 | (3,4):(4,5):0 |
| 523 | (3,4):(3,5):0 |
| 524 | (3,4):(2,4):0 |
| 525 | (3,4):(3,4):1 |
| 526 | |
| 527 | &sqrt |
| 528 | -9:(0,3) |
| 529 | (-100,0):(0,10) |
| 530 | (16,-30):(5,-3) |
| 531 | |
| 532 | &stringify_cartesian |
| 533 | (-100,0):"-100" |
| 534 | (0,1):"i" |
| 535 | (4,-3):"4-3i" |
| 536 | (4,0):"4" |
| 537 | (-4,0):"-4" |
| 538 | (-2,4):"-2+4i" |
| 539 | (-2,-1):"-2-i" |
| 540 | |
| 541 | &stringify_polar |
| 542 | [-1, 0]:"[1,pi]" |
| 543 | [1, pi/3]:"[1,pi/3]" |
| 544 | [6, -2*pi/3]:"[6,-2pi/3]" |
| 545 | [0.5, -9*pi/11]:"[0.5,-9pi/11]" |
| 546 | |
| 547 | { (4,3); [3,2]; (-3,4); (0,2); [2,1] } |
| 548 | |
| 549 | |'z + ~z':'2*Re(z)' |
| 550 | |'z - ~z':'2*i*Im(z)' |
| 551 | |'z * ~z':'abs(z) * abs(z)' |
| 552 | |
| 553 | { (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] } |
| 554 | |
| 555 | |'(root(z, 4))[1] ** 4':'z' |
| 556 | |'(root(z, 5))[3] ** 5':'z' |
| 557 | |'(root(z, 8))[7] ** 8':'z' |
| 558 | |'abs(z)':'r' |
| 559 | |'acot(z)':'acotan(z)' |
| 560 | |'acsc(z)':'acosec(z)' |
| 561 | |'acsc(z)':'asin(1 / z)' |
| 562 | |'asec(z)':'acos(1 / z)' |
| 563 | |'cbrt(z)':'cbrt(r) * exp(i * t/3)' |
| 564 | |'cos(acos(z))':'z' |
| 565 | |'addsq(cos(z), sin(z))':1 |
| 566 | |'cos(z)':'cosh(i*z)' |
| 567 | |'subsq(cosh(z), sinh(z))':1 |
| 568 | |'cot(acot(z))':'z' |
| 569 | |'cot(z)':'1 / tan(z)' |
| 570 | |'cot(z)':'cotan(z)' |
| 571 | |'csc(acsc(z))':'z' |
| 572 | |'csc(z)':'1 / sin(z)' |
| 573 | |'csc(z)':'cosec(z)' |
| 574 | |'exp(log(z))':'z' |
| 575 | |'exp(z)':'exp(a) * exp(i * b)' |
| 576 | |'ln(z)':'log(z)' |
| 577 | |'log(exp(z))':'z' |
| 578 | |'log(z)':'log(r) + i*t' |
| 579 | |'log10(z)':'log(z) / log(10)' |
| 580 | |'logn(z, 2)':'log(z) / log(2)' |
| 581 | |'logn(z, 3)':'log(z) / log(3)' |
| 582 | |'sec(asec(z))':'z' |
| 583 | |'sec(z)':'1 / cos(z)' |
| 584 | |'sin(asin(z))':'z' |
| 585 | |'sin(i * z)':'i * sinh(z)' |
| 586 | |'sqrt(z) * sqrt(z)':'z' |
| 587 | |'sqrt(z)':'sqrt(r) * exp(i * t/2)' |
| 588 | |'tan(atan(z))':'z' |
| 589 | |'z**z':'exp(z * log(z))' |
| 590 | |
| 591 | { (1,1); [1,0.5]; (-2, -1); 2; -3; (-1,0.5); (0,0.5); 0.5; (2, 0); (-1, -2) } |
| 592 | |
| 593 | |'cosh(acosh(z))':'z' |
| 594 | |'coth(acoth(z))':'z' |
| 595 | |'coth(z)':'1 / tanh(z)' |
| 596 | |'coth(z)':'cotanh(z)' |
| 597 | |'csch(acsch(z))':'z' |
| 598 | |'csch(z)':'1 / sinh(z)' |
| 599 | |'csch(z)':'cosech(z)' |
| 600 | |'sech(asech(z))':'z' |
