Merge branch 'master' of ssh://git.code.sf.net/p/maxima/code

[maxima.git] / tests / rtest16.mac

(kill (all), values);
[];

/* make sure things work after a reset().  ID: 1986726 and ID: 2787047
 *
 * display2d is a resettable option variable. We save the value of display2d
 * and restore it after the reset. This allows to run the testsuite in both
 * display modes.
 */
(save:display2d, done);
done$
(reset(),0);
0;
(display2d:save, done);
done$

/* From A. Reiner.  Fixed in buildq.lisp rev. 1.4.
   Exposed a bug caused by the dynamical scope of VARLIST. */
buildq([foo:sin(baz+bar)],1);
1$

buildq([foo:[sin(baz+bar),sin(baz-bar)]],+splice(foo));
sin(baz+bar)+sin(baz-bar)$

/* Verify that extended functionality of rhs/lhs works as advertised */

(kill (x, y, aa, bb, cc), infix("@@"), 0);
0;

map (lhs, [aa < bb, aa <= bb, aa = bb, aa # bb, equal (aa, bb), notequal (aa, bb), aa >= bb, aa > bb]);
[aa, aa, aa, aa, aa, aa, aa, aa];

map (rhs, [aa < bb, aa <= bb, aa = bb, aa # bb, equal (aa, bb), notequal (aa, bb), aa >= bb, aa > bb]);
[bb, bb, bb, bb, bb, bb, bb, bb];

map (lhs, [foo(x) := 2*x, bar(y) ::= 3*y, '(aa : bb), '(aa :: bb), ?marrow(aa, bb)]);
['(foo(x)), '(bar(y)), aa, aa, aa];

map (rhs, [foo(x) := 2*x, bar(y) ::= 3*y, '(aa : bb), '(aa :: bb), ?marrow(aa, bb)]);
[2*x, 3*y, bb, bb, bb];

[lhs (aa @@ bb), lhs (aa @@ bb @@ cc), rhs (aa @@ bb), rhs (aa @@ bb @@ cc)];
[aa, aa @@ bb, bb, cc];

map (lhs, [aa + bb, aa - bb, aa * bb, aa / bb, sin(aa), log(aa)]);
[aa + bb, aa - bb, aa * bb, aa / bb, sin(aa), log(aa)];

map (rhs, [aa + bb, aa - bb, aa * bb, aa / bb, sin(aa), log(aa)]);
[0, 0, 0, 0, 0, 0];

kill ("@@");
done;

/* Verify that grind treats nouns correctly. string calls MSIZE in src/grind.lisp.
 */

kill (all);
done;

string ('(integrate(f(x), x) + integrate(g(x), x, minf, inf) + diff(u, x) + sum(h(x), x, 1, n)));
"sum(h(x),x,1,n)+integrate(g(x),x,minf,inf)+integrate(f(x),x)+diff(u,x)";

string ('integrate(f(x), x) + 'integrate(g(x), x, minf, inf) + 'diff(u, x) + 'sum(h(x), x, 1, n));
"'sum(h(x),x,1,n)+'integrate(g(x),x,minf,inf)+'integrate(f(x),x)+'diff(u,x,1)";

/* GREAT puts nounified atoms before others, it appears ... */
string (%a%a + %b%b + nounify(%c%c) + nounify(%d%d) + %e%e + %f%f);
"%d%d+%c%c+%f%f+%e%e+%b%b+%a%a";

string (sin(x) * cos(x) + tan(x));
"tan(x)+cos(x)*sin(x)";

string ('foo(x, y, z) / bar(a, b, c) + 'baz(%pi - 'quux(%e ^ mumble(%i))));
"'foo(x,y,z)/bar(a,b,c)+'baz(%pi-'quux(%e^mumble(%i)))";

/* It's conceivable that someday nounified arithmetic operators would be treated differently by grind.
 * If/when that happens, revise this example accordingly.
 */
string ('"+"(a, b, '"."(c, d), '"^"(e, f)));
"'?mplus(a,b,'?mnctimes(c,d),'?mexpt(e,f))";

/* Bug 626721 */
logarc(atan2(y,x));
-%i*log((%i*y+x)/sqrt(x^2+y^2));
rectform(ev(%,x=-1,y=1));
3*%pi/4; 

/*
 * Bug [ 1661490 ] An integral gives a wrong result.
 */
(assume(a>0, b>0, sqrt(sqrt(b^2+a^2)-a)*(sqrt(b^2+a^2)+a)^(3/2)-b^2>0),0);
0;
radcan(integrate(exp(-(a+%i*b)*x^2),x,minf,inf)/(sqrt(%pi)/sqrt(a+%i*b)));
1;

/*
 * [ 1663704 ] integrate(sin(r*x)^7/x^4,x,0,inf) -> r^3*false
 *
 * Should return the integral instead of producing false.
 */
integrate(sin(a*x)^7/x^4,x,0,inf);
'integrate(sin(a*x)^7/x^4,x,0,inf);

/* we have assumed a>0 */
integrate(%e^(-a*r)*sin(k*r),r,0,inf);
k/(k^2+a^2);

/*
 * Bug [ 1854888 ] hgfred([5],[5], 1) doesn't simplify
 */
hgfred([5],[5],1);
%e;

/*
 * Bug [ 1858964 ] hgfred([7],[-1], x) --/--> error
 */
hgfred([7],[-1],x);
und;

/*
 * Bug [ 1858939 ] hgfred([-1],[-2],x) --> error
 */
hgfred([-1],[-2],x);
/* Because of revision 1.110 of hyp.lisp gen_laguerre simplifies
   -gen_laguerre(1,-3,x)/2; */
1+x/2;

/*
 * Tests for the :: operator
 */
a:b;
b$

a::3;
3$

b;
3$

p:concat('p,1);
p1$

p::5;
5$

p1;
5$

kill(all);
done$

/* Bug [ 1860250 ] erf(-inf) --> -erf(inf) */

erf(-inf);
-1$

erf(inf);
1$

erf(-x) + erf(x);
0$

erf(a-b) + erf(b-a);
0$

/* Bug [ 1950653 ] bessel_j not simplified 
 * A few additional related tests added too.
 */

bessel_j(1/2,%pi),besselexpand:true;
0;
bessel_y(1/2,%pi/2),besselexpand:true;
0;

/* Bug [ 2149714 ] fpprintprec does not work correctly
*/

fpprec:16;
16;

block([fpprintprec:5], string(1.23b0));
"1.23b0";

block([fpprintprec:5], string(1.2345b0));
"1.2345b0";

block([fpprintprec:5], string(1.23456789b0));
"1.2346b0";

block([fpprintprec:25], string(1.2345678901234567890123456789b0));
"1.234567890123457b0";