| 601 | |'sech(z)':'1 / cosh(z)' |
| 602 | |'sinh(asinh(z))':'z' |
| 603 | |'tanh(atanh(z))':'z' |
| 604 | |
| 605 | { (0.2,-0.4); [1,0.5]; -1.2; (-1,0.5); 0.5; (1.1, 0) } |
| 606 | |
| 607 | |'acos(cos(z)) ** 2':'z * z' |
| 608 | |'acosh(cosh(z)) ** 2':'z * z' |
| 609 | |'acoth(z)':'acotanh(z)' |
| 610 | |'acoth(z)':'atanh(1 / z)' |
| 611 | |'acsch(z)':'acosech(z)' |
| 612 | |'acsch(z)':'asinh(1 / z)' |
| 613 | |'asech(z)':'acosh(1 / z)' |
| 614 | |'asin(sin(z))':'z' |
| 615 | |'asinh(sinh(z))':'z' |
| 616 | |'atan(tan(z))':'z' |
| 617 | |'atanh(tanh(z))':'z' |
| 618 | |
| 619 | &log |
| 620 | (-2.0,0):( 0.69314718055995, 3.14159265358979) |
| 621 | (-1.0,0):( 0 , 3.14159265358979) |
| 622 | (-0.5,0):( -0.69314718055995, 3.14159265358979) |
| 623 | ( 0.5,0):( -0.69314718055995, 0 ) |
| 624 | ( 1.0,0):( 0 , 0 ) |
| 625 | ( 2.0,0):( 0.69314718055995, 0 ) |
| 626 | |
| 627 | &log |
| 628 | ( 2, 3):( 1.28247467873077, 0.98279372324733) |
| 629 | (-2, 3):( 1.28247467873077, 2.15879893034246) |
| 630 | (-2,-3):( 1.28247467873077, -2.15879893034246) |
| 631 | ( 2,-3):( 1.28247467873077, -0.98279372324733) |
| 632 | |
| 633 | &sin |
| 634 | (-2.0,0):( -0.90929742682568, 0 ) |
| 635 | (-1.0,0):( -0.84147098480790, 0 ) |
| 636 | (-0.5,0):( -0.47942553860420, 0 ) |
| 637 | ( 0.0,0):( 0 , 0 ) |
| 638 | ( 0.5,0):( 0.47942553860420, 0 ) |
| 639 | ( 1.0,0):( 0.84147098480790, 0 ) |
| 640 | ( 2.0,0):( 0.90929742682568, 0 ) |
| 641 | |
| 642 | &sin |
| 643 | ( 2, 3):( 9.15449914691143, -4.16890695996656) |
| 644 | (-2, 3):( -9.15449914691143, -4.16890695996656) |
| 645 | (-2,-3):( -9.15449914691143, 4.16890695996656) |
| 646 | ( 2,-3):( 9.15449914691143, 4.16890695996656) |
| 647 | |
| 648 | &cos |
| 649 | (-2.0,0):( -0.41614683654714, 0 ) |
| 650 | (-1.0,0):( 0.54030230586814, 0 ) |
| 651 | (-0.5,0):( 0.87758256189037, 0 ) |
| 652 | ( 0.0,0):( 1 , 0 ) |
| 653 | ( 0.5,0):( 0.87758256189037, 0 ) |
| 654 | ( 1.0,0):( 0.54030230586814, 0 ) |
| 655 | ( 2.0,0):( -0.41614683654714, 0 ) |
| 656 | |
| 657 | &cos |
| 658 | ( 2, 3):( -4.18962569096881, -9.10922789375534) |
| 659 | (-2, 3):( -4.18962569096881, 9.10922789375534) |
| 660 | (-2,-3):( -4.18962569096881, -9.10922789375534) |
| 661 | ( 2,-3):( -4.18962569096881, 9.10922789375534) |
| 662 | |
| 663 | &tan |
| 664 | (-2.0,0):( 2.18503986326152, 0 ) |
| 665 | (-1.0,0):( -1.55740772465490, 0 ) |
| 666 | (-0.5,0):( -0.54630248984379, 0 ) |
| 667 | ( 0.0,0):( 0 , 0 ) |
| 668 | ( 0.5,0):( 0.54630248984379, 0 ) |
| 669 | ( 1.0,0):( 1.55740772465490, 0 ) |
| 670 | ( 2.0,0):( -2.18503986326152, 0 ) |
| 671 | |
| 672 | &tan |
| 673 | ( 2, 3):( -0.00376402564150, 1.00323862735361) |
| 674 | (-2, 3):( 0.00376402564150, 1.00323862735361) |
| 675 | (-2,-3):( 0.00376402564150, -1.00323862735361) |
| 676 | ( 2,-3):( -0.00376402564150, -1.00323862735361) |
| 677 | |
| 678 | &sec |
| 679 | (-2.0,0):( -2.40299796172238, 0 ) |
| 680 | (-1.0,0):( 1.85081571768093, 0 ) |
| 681 | (-0.5,0):( 1.13949392732455, 0 ) |
| 682 | ( 0.0,0):( 1 , 0 ) |
| 683 | ( 0.5,0):( 1.13949392732455, 0 ) |
| 684 | ( 1.0,0):( 1.85081571768093, 0 ) |
| 685 | ( 2.0,0):( -2.40299796172238, 0 ) |
| 686 | |
| 687 | &sec |
| 688 | ( 2, 3):( -0.04167496441114, 0.09061113719624) |
| 689 | (-2, 3):( -0.04167496441114, -0.09061113719624) |
| 690 | (-2,-3):( -0.04167496441114, 0.09061113719624) |
| 691 | ( 2,-3):( -0.04167496441114, -0.09061113719624) |
| 692 | |
| 693 | &csc |
| 694 | (-2.0,0):( -1.09975017029462, 0 ) |
| 695 | (-1.0,0):( -1.18839510577812, 0 ) |
| 696 | (-0.5,0):( -2.08582964293349, 0 ) |
| 697 | ( 0.5,0):( 2.08582964293349, 0 ) |
| 698 | ( 1.0,0):( 1.18839510577812, 0 ) |
| 699 | ( 2.0,0):( 1.09975017029462, 0 ) |
| 700 | |
| 701 | &csc |
| 702 | ( 2, 3):( 0.09047320975321, 0.04120098628857) |
| 703 | (-2, 3):( -0.09047320975321, 0.04120098628857) |
| 704 | (-2,-3):( -0.09047320975321, -0.04120098628857) |
| 705 | ( 2,-3):( 0.09047320975321, -0.04120098628857) |
| 706 | |
| 707 | &cot |
| 708 | (-2.0,0):( 0.45765755436029, 0 ) |
| 709 | (-1.0,0):( -0.64209261593433, 0 ) |
| 710 | (-0.5,0):( -1.83048772171245, 0 ) |
| 711 | ( 0.5,0):( 1.83048772171245, 0 ) |
| 712 | ( 1.0,0):( 0.64209261593433, 0 ) |
| 713 | ( 2.0,0):( -0.45765755436029, 0 ) |
| 714 | |
| 715 | &cot |
| 716 | ( 2, 3):( -0.00373971037634, -0.99675779656936) |
| 717 | (-2, 3):( 0.00373971037634, -0.99675779656936) |
| 718 | (-2,-3):( 0.00373971037634, 0.99675779656936) |
| 719 | ( 2,-3):( -0.00373971037634, 0.99675779656936) |
| 720 | |
| 721 | &asin |
| 722 | (-2.0,0):( -1.57079632679490, 1.31695789692482) |
| 723 | (-1.0,0):( -1.57079632679490, 0 ) |
| 724 | (-0.5,0):( -0.52359877559830, 0 ) |
| 725 | ( 0.0,0):( 0 , 0 ) |
| 726 | ( 0.5,0):( 0.52359877559830, 0 ) |
| 727 | ( 1.0,0):( 1.57079632679490, 0 ) |
| 728 | ( 2.0,0):( 1.57079632679490, -1.31695789692482) |
| 729 | |