/* verify that fpprintprec behavior matches its description */

block ([L1 : [["1.2E-10","1.2E-9","1.2E-8","1.2E-7","1.2E-6","1.2E-5","1.2E-4","0.0012","0.012","0.12","1.2","1.2E+1","1.2E+2",
  "1.2E+3","1.2E+4","1.2E+5","1.2E+6","1.2E+7","1.2E+8","1.2E+9","1.2E+10"],
 ["1.23E-10","1.23E-9","1.23E-8","1.23E-7","1.23E-6","1.23E-5","1.23E-4","0.00123","0.0123","0.123","1.23","12.3",
  "1.23E+2","1.23E+3","1.23E+4","1.23E+5","1.23E+6","1.23E+7","1.23E+8","1.23E+9","1.23E+10"],
 ["1.234E-10","1.234E-9","1.234E-8","1.234E-7","1.234E-6","1.234E-5","1.234E-4","0.001234","0.01234","0.1234","1.234",
  "12.34","123.4","1.234E+3","1.234E+4","1.234E+5","1.234E+6","1.234E+7","1.234E+8","1.234E+9","1.234E+10"],
 ["1.2344E-10","1.2344E-9","1.2344E-8","1.2344E-7","1.2344E-6","1.2344E-5","1.2344E-4","0.0012344","0.012344",
  "0.12344","1.2344","12.344","123.44","1234.4","1.2344E+4","1.2344E+5","1.2344E+6","1.2344E+7","1.2344E+8",
  "1.2344E+9","1.2344E+10"],
 ["1.23443E-10","1.23443E-9","1.23443E-8","1.23443E-7","1.23443E-6","1.23443E-5","1.23443E-4","0.00123443","0.0123443",
  "0.123443","1.23443","12.3443","123.443","1234.43","12344.3","1.23443E+5","1.23443E+6","1.23443E+7","1.23443E+8",
  "1.23443E+9","1.23443E+10"],
 ["1.234432E-10","1.234432E-9","1.234432E-8","1.234432E-7","1.234432E-6","1.234432E-5","1.234432E-4","0.001234432",
  "0.01234432","0.1234432","1.234432","12.34432","123.4432","1234.432","12344.32","123443.2","1.234432E+6",
  "1.234432E+7","1.234432E+8","1.234432E+9","1.234432E+10"],
 ["1.2344321E-10","1.2344321E-9","1.2344321E-8","1.2344321E-7","1.2344321E-6","1.2344321E-5","1.2344321E-4",
  "0.0012344321","0.012344321","0.12344321","1.2344321","12.344321","123.44321","1234.4321","12344.321","123443.21",
  "1234432.1","1.2344321E+7","1.2344321E+8","1.2344321E+9","1.2344321E+10"],
 ["1.23443211E-10","1.23443211E-9","1.23443211E-8","1.23443211E-7","1.23443211E-6","1.23443211E-5","1.23443211E-4",
  "0.00123443211","0.0123443211","0.123443211","1.23443211","12.3443211","123.443211","1234.43211","12344.3211",
  "123443.211","1234432.11","1.23443211E+7","1.23443211E+8","1.23443211E+9","1.23443211E+10"],
 ["1.234432112E-10","1.234432112E-9","1.234432112E-8","1.234432112E-7","1.234432112E-6","1.234432112E-5",
  "1.234432112E-4","0.001234432112","0.01234432112","0.1234432112","1.234432112","12.34432112","123.4432112",
  "1234.432112","12344.32112","123443.2112","1234432.112","1.234432112E+7","1.234432112E+8","1.234432112E+9",
  "1.234432112E+10"],
 ["1.2344321123E-10","1.2344321123E-9","1.2344321123E-8","1.2344321123E-7","1.2344321123E-6","1.2344321123E-5",
  "1.2344321123E-4","0.0012344321123","0.012344321123","0.12344321123","1.2344321123","12.344321123","123.44321123",
  "1234.4321123","12344.321123","123443.21123","1234432.1123","1.2344321123E+7","1.2344321123E+8","1.2344321123E+9",
  "1.2344321123E+10"],
 ["1.23443211234E-10","1.23443211234E-9","1.23443211234E-8","1.23443211234E-7","1.23443211234E-6","1.23443211234E-5",
  "1.23443211234E-4","0.00123443211234","0.0123443211234","0.123443211234","1.23443211234","12.3443211234",
  "123.443211234","1234.43211234","12344.3211234","123443.211234","1234432.11234","1.23443211234E+7",
  "1.23443211234E+8","1.23443211234E+9","1.23443211234E+10"],
 ["1.234432112344E-10","1.234432112344E-9","1.234432112344E-8","1.234432112344E-7","1.234432112344E-6",
  "1.234432112344E-5","1.234432112344E-4","0.001234432112344","0.01234432112344","0.1234432112344","1.234432112344",
  "12.34432112344","123.4432112344","1234.432112344","12344.32112344","123443.2112344","1234432.112344",
  "1.234432112344E+7","1.234432112344E+8","1.234432112344E+9","1.234432112344E+10"],
 ["1.2344321123443E-10","1.2344321123443E-9","1.2344321123443E-8","1.2344321123443E-7","1.2344321123443E-6",
  "1.2344321123443E-5","1.2344321123443E-4","0.0012344321123443","0.012344321123443","0.12344321123443",
  "1.2344321123443","12.344321123443","123.44321123443","1234.4321123443","12344.321123443","123443.21123443",
  "1234432.1123443","1.2344321123443E+7","1.2344321123443E+8","1.2344321123443E+9","1.2344321123443E+10"],
 ["1.23443211234432E-10","1.23443211234432E-9","1.23443211234432E-8","1.23443211234432E-7","1.23443211234432E-6",
  "1.23443211234432E-5","1.23443211234432E-4","0.00123443211234432","0.0123443211234432","0.123443211234432",
  "1.23443211234432","12.3443211234432","123.443211234432","1234.43211234432","12344.3211234432","123443.211234432",
  "1234432.11234432","1.23443211234432E+7","1.23443211234432E+8","1.23443211234432E+9","1.23443211234432E+10"],
 ["1.234432112344321E-10","1.234432112344321E-9","1.234432112344321E-8","1.234432112344321E-7","1.234432112344321E-6",
  "1.234432112344321E-5","1.234432112344321E-4","0.001234432112344321","0.01234432112344321","0.1234432112344321",
  "1.234432112344321","12.34432112344321","123.4432112344321","1234.432112344321","12344.32112344321",
  "123443.2112344321","1234432.112344321","1.234432112344321E+7","1.234432112344321E+8","1.234432112344321E+9",
  "1.234432112344321E+10"],
 ["1.234432112344321E-10","1.234432112344321E-9","1.234432112344321E-8","1.234432112344321E-7","1.234432112344321E-6",
  "1.234432112344321E-5","1.234432112344321E-4","0.001234432112344321","0.01234432112344321","0.1234432112344321",
  "1.234432112344321","12.34432112344321","123.4432112344321","1234.432112344321","12344.32112344321",
  "123443.2112344321","1234432.112344321","1.234432112344321E+7","1.234432112344321E+8","1.234432112344321E+9",
  "1.234432112344321E+10"],
 ["1.234432112344321E-10","1.234432112344321E-9","1.234432112344321E-8","1.234432112344321E-7","1.234432112344321E-6",
  "1.234432112344321E-5","1.234432112344321E-4","0.001234432112344321","0.01234432112344321","0.1234432112344321",
  "1.234432112344321","12.34432112344321","123.4432112344321","1234.432112344321","12344.32112344321",
  "123443.2112344321","1234432.112344321","1.234432112344321E+7","1.234432112344321E+8","1.234432112344321E+9",
  "1.234432112344321E+10"],
 ["1.234432112344321E-10","1.234432112344321E-9","1.234432112344321E-8","1.234432112344321E-7","1.234432112344321E-6",
  "1.234432112344321E-5","1.234432112344321E-4","0.001234432112344321","0.01234432112344321","0.1234432112344321",
  "1.234432112344321","12.34432112344321","123.4432112344321","1234.432112344321","12344.32112344321",
  "123443.2112344321","1234432.112344321","1.234432112344321E+7","1.234432112344321E+8","1.234432112344321E+9",
  "1.234432112344321E+10"],
 ["1.234432112344321E-10","1.234432112344321E-9","1.234432112344321E-8","1.234432112344321E-7","1.234432112344321E-6",
  "1.234432112344321E-5","1.234432112344321E-4","0.001234432112344321","0.01234432112344321","0.1234432112344321",
  "1.234432112344321","12.34432112344321","123.4432112344321","1234.432112344321","12344.32112344321",
  "123443.2112344321","1234432.112344321","1.234432112344321E+7","1.234432112344321E+8","1.234432112344321E+9",
  "1.234432112344321E+10"]],
 L2 : block ([foo : 1.2344321123443211234],
             makelist (block ([fpprintprec : m], makelist (string (foo*10^n), n, -10, 10)), m, 2, 20))],
 map (lambda ([s1, s2], if sequalignore (s1, s2) then true else s2 # s1), flatten (L1), flatten (L2)),
 delete (true, %%));
[];

/* SF bug #3213: "fpprintprec do not round bfloat correctly." */

(reset (fpprec, bftrunc),
 fpprintprec:4,
 string (1.23456789b0));
"1.235b0";

/* default fpprintprec => print all digits */

(reset (fpprintprec), 0);
0;

string (bfloat (1000000000000*(1 + 8/9)));
"1.888888888888889b12";

string (bfloat ((1 + 8/9)/1000000000000)), fpprec=50;
"1.8888888888888888888888888888888888888888888888889b-12";

string (bfloat (1000000000000*(1 + 8/9))), fpprec=10;
"1.888888889b12";

string (bfloat ((1 + 8/9)/1000000000000)), fpprec=2;
"1.9b-12";

/* expect to see bigfloat ending in 5 to round to even */

map (string, map (bfloat, 1 + [1, 2, 3, 4, 5, 6, 7]/8)), fpprintprec=3;
["1.12b0", "1.25b0", "1.38b0", "1.5b0", "1.62b0", "1.75b0", "1.88b0"];

map (string, map (bfloat, 100 + makelist (i/64, i, 1, 63))), fpprintprec=8;
["1.0001562b2", "1.0003125b2", "1.0004688b2", "1.000625b2", "1.0007812b2", "1.0009375b2", "1.0010938b2", "1.00125b2",
 "1.0014062b2", "1.0015625b2", "1.0017188b2", "1.001875b2", "1.0020312b2", "1.0021875b2", "1.0023438b2", "1.0025b2",
 "1.0026562b2", "1.0028125b2", "1.0029688b2", "1.003125b2", "1.0032812b2", "1.0034375b2", "1.0035938b2", "1.00375b2",
 "1.0039062b2", "1.0040625b2", "1.0042188b2", "1.004375b2", "1.0045312b2", "1.0046875b2", "1.0048438b2", "1.005b2",
 "1.0051562b2", "1.0053125b2", "1.0054688b2", "1.005625b2", "1.0057812b2", "1.0059375b2", "1.0060938b2", "1.00625b2",
 "1.0064062b2", "1.0065625b2", "1.0067188b2", "1.006875b2", "1.0070312b2", "1.0071875b2", "1.0073438b2", "1.0075b2",
 "1.0076562b2", "1.0078125b2", "1.0079688b2", "1.008125b2", "1.0082812b2", "1.0084375b2", "1.0085938b2", "1.00875b2",
 "1.0089062b2", "1.0090625b2", "1.0092188b2", "1.009375b2", "1.0095312b2", "1.0096875b2", "1.0098438b2"];