| 730 | &asin |
| 731 | ( 2, 3):( 0.57065278432110, 1.98338702991654) |
| 732 | (-2, 3):( -0.57065278432110, 1.98338702991654) |
| 733 | (-2,-3):( -0.57065278432110, -1.98338702991654) |
| 734 | ( 2,-3):( 0.57065278432110, -1.98338702991654) |
| 735 | |
| 736 | &acos |
| 737 | (-2.0,0):( 3.14159265358979, -1.31695789692482) |
| 738 | (-1.0,0):( 3.14159265358979, 0 ) |
| 739 | (-0.5,0):( 2.09439510239320, 0 ) |
| 740 | ( 0.0,0):( 1.57079632679490, 0 ) |
| 741 | ( 0.5,0):( 1.04719755119660, 0 ) |
| 742 | ( 1.0,0):( 0 , 0 ) |
| 743 | ( 2.0,0):( 0 , 1.31695789692482) |
| 744 | |
| 745 | &acos |
| 746 | ( 2, 3):( 1.00014354247380, -1.98338702991654) |
| 747 | (-2, 3):( 2.14144911111600, -1.98338702991654) |
| 748 | (-2,-3):( 2.14144911111600, 1.98338702991654) |
| 749 | ( 2,-3):( 1.00014354247380, 1.98338702991654) |
| 750 | |
| 751 | &atan |
| 752 | (-2.0,0):( -1.10714871779409, 0 ) |
| 753 | (-1.0,0):( -0.78539816339745, 0 ) |
| 754 | (-0.5,0):( -0.46364760900081, 0 ) |
| 755 | ( 0.0,0):( 0 , 0 ) |
| 756 | ( 0.5,0):( 0.46364760900081, 0 ) |
| 757 | ( 1.0,0):( 0.78539816339745, 0 ) |
| 758 | ( 2.0,0):( 1.10714871779409, 0 ) |
| 759 | |
| 760 | &atan |
| 761 | ( 2, 3):( 1.40992104959658, 0.22907268296854) |
| 762 | (-2, 3):( -1.40992104959658, 0.22907268296854) |
| 763 | (-2,-3):( -1.40992104959658, -0.22907268296854) |
| 764 | ( 2,-3):( 1.40992104959658, -0.22907268296854) |
| 765 | |
| 766 | &asec |
| 767 | (-2.0,0):( 2.09439510239320, 0 ) |
| 768 | (-1.0,0):( 3.14159265358979, 0 ) |
| 769 | (-0.5,0):( 3.14159265358979, -1.31695789692482) |
| 770 | ( 0.5,0):( 0 , 1.31695789692482) |
| 771 | ( 1.0,0):( 0 , 0 ) |
| 772 | ( 2.0,0):( 1.04719755119660, 0 ) |
| 773 | |
| 774 | &asec |
| 775 | ( 2, 3):( 1.42041072246703, 0.23133469857397) |
| 776 | (-2, 3):( 1.72118193112276, 0.23133469857397) |
| 777 | (-2,-3):( 1.72118193112276, -0.23133469857397) |
| 778 | ( 2,-3):( 1.42041072246703, -0.23133469857397) |
| 779 | |
| 780 | &acsc |
| 781 | (-2.0,0):( -0.52359877559830, 0 ) |
| 782 | (-1.0,0):( -1.57079632679490, 0 ) |
| 783 | (-0.5,0):( -1.57079632679490, 1.31695789692482) |
| 784 | ( 0.5,0):( 1.57079632679490, -1.31695789692482) |
| 785 | ( 1.0,0):( 1.57079632679490, 0 ) |
| 786 | ( 2.0,0):( 0.52359877559830, 0 ) |
| 787 | |
| 788 | &acsc |
| 789 | ( 2, 3):( 0.15038560432786, -0.23133469857397) |
| 790 | (-2, 3):( -0.15038560432786, -0.23133469857397) |
| 791 | (-2,-3):( -0.15038560432786, 0.23133469857397) |
| 792 | ( 2,-3):( 0.15038560432786, 0.23133469857397) |
| 793 | |
| 794 | &acot |
| 795 | (-2.0,0):( -0.46364760900081, 0 ) |
| 796 | (-1.0,0):( -0.78539816339745, 0 ) |
| 797 | (-0.5,0):( -1.10714871779409, 0 ) |
| 798 | ( 0.5,0):( 1.10714871779409, 0 ) |
| 799 | ( 1.0,0):( 0.78539816339745, 0 ) |
| 800 | ( 2.0,0):( 0.46364760900081, 0 ) |
| 801 | |
| 802 | &acot |
| 803 | ( 2, 3):( 0.16087527719832, -0.22907268296854) |
| 804 | (-2, 3):( -0.16087527719832, -0.22907268296854) |
| 805 | (-2,-3):( -0.16087527719832, 0.22907268296854) |
| 806 | ( 2,-3):( 0.16087527719832, 0.22907268296854) |
| 807 | |
| 808 | &sinh |
| 809 | (-2.0,0):( -3.62686040784702, 0 ) |
| 810 | (-1.0,0):( -1.17520119364380, 0 ) |
| 811 | (-0.5,0):( -0.52109530549375, 0 ) |
| 812 | ( 0.0,0):( 0 , 0 ) |
| 813 | ( 0.5,0):( 0.52109530549375, 0 ) |
| 814 | ( 1.0,0):( 1.17520119364380, 0 ) |
| 815 | ( 2.0,0):( 3.62686040784702, 0 ) |
| 816 | |
| 817 | &sinh |
| 818 | ( 2, 3):( -3.59056458998578, 0.53092108624852) |
| 819 | (-2, 3):( 3.59056458998578, 0.53092108624852) |
| 820 | (-2,-3):( 3.59056458998578, -0.53092108624852) |
| 821 | ( 2,-3):( -3.59056458998578, -0.53092108624852) |
| 822 | |
| 823 | &cosh |
| 824 | (-2.0,0):( 3.76219569108363, 0 ) |
| 825 | (-1.0,0):( 1.54308063481524, 0 ) |
| 826 | (-0.5,0):( 1.12762596520638, 0 ) |
| 827 | ( 0.0,0):( 1 , 0 ) |
| 828 | ( 0.5,0):( 1.12762596520638, 0 ) |
| 829 | ( 1.0,0):( 1.54308063481524, 0 ) |
| 830 | ( 2.0,0):( 3.76219569108363, 0 ) |
| 831 | |
| 832 | &cosh |
| 833 | ( 2, 3):( -3.72454550491532, 0.51182256998738) |
| 834 | (-2, 3):( -3.72454550491532, -0.51182256998738) |
| 835 | (-2,-3):( -3.72454550491532, 0.51182256998738) |
| 836 | ( 2,-3):( -3.72454550491532, -0.51182256998738) |
| 837 | |
| 838 | &tanh |
| 839 | (-2.0,0):( -0.96402758007582, 0 ) |
| 840 | (-1.0,0):( -0.76159415595576, 0 ) |
| 841 | (-0.5,0):( -0.46211715726001, 0 ) |
| 842 | ( 0.0,0):( 0 , 0 ) |
| 843 | ( 0.5,0):( 0.46211715726001, 0 ) |
| 844 | ( 1.0,0):( 0.76159415595576, 0 ) |
| 845 | ( 2.0,0):( 0.96402758007582, 0 ) |
| 846 | |
| 847 | &tanh |
| 848 | ( 2, 3):( 0.96538587902213, -0.00988437503832) |
| 849 | (-2, 3):( -0.96538587902213, -0.00988437503832) |
| 850 | (-2,-3):( -0.96538587902213, 0.00988437503832) |
| 851 | ( 2,-3):( 0.96538587902213, 0.00988437503832) |
| 852 | |
| 853 | &sech |
| 854 | (-2.0,0):( 0.26580222883408, 0 ) |