/* bftrunc=false => don't strip trailing 0's */

map (string, map (bfloat, 1 + [1, 2, 3, 4, 5, 6, 7]/8)), fpprintprec=3, bftrunc=false;
["1.12b0", "1.25b0", "1.38b0", "1.50b0", "1.62b0", "1.75b0", "1.88b0"];

map (string, map (bfloat, 100 + makelist (i/64, i, 1, 63))), fpprintprec=8, bftrunc=false;
["1.0001562b2", "1.0003125b2", "1.0004688b2", "1.0006250b2", "1.0007812b2", "1.0009375b2", "1.0010938b2", "1.0012500b2",
 "1.0014062b2", "1.0015625b2", "1.0017188b2", "1.0018750b2", "1.0020312b2", "1.0021875b2", "1.0023438b2", "1.0025000b2",
 "1.0026562b2", "1.0028125b2", "1.0029688b2", "1.0031250b2", "1.0032812b2", "1.0034375b2", "1.0035938b2", "1.0037500b2",
 "1.0039062b2", "1.0040625b2", "1.0042188b2", "1.0043750b2", "1.0045312b2", "1.0046875b2", "1.0048438b2", "1.0050000b2",
 "1.0051562b2", "1.0053125b2", "1.0054688b2", "1.0056250b2", "1.0057812b2", "1.0059375b2", "1.0060938b2", "1.0062500b2",
 "1.0064062b2", "1.0065625b2", "1.0067188b2", "1.0068750b2", "1.0070312b2", "1.0071875b2", "1.0073438b2", "1.0075000b2",
 "1.0076562b2", "1.0078125b2", "1.0079688b2", "1.0081250b2", "1.0082812b2", "1.0084375b2", "1.0085938b2", "1.0087500b2",
 "1.0089062b2", "1.0090625b2", "1.0092188b2", "1.0093750b2", "1.0095312b2", "1.0096875b2", "1.0098438b2"];

/* fpprintprec <= fpprec */

every (lambda ([s], s="5.6b2"), map (string, makelist (bfloat (5000/9), fpprec, fpprintprec, 40))), fpprintprec=2;
true;

every (lambda ([s], s="5.56b-4"), map (string, makelist (bfloat (5/9000), fpprec, fpprintprec, 40))), fpprintprec=3;
true;

every (lambda ([s], s="5.556b2"), map (string, makelist (bfloat (5000/9), fpprec, fpprintprec, 40))), fpprintprec=4;
true;

every (lambda ([s], s="5.5556b-4"), map (string, makelist (bfloat (5/9000), fpprec, fpprintprec, 40))), fpprintprec=5;
true;

every (lambda ([s], s="5.55556b2"), map (string, makelist (bfloat (5000/9), fpprec, fpprintprec, 40))), fpprintprec=6;
true;

every (lambda ([s], s="5.555556b-4"), map (string, makelist (bfloat (5/9000), fpprec, fpprintprec, 40))), fpprintprec=7;
true;

every (lambda ([s], s="5.5555556b2"), map (string, makelist (bfloat (5000/9), fpprec, fpprintprec, 40))), fpprintprec=8;
true;

/* fpprintprec >= fpprec */

/* bfloat generally produces a number which is not exactly the same as its argument.
 * When bfloat returns a bigger number, it should round up when formatted,
 * and when it's smaller, it should round down.
 * When bfloat returns an exact result, in this case it should round up (to 6).
 * Consider this when figuring out the correct output for these examples.
 */

(fpprec : 2, 
 mybf : bfloat(5/9000),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))/1000));
true;

every (lambda ([s], s="5.6b-4"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(fpprec : 3,
 mybf : bfloat(5000/9),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))*1000));
true;

every (lambda ([s], s="5.56b2"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(fpprec : 4,
 mybf : bfloat(5/9000),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))/1000));
true;

every (lambda ([s], s="5.556b-4"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(fpprec : 5,
 mybf : bfloat(5000/9),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))*1000));
false;

every (lambda ([s], s="5.5555b2"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(fpprec : 10,
 mybf : bfloat(5/9000),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))/1000));
true;

every (lambda ([s], s="5.555555556b-4"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(fpprec : 20,
 mybf : bfloat(5000/9),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))*1000));
true;

every (lambda ([s], s="5.5555555555555555556b2"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(fpprec : 40,
 mybf : bfloat(5/9000),
 is (rationalize (mybf) >= (5*sum(10^i, i, 0, fpprec)/10^(fpprec + 1))/1000));
false;

every (lambda ([s], s="5.555555555555555555555555555555555555555b-4"), map (string, makelist (mybf, fpprintprec, fpprec, 40)));
true;

(reset (fpprec), 0);
0;

/*
 * Bug 2142758: integrate(sqrt(2-2*x^2)*(sqrt(2)*x^2+sqrt(2))/(4-4*x^2),x,0,1)
 */
integrate(sqrt(2-2*x^2)*(sqrt(2)*x^2+sqrt(2))/(4-4*x^2),x,0,1);
3*%pi/8;


integrate(sqrt(1-x^2)*(x^2+1)/(2-2*x^2),x,0,1);
3*%pi/8;

integrate(sqrt(1-x^2)*(x^2+1)/(1-x^2),x,0,1);
3*%pi/4;

/*
 * Bug [ 2208303 ] Problem with jacobi_dn and elliptic_kc
 */
jacobi_dn(elliptic_kc(m)*t,m);
jacobi_dn(elliptic_kc(m)*t,m);

/*
 * Bug [ 2180110 ] GCL do not signal an overflow converting bigfloat to float
 */
errcatch(float(2b400));
[];

errcatch(float(bfloat(2^1024)));
[];

/*
 * Bug [ 2055235 ] Plot leaves range with jacobi functions
 *
 * Actually jacobi_cn(100, .7) is computed inaccurately.  Just check that abs(jacobi_cn(100,.7)) < 1
 */

is(abs(jacobi_cn(100.0, 0.7)) < 1);
true$

/*
 * Bug [ 1658067 ] jacobi_sn(elliptic_kc(1-m)*%i/2,m) isn't simplified
 *
 * This test (and Maxima) used to be wrong.  This is related to the
 * jacobi_sc(elliptic_kc(m)/2,m) test below.
 */
jacobi_sn(elliptic_kc(1-m)*%i/2,m);
%i/m^(1/4)$

jacobi_sc(elliptic_kc(m)+u,m);
-jacobi_cs(u,m)/sqrt(1-m)$

/*
 * Maxima used to get this wrong by returning the reciprocal instead
 */
jacobi_sc(elliptic_kc(m)/2,m);
1/(1-m)^(1/4);

/*
 * Bug [ 2505945 ] - hgfred([2,-1/2],[3],-x^2);
 *
 * Shouldn't signal from diff about non-variable second arg.
 *
 * The expected value here is computed from
 * factor(ratsimp(subst([z=-x^2],hgfred([2,-1/2],[3],z))))
 */
factor(ratsimp(hgfred([2,-1/2],[3],-x^2)));
4*(3*x^4*sqrt(x^2+1)+x^2*sqrt(x^2+1)-2*sqrt(x^2+1)+2)/(15*x^4)$

/*
 * Bug 2534420: asinh(%i*2b0) causes error
 */
is(abs(asinh(%i*2b0)-expand(bfloat(asinh(%i*2)))) < 3b-17);
true;

/*
 * Bug 2543079: bfloat(gamma(3/4)/gamma(1/4)) is wrong.
 */
bfloat(gamma(3/4)/gamma(1/4));
3.379891200336424b-1;

/*
 * Bug 2582034 - hgfred([a/2,-a/2],[1/2],z) causes error
 */
(assume(zn<0), done);
done;

hgfred([a/2,-a/2],[1/2],zn);
((%i*sqrt(zn)+sqrt(1-zn))^a+(sqrt(1-zn)-%i*sqrt(zn))^a)/2$
 
/*
 * Bug 2618401 - bfloat produces incorrect answer
 */
is(abs(bfloat((sqrt(2)+2)*%pi^(3/2)/(8*gamma(3/4)^2))-float((sqrt(2)+2)*%pi^(3/2)/(8*gamma(3/4)^2))) < 1d-15);
true;

/* (-1.0b0)^(1/3) vs (-1.0d0)^(1/3) - ID: 619927 */
(-1b0)^(1/3);
-1.0b0;

(-1.0)^(1/3);
-1.0;

domain:complex;
complex;

(-1b0)^(1/3);
1.0b0*(-1)^(1/3);

(-1.0)^(1/3);
1.0*(-1)^(1/3);

domain:real;
real;

(-1b0)^(1/3),domain:complex,m1pbranch:true;
1.0b0*(sqrt(3)*%i/2+1/2);

(-1.0)^(1/3),domain:complex,m1pbranch:true;
 1.0*(sqrt(3)*%i/2+1/2);


/*
 * Bug [ 2688847 ] float of rats rounds incorrectly
 */
float((2^60-1)/2^60)-1;
0.0;
float((2^1000-1)/2^1000)-1;
0.0;

/*
 * Bug [ 2687962 ] hgfred([-3/2,1],[-1/2],-t) division by zero
 *
 * Solution from functions.wolfram.com
 */
ratsimp(hgfred([-3/2,1],[-1/2], t));
1+3*t-3*t^(3/2)*atanh(sqrt(t));

/*
 * Bug 2793827: internal error in integrate
 */
(assume(n>0),declare(n,integer),0);
0;

integrate((g32475^n*(g32475*n-n-1)/(g32475-1)^2+1/(g32475-1)^2)/(1-g32475)
 -(g32475^(2*n+1)*(g32475*n-n-1)/(g32475-1)^2+g32475^(n+1)/(g32475-1)^2)
  /(1-g32475),g32475,0,1);
(n+1)^2/2-1/2;

kill(all);
done;

/*
 * Bug 609464 : 1+%e,numer and %e^%e,numer
 *
 * The simplifier has been extended to handle %e like other constants.
 * In addition functions with arguments which involve %e simplify
 * accordingly.
 */

%e,numer;
2.7182818284590451;

%e+1,numer;
3.7182818284590451;

%e^%e,numer;
15.154262241479262;

%e^x,numer;
%e^x;

sin(%e),numer;
0.41078129050290885;

sin(%e+1),numer;
-0.54525155669233449;

/* Do not simplify, when %e is the base of an expression and %enumer FALSE*/

sin(%e^(2*x+1)),numer;
sin(%e^(2*x+1));

sin(%e^(%e^(2*x+1))),numer;
sin(%e^(%e^(2*x+1)));

/* Additionally simplifications when %enumer TRUE */

%enumer:true;
true;

sin(%e^x),numer;
sin(2.7182818284590451^x);

sin(%e^(%e^(2*x+1))),numer;
sin(2.7182818284590451^(2.7182818284590451^(2*x+1)));

%enumer:false;
false;

/*
 * Bug ID: 2797885 - "problem with integration"
 *
 * integrate(exp(%i*x)*sin(x),x) generates a Lisp error.
 *
 * This is a special case for the integrand: exp(a*x)*sin(b*x),
 * with a^2+b^2 equal to zero.
 */

/* This is the general case for an integral with exp and sin or cos */
integrate(exp(a*x)*sin(b*x),x);
%e^(a*x)*(a*sin(b*x)-b*cos(b*x))/(b^2+a^2);

integrate(exp(a*x)*cos(b*x),x);
%e^(a*x)*(b*sin(b*x)+a*cos(b*x))/(b^2+a^2);

/* Now the special case with a=%i and b=1 */
expand(integrate(exp(%i*x)*sin(x),x));
%i*x/2-%e^(2*%i*x)/4;

expand(integrate(exp(x)*sin(%i*x),x));
%i*%e^(2*x)/4-%i*x/2;

expand(integrate(exp(%i*x)*cos(x),x));
x/2-%i*%e^(2*%i*x)/4;

expand(integrate(exp(x)*cos(%i*x),x));
%e^(2*x)/4+x/2;

/* Bug ID: 932076 -  ode2( 'diff(y,x)=%i*y+sin(x), y, x) => div by 0
 *
 * This bug is related to the Bug ID: 2797885 - "problem with integration"
 */

ode2('diff(y,x)-%i*y-sin(x),y,x);
y = (%c-%i*(x-%i*%e^-(2*%i*x)/2)/2)*%e^(%i*x);

/*
 * Bug ID: 826623 "simplifer returns %i*%i"
 * 
 * Some examples to show simplification of expressions of the form
 * (a*b*...)^q*(a*b*...)^r, where q+r=1
 */

sqrt(-%i)*sqrt(-%i)*%i;
1;

sqrt(a*b)*sqrt(a*b)*a*b;
a^2*b^2;

(a*b*c)^(3/4)*(a*b*c)^(1/4)*c;
a*b*c^2;

/*
 * Bug ID: 2792493 "hgfred([1],[-5.2],x);"
 */
hgfred([1],[-5.2],x);
%f[1,1]([-6.2],[-5.2],-x)*%e^x$

/* BUG ID: 721575 2/sqrt(2) doesn\'t simplify */
2/sqrt(2);
sqrt(2);

(1/2)*sqrt(2);
1/sqrt(2);

sqrt(2)*(1/2);
1/sqrt(2);

/* BUG ID 2029041 a*sqrt(2)/2 unsimplified */

a*sqrt(2)/2;
a/sqrt(2);

/* BUG ID 1923119 1/sqrt(8)-sqrt(8)/8 */

1/sqrt(8)-sqrt(8)/8;
0;

/* BUG ID 1927178 integrate(sin(t),t,%pi/4,3*%pi/4) */

integrate(sin(t),t,%pi/4,3*%pi/4);
sqrt(2);

/* BUG ID: 1480562 2*a*2^k isn't simplified to a*2^(k+1) */

2*a*2^k;
a*2^(k+1);

a*2^k*2;
a*2^(k+1);

/* Some examples to show simplification of expressions
 * with floating point and bigfloat numbers after improvement
 * of plusin
 */

(4.0*x-4.0*x);
0.0;
(4.0*x-3.0*x);
1.0*x;
(4.0*x-3.0*x)/2;
0.5*x;

(4.0b0*x-4.0*x);
0.0b0;
(4.0b0*x-3.0*x);
1.0b0*x;
(4.0b0*x-3.0*x)/2;
0.5b0*x;

/* BUG ID: 1996354 unsimplifed result from expand */

expand((%e^(-2*sqrt(2))*(%e^(2*sqrt(2))+2*%e^sqrt(2)+1)^2)/16
       +(%e^(-2*sqrt(2))*(%e^(2*sqrt(2))-2*%e^sqrt(2)+1)^2)/16
       -(%e^(-2*sqrt(2))*(%e^(2*sqrt(2))-1)^2)/8);
1;

/* BUG ID: 631216 - "horner([...],x)/FIX" 
   horner now maps over lists, matrices and equations.
 */

horner(x^2+x=a*x^2+b*x);
x*(x+1) = x*(a*x+b);
horner([x^2+x,x^3+x,x^4+x]);
[x*(x+1),x*(x^2+1),x*(x^3+1)];

/* BUG ID: 2699862 "derivative of polylogarithm"
 * The noun form is not put on the property list, but NIL. The routine 
 * sdiffgrad generates a noun form, when the derivative is not known.
 */

diff(li[n](x),n);
'diff(li[n](x),n);

diff(li[n*x](x),x);
'diff(li[n*x](x),x);

diff(li[n](x),x,1,n,1);
'diff(li[n-1](x),n)/x;

/* Not reported as a bug, but the same problem for the function psi */

diff(psi[n](x),n);
'diff(psi[n](x),n);

diff(psi[n*x](x),x);
'diff(psi[n*x](x),x);

diff(psi[n](x),x,1,n,1);
'diff(psi[n+1](x),n);

/* BUG ID: 2824909 " exp(%i*%pi/4) not simplified" 
 * Check the simplification of exp(%i*%pi/4) and exp(-%i*pi/4)
 */

exp(%i*%pi/4);
1/sqrt(2)+%i/sqrt(2);
exp(-%i*%pi/4);
1/sqrt(2)-%i/sqrt(2);

/*
 * Bug ID: 2831259 - bfloat() underflow bug
 */
fpprec:500;
500;
float(0b0);
0.0;

/*
 * BUG ID: 2835098 - SIGN-PREP strangeness
 */
 
block ([?limitp : true], sign (foo (x)));
pnz;

integrate(sqrt(2*m*(E[n]-U(x))),x,-x[0],x[0])=(n-1/2)*%pi*hbar;
sqrt(2)*'integrate(sqrt(m*(E[n]-U(x))),x,-x[0],x[0]) = %pi*hbar*(n-1/2);

integrate(f(x),x,x[0],x[1]);
'integrate(f(x),x,x[0],x[1]);

/*
 * BUG ID: 2840566 - defint fails to determine if one of its limit is real
 */
 
(assume(b>0,c>0),done);
done;
 
integrate(x,x,0,sqrt(b^2+(b-c)^2));
(c^2-2*b*c+2*b^2)/2;

/*
 * BUG ID: 2842060 - unsimplified result from integrate
 */
 
/* The result for a general symbol x */
integrate(1/x/sqrt(x^2-1),x);
-asin(1/abs(x));

(assume(x>0), done);
done;
 