| 855 | (-1.0,0):( 0.64805427366389, 0 ) |
| 856 | (-0.5,0):( 0.88681888397007, 0 ) |
| 857 | ( 0.0,0):( 1 , 0 ) |
| 858 | ( 0.5,0):( 0.88681888397007, 0 ) |
| 859 | ( 1.0,0):( 0.64805427366389, 0 ) |
| 860 | ( 2.0,0):( 0.26580222883408, 0 ) |
| 861 | |
| 862 | &sech |
| 863 | ( 2, 3):( -0.26351297515839, -0.03621163655877) |
| 864 | (-2, 3):( -0.26351297515839, 0.03621163655877) |
| 865 | (-2,-3):( -0.26351297515839, -0.03621163655877) |
| 866 | ( 2,-3):( -0.26351297515839, 0.03621163655877) |
| 867 | |
| 868 | &csch |
| 869 | (-2.0,0):( -0.27572056477178, 0 ) |
| 870 | (-1.0,0):( -0.85091812823932, 0 ) |
| 871 | (-0.5,0):( -1.91903475133494, 0 ) |
| 872 | ( 0.5,0):( 1.91903475133494, 0 ) |
| 873 | ( 1.0,0):( 0.85091812823932, 0 ) |
| 874 | ( 2.0,0):( 0.27572056477178, 0 ) |
| 875 | |
| 876 | &csch |
| 877 | ( 2, 3):( -0.27254866146294, -0.04030057885689) |
| 878 | (-2, 3):( 0.27254866146294, -0.04030057885689) |
| 879 | (-2,-3):( 0.27254866146294, 0.04030057885689) |
| 880 | ( 2,-3):( -0.27254866146294, 0.04030057885689) |
| 881 | |
| 882 | &coth |
| 883 | (-2.0,0):( -1.03731472072755, 0 ) |
| 884 | (-1.0,0):( -1.31303528549933, 0 ) |
| 885 | (-0.5,0):( -2.16395341373865, 0 ) |
| 886 | ( 0.5,0):( 2.16395341373865, 0 ) |
| 887 | ( 1.0,0):( 1.31303528549933, 0 ) |
| 888 | ( 2.0,0):( 1.03731472072755, 0 ) |
| 889 | |
| 890 | &coth |
| 891 | ( 2, 3):( 1.03574663776500, 0.01060478347034) |
| 892 | (-2, 3):( -1.03574663776500, 0.01060478347034) |
| 893 | (-2,-3):( -1.03574663776500, -0.01060478347034) |
| 894 | ( 2,-3):( 1.03574663776500, -0.01060478347034) |
| 895 | |
| 896 | &asinh |
| 897 | (-2.0,0):( -1.44363547517881, 0 ) |
| 898 | (-1.0,0):( -0.88137358701954, 0 ) |
| 899 | (-0.5,0):( -0.48121182505960, 0 ) |
| 900 | ( 0.0,0):( 0 , 0 ) |
| 901 | ( 0.5,0):( 0.48121182505960, 0 ) |
| 902 | ( 1.0,0):( 0.88137358701954, 0 ) |
| 903 | ( 2.0,0):( 1.44363547517881, 0 ) |
| 904 | |
| 905 | &asinh |
| 906 | ( 2, 3):( 1.96863792579310, 0.96465850440760) |
| 907 | (-2, 3):( -1.96863792579310, 0.96465850440761) |
| 908 | (-2,-3):( -1.96863792579310, -0.96465850440761) |
| 909 | ( 2,-3):( 1.96863792579310, -0.96465850440760) |
| 910 | |
| 911 | &acosh |
| 912 | (-2.0,0):( 1.31695789692482, 3.14159265358979) |
| 913 | (-1.0,0):( 0, 3.14159265358979) |
| 914 | (-0.5,0):( 0, 2.09439510239320) |
| 915 | ( 0.0,0):( 0, 1.57079632679490) |
| 916 | ( 0.5,0):( 0, 1.04719755119660) |
| 917 | ( 1.0,0):( 0 , 0 ) |