/* abs(x) simplifies to x for x>0 */
integrate(1/x/sqrt(x^2-1),x);
-asin(1/x);

(forget(x>0), done);
done;

/* 
 * Bug ID: 2820202 - rootscontract(%i/2) 
 */
rootscontract(%i/2);
%i/2;

/* Bug ID: 2872738 - sign(-(1/n)*(-1)^n)
 * We got the error because of the simplification
 *  (-1)^n*(-1) -> (-1)^(n+1) and not -(-1)^n
 * The other case 
    (-1)*(-1)^n simplifies already to -(-1)^n
 * Adding tests for both cases.
 */
kill(all);
done;
sign(-(1/n)*(-1)^n);
pn;
(-1)*(-1)^n;
-(-1)^n;
(-1)^n*(-1);
-(-1)^n;

/* Bug ID: 2835634 - logcontract broken
 * Bug ID: 1467368 - logcontract returns unsimplified expr
 */
logcontract(log(x)-log(2));
log(x/2);
/* Check that we do not break the following again */
logcontract(log(%e*k)-log(%e^-1*k));
2;
log(%e^2),logexpand:false;
2;

/* Bug ID: 2880923 - realpart --> floating-point-overflow
 */
sign(exp(2009));
pos;
realpart(sqrt(4*%e^2009-3)-1);
sqrt(4*%e^2009-3)-1;
sqrt(4*exp(2009));
2*%e^(2009/2);

/* Bug ID: 640332 - Need to specdisrep more systematically
   Add the examples of the bug report.
 */
ratdisrep(diff(rat(x),rat(x)));
1;
diff(x,rat(x));
1;
outofpois(diff(intopois(sin(x)),x));
cos(x);
taylor(intopois(sin(x)),x,0,3);
x-x^3/6;
ratsimp(intopois(sin(x)));
sin(x);

/* Bug ID: 627759 - Ratdisrep of aggregates 
 */
ratdisrep(rat(x=y));
x = y;
ratdisrep(rat([x=a,y=b]));
[x = a,y = b];
ratdisrep(rat(matrix([a,b],[c,d])));
matrix([a,b],[c,d]);

/* Bug ID: 711885 - Rootscontract with imaginaries fails
 */
(oldvalue:radexpand, radexpand:false, done);
done;

rootscontract(((sqrt(3)*%i+1)^(3/2)-4*%i)/sqrt(sqrt(3)*%i+1));
((sqrt(-3)+1)^(3/2)-4*%i)/sqrt(sqrt(-3)+1);

/* It is a problem of the simplifier. Show that it works */
sqrt(1/(1+sqrt(-3)));
1/sqrt(sqrt(-3)+1);

(radexpand:oldvalue, done);
done;

/* BUG ID: 767556 - Distributing operations over =
 * The operators "." and "^^" distribute over equations.
 */

x . (a=b);
x . a = x . b;

(a=b)^^x;
a^^x = b^^x;

/* A more complicated example */
x . ((2*a+b . c) = x . (y + z))^^w;
x . (b . c+2*a)^^w = x . (x . (z+y))^^w;

/* Bug ID: 2914176 - Conversion of rational to bfloat is inaccurate
 *
 * The difference should be 1/262144, but we don't check for that.
 */
(oldfpprec:fpprec, fpprec:5, done);
done;
is(bfloat((2^20+1)/(2^20-1)) - 1b0 > 0);
true;

/* Related to the fix for 2914176.  Didn't handle the ratio 0/1 */
is(equal(0b0, 0));
true;

(fpprec:oldfpprec, done);
done;

/* Bug ID:2933882 - Power function: 0^a not fully implemented
 * Show some simplifications of 0^a
 */

assume(a>0);
[a>0];
0^a;
0;
errcatch(0^-a);
[];
0^(a+%i);
0;
0^(1/2+%i);
0;
errcatch(0^(-1/2+%));
[];
errcatch(0^%i);
[]; 

forget(a>0);
[a>0];

/* Bug ID: 2938078 - Crash on attached input
 */
 
declare(n,integer, j,noninteger);
done;
assume(equal(x,n), equal(y,j), equal(z,i));
[equal(x, n), equal(y, j), equal(z,i)];

featurep(x,integer);
true;
featurep(x,noninteger);
false;
featurep(y,integer);
false;
featurep(y,noninteger);
true;

diff(z+1,z);
1;

remove(n,integer, j,noninteger);
done;
forget(equal(x,n), equal(y,j), equal(z,i));
[equal(x, n), equal(y, j), equal(z, i)];

/* Bug ID:  2948800 - integrate((1-cos(2*x)^2)^2/x^4,x,0,inf) wrong
 */
 
integrate((1-cos(2*x)^2)^2/x^4,x,0,inf);
8*%pi/3;

assume(a>0);
[a>0];

/* The more general type with an argument a*x and a positive */
integrate((1-cos(a*x)^2)^2/x^4,x,0,inf);
%pi*a^3/3;

forget(a>0);
[a>0];

/* Bug ID: 777564 - subtraction "-"(a,b) should work/FIX */

"-"();
0;

"-"(a);
-a;

"-"(2*a);
-2*a;

"-"(a+b);
-b-a;

"-"(a+b+c);
-c-b-a;

"-"(100,20,10);
70;

map("-",[a,x,100],[b,y,20]);
[a-b,x-y,80];

map("-",[a,x,100],[b,y,20],[c,z,10]);
[-c-b+a,-z-y+x,70];

/* Bug ID: 910270 - 1/+3*x parses as 1/(+3*x)
 * Show that the "+" operator can be used as a prefix operator too.
 */
1/+3*x;
x*1/3;
1/+x/3;
1/(3*x);
a^+b*c;
c*a^b;

/* Bug ID: 2961822 - sinh(0.0b0) causes Maxima to abort
 */
sinh(0.0b0);
0.0b0;

/* Bug ID: 1219846 - properties of translated functions
 * The property noun is already present
 */
kill(f);
done;
f(x):=x;
f(x):=x;
properties(f);
[function,noun];
translate(f);
[f];
properties(f);
[transfun,function,noun];
kill(f);
done;

/* Bug ID: 2968344 - gamma_incomplete(1.0, 4.368265444147715e+19) fails
 */
gamma_incomplete(1.0, 4.368265444147715e+19);
0.0;

/* Bug ID: 643254 - orderlessp([rat(x)], [rat(x)])
 */
orderlessp([rat(x)],[rat(x)]);
false;

/* Bug ID: 781657 - binomial(x,x) => 1, but binomial(-1,-1) => 0
 * binomial(x,x) simplifies to 1 only if the sign of x is known not to be 
 * negative.
 */
is(equal(binomial(x,x),1));
'unknown;
is(equal(binomial(x^2,x^2),1));
true;

/* Bug ID:856209 - inconsistency between facts() and facts(v)
 * Show that facts(expr) now works more general.
 */
assume(z+a>0,b>z);
[a+z>0,b>z];
facts(a);
[a+z>0];
facts(b);
[b>z];
facts(z);
[a+z>0,b>z];
facts(a+z);
[a+z>0];
forget(z+a>0,b>z);
[a+z>0,b>z];

/* Bug ID: 840848 - trigreduce doesn't enter unknown functions
 */
trigexpand(f(sin(2*x)));
f(2*cos(x)*sin(x));
trigreduce(%);
f(sin(2*x));

/* Bug ID: 2954472 - rectform with large floats gives bad answer
*/
is(abs(rectform(1e160/(1e160+%i))-1) < 1e-160);
true;

is(abs(rectform(1e160/(1e160+3/2*%i))-1) < 1.5e-160);
true;

/* Bug ID: 2953369 - Definite Integration of 1/(a-b*cos(x)) wrong
 *
 * For simplicity we test the equivalent integrate(1/(1-r*cos(x)),x,0,%pi).
 */
/* These assumes are to answer the questions integrate (from routine unitcir) will ask */
(assume(r>0,r<1,abs(sqrt(1-r^2)-1)/r-1 < 0, sqrt(1-r^2)-r+1>0), 0);
0;
integrate(1/(1-r*cos(x)),x,0,%pi);
%pi/sqrt(1-r^2);

/* Bug ID: 2907727 - Incorrect Integral with option integrate_use_rootsof
 * :true
 */
%rnum:0;
0;
integrate((d*x^2+2*c*x+3*b)/(g*r*x^3+d*x^2+c*x+b), x), integrate_use_rootsof:true;
lsum((%r1^2*d+2*%r1*c+3*b)*log(x-%r1)/(3*%r1^2*g*r+2*%r1*d+c),%r1,
             rootsof(g*r*%r1^3+d*%r1^2+c*%r1+b,%r1));

/* Bug ID: 2880797 - bad answer in integrate(sqrt(sin(t)^2+cos(t)^2),t,0,2*%pi)
 *
 */
integrate(sqrt(sin(t)^2+cos(t)^2),t,0,2*%pi);
2*%pi;