| 918 | ( 2.0,0):( 1.31695789692482, 0 ) |
| 919 | |
| 920 | &acosh |
| 921 | ( 2, 3):( 1.98338702991654, 1.00014354247380) |
| 922 | (-2, 3):( 1.98338702991653, 2.14144911111600) |
| 923 | (-2,-3):( 1.98338702991653, -2.14144911111600) |
| 924 | ( 2,-3):( 1.98338702991654, -1.00014354247380) |
| 925 | |
| 926 | &atanh |
| 927 | (-2.0,0):( -0.54930614433405, 1.57079632679490) |
| 928 | (-0.5,0):( -0.54930614433405, 0 ) |
| 929 | ( 0.0,0):( 0 , 0 ) |
| 930 | ( 0.5,0):( 0.54930614433405, 0 ) |
| 931 | ( 2.0,0):( 0.54930614433405, 1.57079632679490) |
| 932 | |
| 933 | &atanh |
| 934 | ( 2, 3):( 0.14694666622553, 1.33897252229449) |
| 935 | (-2, 3):( -0.14694666622553, 1.33897252229449) |
| 936 | (-2,-3):( -0.14694666622553, -1.33897252229449) |
| 937 | ( 2,-3):( 0.14694666622553, -1.33897252229449) |
| 938 | |
| 939 | &asech |
| 940 | (-2.0,0):( 0 , 2.09439510239320) |
| 941 | (-1.0,0):( 0 , 3.14159265358979) |
| 942 | (-0.5,0):( 1.31695789692482, 3.14159265358979) |
| 943 | ( 0.5,0):( 1.31695789692482, 0 ) |
| 944 | ( 1.0,0):( 0 , 0 ) |
| 945 | ( 2.0,0):( 0 , 1.04719755119660) |
| 946 | |
| 947 | &asech |
| 948 | ( 2, 3):( 0.23133469857397, -1.42041072246703) |
| 949 | (-2, 3):( 0.23133469857397, -1.72118193112276) |
| 950 | (-2,-3):( 0.23133469857397, 1.72118193112276) |
| 951 | ( 2,-3):( 0.23133469857397, 1.42041072246703) |
| 952 | |
| 953 | &acsch |
| 954 | (-2.0,0):( -0.48121182505960, 0 ) |
| 955 | (-1.0,0):( -0.88137358701954, 0 ) |
| 956 | (-0.5,0):( -1.44363547517881, 0 ) |
| 957 | ( 0.5,0):( 1.44363547517881, 0 ) |
| 958 | ( 1.0,0):( 0.88137358701954, 0 ) |
| 959 | ( 2.0,0):( 0.48121182505960, 0 ) |
| 960 | |
| 961 | &acsch |
| 962 | ( 2, 3):( 0.15735549884499, -0.22996290237721) |
| 963 | (-2, 3):( -0.15735549884499, -0.22996290237721) |
| 964 | (-2,-3):( -0.15735549884499, 0.22996290237721) |
| 965 | ( 2,-3):( 0.15735549884499, 0.22996290237721) |
| 966 | |
| 967 | &acoth |
| 968 | (-2.0,0):( -0.54930614433405, 0 ) |
| 969 | (-0.5,0):( -0.54930614433405, 1.57079632679490) |
| 970 | ( 0.5,0):( 0.54930614433405, 1.57079632679490) |
| 971 | ( 2.0,0):( 0.54930614433405, 0 ) |
| 972 | |
| 973 | &acoth |
| 974 | ( 2, 3):( 0.14694666622553, -0.23182380450040) |
| 975 | (-2, 3):( -0.14694666622553, -0.23182380450040) |
| 976 | (-2,-3):( -0.14694666622553, 0.23182380450040) |
| 977 | ( 2,-3):( 0.14694666622553, 0.23182380450040) |
| 978 | |
| 979 | # eof |