/* Bug ID: 2980551 - Inconsistent simplification of exp(x*%i*%pi)
 *
 * These examples show consistent simplification for x an expression which
 * can contain float or bigfloat values
 */

exp(2*%i*%pi);
1;

exp((2+x)*%i*%pi);
exp(x*%i*%pi);

exp(2*%i*%pi+x*%i*%pi);
exp(x*%i*%pi);

log(exp((2+x)^2*%i*%pi));
(2+x)^2*%i*%pi;

exp(2.0*%i*%pi);
1.0;

exp((2.0+x)*%i*%pi);
exp(1.0*x*%i*%pi);

exp(2.0*%i*%pi+x*%i*%pi);
exp(1.0*x*%i*%pi);

log(exp((2.0+x)^2*%i*%pi));
(2.0+x)^2*%i*%pi;

exp(2.0b0*%i*%pi);
1.0b0;

exp((2.0b0+x)*%i*%pi);
exp(1.0b0*x*%i*%pi);

exp(2.0b0*%i*%pi+x*%i*%pi);
exp(1.0b0*x*%i*%pi);

log(exp((2.0b0+x)^2*%i*%pi));
(2.0b0+x)^2*%i*%pi;

exp(3/2*%pi*%i);
-%i;
exp(1.5*%pi*%i);
-1.0*%i;
exp(1.5b0*%pi*%i);
-1.0b0*%i;

exp((3/2+x)*%pi*%i);
-%i*exp(%i*%pi*x);
exp((1.5+x)*%pi*%i);
-%i*exp(1.0*%i*%pi*x);
exp((1.5b0+x)*%pi*%i);
-%i*exp(1.0b0*%i*%pi*x);

/* Bug ID: 2781127 - bfpsi0 of complex
 *
 * (The result was not in rectangular form but it should be.)
 */
bfpsi0(4.5 + %i,15);
2.43845477527606b-1*%i + 1.41875534014717b0;

/* Bug ID: 2988190 - atan2(1b20,-1b0); badly wrong
 * It's really a bug in atan for x < -1, so test both.
 */
(fpprec:16, atan(-1b20));
-1.570796326794897b0;
atan2(1b20,-1b0);
1.570796326794897b0;

/*
 * Bug ID: 2991924 - Incorrect integration of rational functions
 */
integrate(1/(x^4-2),x,0,1) - integrate(1/(x^2-sqrt(2))/(x^2+sqrt(2)),x,0,1);
0;

integrate(1/(x^6-4),x,0,1) - integrate(1/(x^3-2)/(x^3+2),x,0,1);
0;

/* BUG ID:  2113751 - Incomprehensible behavior of coeff()
 */
coeff(2*%e^x, x, 0);
2*%e^x;

/* For numerical tests */
closeto(e,tol):=block([numer:true,abse],abse:abs(e),if(abse<tol) then true else abse);
closeto(e,tol):=block([numer:true,abse],abse:abs(e),if(abse<tol) then true else abse);

/* Bug ID: 2997276 - zeta(3),numer; gives Lisp error 
 * 
 * Also add a test for complex rational argument, which wasn't handled
 * correctly either.
 *
 * Some Lisp implementations fail these tests because things like
 * (cl:expt 2d0 3) only gives single-float accuracy (but with
 * double-float precision).
 */
closeto(zeta(3)-1.202056903159594,1e-15), numer:true;
true;
closeto(zeta(3+%i)-(1.10721440843141 - .1482908671781754*%i), 1e-15);
true;
/*
 * Reported on mailing list 2011-05-22 by Thomas Dean:
 *
 * plot2d(abs(zeta(1/2+x*%i)),[x,0,36]) causes a Lisp error with
 * clisp.
 *
 */
closeto(abs(zeta(1/2+.5*%i)) - 1.06534921249378, 1e-14);
true;

/* Bug ID: 2997401 - float(log(200!)) produces an error
 *
 */
closeto(float(log(200!))-863.2319871924054, 1e-15);
true;

closeto(float(log((1+200!)/7))-861.2860770433501, 1e-15);
true;

/* Additional tests */
closeto(float(log(-1))-float(%pi)*%i, 1e-15);
true;

closeto(float(log((1+200!)/(-7))) - (3.141592653589793*%i + 861.2860770433501), 1e-14);
true;

closeto(float(log((1+200!)+(1+199!)*%i))- (.004999958333958322*%i + 863.2319996922491), 1e-15);
true;

closeto(float(log((1+200!)/7+(1+199!)/11*%i)) - (.003181807444342708*%i + 869.9736929490153), 1e-15);
true;

/* Bug ID: 2306402 - scalarp bug
 * Bug ID: 1985748 - array and scalar declarations yield inconsistent results
 * Examples from the bug report to show consistent behavior of scalarp
 */
 
declare(x,scalar);
done;
scalarp(foo(x));
true;
scalarp(foo(1));
true;
scalarp(foo(x,1));
true;

scalarp(x);
true;
scalarp(x[1]);
true;
array(x,5);
x;
scalarp(x);
true;
scalarp(x[1]);
true;
nonscalarp(x);
false;

kill(x);
done;

/* Bug ID:1723548 - gradef for variables: not used in diff
 * Show that the total differential of f works in expressions too.
 */
depends(f,[x,y]);
[f(x,y)];
diff(f);
'diff(f,y,1)*del(y)+'diff(f,x,1)*del(x)$
diff(3*f);
3*'diff(f,y,1)*del(y)+3*'diff(f,x,1)*del(x)$
diff(a*f);
a*'diff(f,y,1)*del(y)+a*'diff(f,x,1)*del(x)+f*del(a)$
remove(f,dependency);
done;

/* Bug ID: 1089719 addrow creates strange matrix
 */
m:matrix([0,0]);
matrix([0,0]);
m:addrow(m,m);
matrix([0,0],[0,0])$
m[1,1]:11;
11;
m;
matrix([11,0],[0,0])$
kill(m);
done;

/* Bug ID: 1663385 - declare multiplicative - wrong simplification
 */
declare(f,additive,f,multiplicative);
done;
f(a*b+c);
f(a)*f(b)+f(c);
kill(f);
done;

/* Bug ID: 816808 - subst(in)part of rat -- internal errs
 */
substpart(x,2/3,2);
2/x;
substinpart(4,2/3,2);
1/2;

/* Bug ID: 1117533 - letsimp complains about assignment to %pi
 */
matchdeclare(a,true);
done;
(let(%pi*a,foo(a)),done);
done;
letsimp(%pi*x);
foo(x);
remlet(%pi*a);
done;

/* Bug ID: 2805600 depends() partially prevents diff() to work
 */
depends(t,x);
[t(x)];
diff(f(t),z);
0;
remove(t,dependency);
done;

/* Bug ID: 1184718 - AT needs soime basic simplifications
 */
'at(2,x=0);
2;

/* Bug ID: 2998227 - spurious at(0,A=0)
 */
taylor(integrate(gamma(x+1),x,0,A),A,0,3),nouns;
A-%gamma*A^2/2+(6*%gamma^2+%pi^2)*A^3/36;

/* Bug ID: 3010829 - numerical evaluation of elliptic_ec fails for argument > 1
 */
closeto(elliptic_ec(2.0)-(.5990701173677959*%i+0.599070117367796), 1.5e-15);
true;

/* Bug ID: 1929287 - 0.0 + [0] ---> [0]
 */
0.0+[0];
[0.0];
0.0b0+[0];
[0.0b0];
0.0+matrix([0,1/2,1,x]);
matrix([0.0,0.5,1.0,x]);
0.0b0+matrix([0,1/2,1,x]);
matrix([0.0b0,5.0b-1,1.0b0,x]);

/* Bug ID: 2996106 - at(diff(f(x,y),x,1,y,1),[x=a,y=b]) is wrong
 */
at(diff(f(x,y),x,1,y,1),[x=a,y=b]);
'at('diff(f(x,y),x,1,y,1),[x = a,y = b]);

/* Bug report ID: 2556133 - "at" should do parallel substitutions
 */
errcatch(at(atan2(y^2+1,x),[y=%i,x=0]));
[];
errcatch(at(atan2(y^2+1,x),[x=0,y=%i]));
[];

/* Bug report ID: 2014941 - compositions of 'at'
 */
at(at(diff(f(x),x),[x=b]),[b=y]);
'at('diff(f(x),x,1),[x = y]);

at(diff(f(x,y),x,1,y,1),[x=a,y=b]) - at(diff(f(x,y),x,1,y,1),[y=b,x=a]);
0;

/* Bug report ID: 1677217 - composistions of 'at'
 */
depends(y,[x,z]);
[y(x,z)];
at(at(diff(y,x),x=a),z=b);
'at('at('diff(y,x,1),x = a),z = b);
remove(y,dependency);
done;

/* Bug report ID: 3023978 - integrate(x^x+x,x) is wrong
 */
integrate(x^x+x,x);
'integrate(exp(x*log(x)),x)+x^2/2;

/* Bug report ID: 2465066 - unsimplified result from integrate
 */
matchdeclare(x, symbolp);
done;
(tellsimpafter('integrate(f(x),x), g(x)),done);
done;
integrate(5*f(x) + 7,x);
5*g(x)+7*x;
kill(rules);
done;

/* Bug report ID: 2789110 - solve, tan and atan depend on order of variables
 */
solve(tan(x - atan(a/b)) = 0, x);
[x = atan(a/b)];
solve(tan(x - atan(b/a)) = 0, x);
[x = atan(b/a)];

/* Bug report ID: 1961494 - translated functions & values list
 */
(kill(all), f():= x:2, translate(f));
[f];
f();
2;
values;
[x];
kill(f,x);
done;
/* The value of x has been removed. */
x;
'x;

/* Bug report ID: 3025038 - gruntz needs logexpand:true
 */
gruntz( (x + 2^x) / 3^x, x, inf),logexpand:false;
0;

/* Bug report ID: 2977217 - maxima can not integrate x*exp(-1/2*(x-m)^2)
 */
integrate(x*exp(-1/2*(x-m)^2),x);
%i*(sqrt(2)*%i*gamma_incomplete(1,(m-x)^2/2)*(m-x)^2/(x-m)^2
   -%i*gamma_incomplete(1/2,(m-x)^2/2)*m*(m-x)/abs(x-m))/sqrt(2);

/* Bug report ID: 2996542 - log(x) integration is incorrect
 */
assume(a>0);
[a>0];
integrate(log(x),x,0,a);
a*log(a)-a;
forget(a>0);
[a>0];

/* Bug report ID: 3062883 - diff does not recognize indirect dependencies 
 *                          in expressions
 */

depends([a,b],x,x,t);
[a(x),b(x),x(t)];
diff(-a,t);
-'diff(a,x,1)*'diff(x,t,1);
diff(a*b,t);
a*'diff(b,x,1)*'diff(x,t,1)+'diff(a,x,1)*b*'diff(x,t,1);
remove([a,b,x],dependency);
done;

/* Bug report ID: 3080397 - laplace(unit_step(-t),t,s) generates an error.
 */
laplace(unit_step(-t),t,s);
0;

/* Bug report ID: 3081820 - lbfgs causes error
 *
 * Still generates an error, but a different error that maxima
 * signals.
 */
block([V:0.75, a:24, b:68, e],
  C(r) := 2*%pi*b*r^2 + 4*a*%pi*r + 2*b*V/r + a*V/(%pi*r^2),
  load(lbfgs),
  /* This should signal an error that we catch */
  e : errcatch(lbfgs(C(r), [r], [1], 1e-4, [1,0])),
  [e, error]);
[[], ["Evaluation of gradient at ~M failed.  Bad initial point?~%", [0.0]]];

/* Bug report ID: 875089 - defint(f(x)=g(x),x,0,1) -> false = false
 *
 * We distribute defint more early in the code of bags to get a correct result.
 */
 defint(f(x)=g(x),x,0,1);
 'integrate(f(x),x,0,1)='integrate(g(x),x,0,1);
 
/* Bug report ID: 2796194 - error doing a Fourier transform */
(assume(equal(x,0)),done);
done;

errcatch(integrate(%pi*exp(-2*%pi*t)*exp(2*%pi*x*t*%i),t,minf,inf));
[];

error;
["defint: integral is divergent."];

(forget(equal(x,0)),done);
done;

/* Bug reported on the mailing list
 * <http://www.math.utexas.edu/pipermail/maxima/2010/023024.html>
 * integrate(cos(2*x)*cos(x),x) is wrong.
 *
 * Add a few more test that are similar to test the part of
 * monstertrig that deals with trig1(m*x)*trig2(n*x) where trig1 and
 * trig2 are either sin or cos.
 */
integrate(cos(2*x)*cos(x),x);
sin(3*x)/6+sin(x)/2;

integrate(sin(2*x)*sin(x),x);
sin(x)/2-sin(3*x)/6;

integrate(cos(2*x)*sin(x),x);
cos(x)/2-cos(3*x)/6;

integrate(sin(x)*cos(2*x),x);
cos(x)/2-cos(3*x)/6;

/* Bug ID: 3111568 - subsequent calls to gradef hide variable dependencies
 */
gradef(f,x,g);
f;
gradef(f,y,h);
f;
dependencies;
[f(y,x)];
kill(f);
done;

/* Bug ID: 3118770 - %edispflag:true causes a bug
 */
%edispflag:true;
true;
integrate(x/(%e)^(2*x), x, 0, 1);
1/4-3/(4*%e^2);
reset(%edispflag);
[%edispflag];

/* Bug ID: 3067098 - The command timer for a Lisp function
 * Check that the function trisplit does not go away, when we collect
 * timing statistics for this function and call later kill(all).
 */
timer(?trisplit);
[?trisplit];
kill(all);
done;
rectform(1+%i);
1+%i;

/* Bug ID: 3133916 - scanmap(minfactorial,a!) infinite loop
 */
scanmap(minfactorial, a!);
a!;

/* Bug ID: 3131324 - simplification of sqrt
 */
sqrt(x^3)/sqrt(x^3);
1;

/* Bug ID: 1285104 - trigsimp and trigreduce & square roots
 */

trigreduce(sqrt(r^2*sin(x)^2+r^2*cos(x)^2));
abs(r);
trigreduce(sqrt(r^2*sin(x)^2+r^2*cos(x)^2)),radexpand:all;
r;
radexpand:false;
false;
trigreduce(sqrt(r^2*sin(x)^2+r^2*cos(x)^2));
sqrt(r^2);
reset(radexpand);
[radexpand];

/* Bug ID: 917283 - Comment syntax confused
 * Show that nested comments work as expected.
 */
a/*/**/*/+b;
a+b;
a/*/**/*/+/*/**/*/b;
a+b;

/* Bug ID: 3138054 -  bfloat problem / FIX -
 */
exp(gamma(1/3)),bfloat;
1.45696199392313b1;

/* Bug ID: 3288989 - Lisp functions and linear display
 * Show that we do not get a Lisp error.
 */
grind(?cdr([a,b,c]));
done;

/* Bug ID: 3291590 - Problems with fast arrays
 */
(a:make_array(hashed), done);
done;
a[100]:100;
100;
a[x]:sin(x);
sin(x);
a[x*y]:x^2+y;
y+x^2;

/* Cutting out these two examples.
 * The ordering of the lists is different depending on the underlying Lisp.
 *
 * arrayinfo(a);
 * [hash_table,1,100,x,x*y];
 * listarray(a);
 * [100,sin(x),y+x^2];
 */

(f:make_array(functional, 'factorial, hashed), done);
done;
f[10];
3628800;

(kill(f), 0);
0$

(a: make_array(fixnum, 2, 2), done);
done;
listarray(a);
[0, 0, 0, 0];

use_fast_arrays:true;
true;

(array(a, any, 2, 2), done);
done;
arrayinfo(a);
[declared, 2, [2, 2]];
(array(a, fixnum, 2, 2), done);
done;
arrayinfo(a);
[declared, 2, [2, 2]];
(array(a, flonum, 2, 2), done);
done;
arrayinfo(a);
[declared, 2, [2, 2]];
(array(a, hashed), done);
done;
arrayinfo(a);
[hash_table, 1];

reset(use_fast_arrays);
[use_fast_arrays];
kill(a);
done;

/* Bug ID: 3247367 - expand returns unsimplified
 */
sqrt(2)+sqrt(2);
2^(3/2);
sqrt(2)+sqrt(2)+sqrt(2);
3*sqrt(2);
sqrt(2)+sqrt(2)+sqrt(2)+sqrt(2);
2^(5/2);
sqrt(2)+sqrt(2)+sqrt(2)+sqrt(2)+sqrt(2);
5*sqrt(2);

2*sqrt(2)+3*sqrt(2);
5*sqrt(2);
3*sqrt(2)+2*sqrt(2);
5*sqrt(2);
3*sqrt(2)+2*sqrt(2)+sqrt(2);
3*2^(3/2);
sqrt(2)+3*sqrt(2)+2*sqrt(2);
3*2^(3/2);

sqrt(1/2)+sqrt(1/2);
sqrt(2);
sqrt(1/2)+sqrt(1/2)+sqrt(1/2);
3/sqrt(2);
sqrt(1/2)+sqrt(1/2)+sqrt(1/2)+sqrt(1/2);
2^(3/2);
sqrt(1/2)+sqrt(1/2)+sqrt(1/2)+sqrt(1/2)+sqrt(1/2);
5/sqrt(2);

2*sqrt(1/2)+3*sqrt(1/2);
5/sqrt(2);
3*sqrt(1/2)+2*sqrt(1/2);
5/sqrt(2);
3*sqrt(1/2)+2*sqrt(1/2)+sqrt(1/2);
3*sqrt(2);
sqrt(1/2)+3*sqrt(1/2)+2*sqrt(1/2);
3*sqrt(2);

2^(1/5)+2^(1/5);
2^(6/5);
2^(1/5)+2^(1/5)+2^(1/5);
3*2^(1/5);
2^(1/5)+2^(1/5)+2^(1/5)+2^(1/5);
2^(11/5);
2^(1/5)+2^(1/5)+2^(1/5)+2^(1/5)+2^(1/5);
5*2^(1/5);

(1/2)^(1/5)+(1/2)^(1/5);
2^(4/5);
(1/2)^(1/5)+(1/2)^(1/5)+(1/2)^(1/5);
3/2^(1/5);
(1/2)^(1/5)+(1/2)^(1/5)+(1/2)^(1/5)+(1/2)^(1/5);
2^(9/5);
(1/2)^(1/5)+(1/2)^(1/5)+(1/2)^(1/5)+(1/2)^(1/5)+(1/2)^(1/5);
5/2^(1/5);

2*(1/2)^(1/5)+3*(1/2)^(1/5);
5/2^(1/5);
3*(1/2)^(1/5)+2*(1/2)^(1/5);
5/2^(1/5);

2^sin(x)+2^sin(x);
2^(sin(x)+1);
2^sin(x)+2^sin(x)+2^sin(x);
3*2^sin(x);
2^sin(x)+2^sin(x)+2^sin(x)+2^sin(x);
2^(sin(x)+2);
2^sin(x)+2^sin(x)+2^sin(x)+2^sin(x)+2^sin(x);
5*2^sin(x);

(1/2)^sin(x)+(1/2)^sin(x);
2^(1-sin(x));
(1/2)^sin(x)+(1/2)^sin(x)+(1/2)^sin(x);
3/2^sin(x);
(1/2)^sin(x)+(1/2)^sin(x)+(1/2)^sin(x)+(1/2)^sin(x);
2^(2-sin(x));
(1/2)^sin(x)+(1/2)^sin(x)+(1/2)^sin(x)+(1/2)^sin(x)+(1/2)^sin(x);
5/2^sin(x);

(1-sqrt(5))^3-4*(1-sqrt(5))^2+8, expand;
0;
1/sqrt(2)+1/sqrt(2)+1/sqrt(2);
3/sqrt(2);
2^(9/5)+2^(4/5);
3*2^(4/5);
3*sqrt(2)+2*sqrt(2);
5*sqrt(2);
2*sqrt(2)+3*sqrt(2);
5*sqrt(2);
(1-sqrt(5))^3, expand;
16-8*sqrt(5);
p : z^3-2^(3/2)*%i*z^2-4*z^2+2^(5/2)*%i*z+2*z;
z^3-2^(3/2)*%i*z^2-4*z^2+2^(5/2)*%i*z+2*z;
divide(p, (z-2-sqrt(2)*%i),z);
[z^2+(-sqrt(2)*%i-2)*z,0];
2^a + 3*2^(a+1);
7*2^a;
2^(3/5)+2^(-2/5);
3/2^(2/5);
2^(3/5+x)+2^(-2/5+x);
3*2^(x-2/5);

/* -----------------------------------------------------------------------------
 * Bug ID: 1439566 - zerobern & bernpoly
 * Show that the option variable zerobern does not change the results.
 * -------------------------------------------------------------------------- */
zerobern:false;
false;
bernpoly(x,3);
x^3-3*x^2/2+x/2$
bernpoly(x,5);
x^5-5*x^4/2+5*x^3/3-x/6$
reset(zerobern);
[zerobern];

/* Show that bern no longer fails with zerobern:false.
 *
 * The compared values are from A&S p810, Table 23.2.
 */
bern(28);
-23749461029/870$
bern(40);
-261082718496449122051/13530$
euler(16);
19391512145$

zerobern:false;
false$
bern(17);
-7709321041217/510$
bern(24);
596451111593912163277961/282$
euler(10);
370371188237525$

reset(zerobern);
[zerobern]$

/* -----------------------------------------------------------------------------
 * Bug ID: 2905929 - gcdex
 * -------------------------------------------------------------------------- */
q0[2] : 6;
6$
ratdisrep(gcdex(x-7, x-8));
[1,-1,1]$

is(equal(gcdex(z^2-1, 0, z), [1,0,z^2-1]));
true;

is(equal(gcdex(0, z^2-1, z), [0, 1, z^2-1]));
true;

/* Examples from the Maxima Manual */
is(equal(gcdex(x^2+1, x^3+4), [-(x^2+4*x-1)/17,(x+4)/17,1]));
true$
is(equal(gcdex(x*(y+1), y^2-1, x), [0,1/(y^2-1),1]));
true$

kill(q0);
done;

/* -----------------------------------------------------------------------------
 * Bug ID: 3389830 - Error break in rtest15 with linear display
 *
 * Show that we do not get an error with grind for a prefix and a postfix
 * Operator when displaying the definition of a function or a macro.
 * This is a test of the function msz-mdef in grind.lisp.
 * -------------------------------------------------------------------------- */
(postfix("f"), prefix("g"), done);
done$
grind("f"(x):=x);
done$
grind("f"(x)::=x);
done$
grind("g"(x):=x);
done$
grind("g"(x)::=x);
done$

kill ("f", "g");
done;

/* -----------------------------------------------------------------------------
 * Bug ID: 3396631 - equal terms produce different results
 * Correcting a bug in plusin revision 23.08.2011
 * -------------------------------------------------------------------------- */
5*sqrt(5)+2*sqrt(3)+6*sqrt(5);
11*sqrt(5)+2*sqrt(3)$
5*sqrt(5)+4*sqrt(3)+6*sqrt(5);
11*sqrt(5)+4*sqrt(3)$
5*sqrt(5)+3*sqrt(5)+5*sqrt(3)+3*sqrt(3)+2*sqrt(75)+2*sqrt(45);
14*sqrt(5)+2*3^(5/2)$
5*sqrt(5)+5*sqrt(3)+3*sqrt(3)+2*sqrt(75)+2*sqrt(45)+3*sqrt(5);
14*sqrt(5)+2*3^(5/2)$
5*sqrt(5)+5*sqrt(3)+2*sqrt(75)+2*sqrt(45)+3*sqrt(5)+3*sqrt(3);
14*sqrt(5)+2*3^(5/2)$

/* -----------------------------------------------------------------------------
 * Bug ID 3437268: expand doesn't fully expand
 * Check the expected results after revision 14.11.2011 of simp.lisp.
 * -------------------------------------------------------------------------- */
-3*a/sqrt(2)+sqrt(2)*a+a/sqrt(2);
0$
expand(-(atan((sqrt(2)*sin(8))/cos(8))+3*%pi)/sqrt(2)
        +atan((sqrt(2)*sin(8))/cos(8))/sqrt(2)
        +sqrt(2)*%pi
        +%pi/sqrt(2));
0$

/*
 * SF Bug: elliptic_e error observed - ID: 3438911
 *
 * Test the bug by using the identity:
 * elliptic_e(%pi,m) = 2*elliptic_ec(m)
 *
 * There's also a bug in elliptic_f.  We test this using the identity:
 * elliptic_f(%pi,m) = 2*elliptic_kc(m)
 */

closeto(e,tol):=block([numer:true,abse],abse:abs(e),if(abse<tol) then true else abse);
closeto(e,tol):=block([numer:true,abse],abse:abs(e),if(abse<tol) then true else abse);

closeto(elliptic_e(float(%pi), .1) - 2*elliptic_ec(.1), 1e-15);
true;
closeto(elliptic_e(float(%pi), .9) - 2*elliptic_ec(.9), 1e-15);
true;
closeto(elliptic_e(bfloat(%pi), .1b0) - 2*elliptic_ec(.1b0), 1e-16);
true;
closeto(elliptic_e(bfloat(%pi), .9b0) - 2*elliptic_ec(.9b0), 1e-16);
true;

closeto(elliptic_f(float(%pi), .1) - 2*elliptic_kc(.1), 1e-15);
true;
closeto(elliptic_f(float(%pi), .9) - 2*elliptic_kc(.9), 1e-15);
true;
closeto(elliptic_f(bfloat(%pi), .1b0) - 2*elliptic_kc(.1b0), 1e-16);
true;
closeto(elliptic_f(bfloat(%pi), .9b0) - 2*elliptic_kc(.9b0), 1e-16);
true;

/*
 * Bug 3526111 - float erf (%i) not working
 */

closeto(float(erf(%i)) - 1.650425758797543*%i, 1e-15);
true;

/*
 * Bug 3529992: Shi (sinh integral) wrong branch, integrate inconsistent
 */
closeto(float(expintegral_shi(1/2) - 0.50699674981966719583), 3e-16);
true;

/* integrate changes k[0] --> k(0) - ID: 3530767 */
integrate(x * (x^2 + k[0])/(1 + x^2),x);
((k[0]-1)*log(x^2+1))/2+x^2/2$

/* polarform error on simple case - ID: 3517034 */
polarform((a+1)/2 - a/2 - 1/2);
0$

polarform((a+1)/2 - a/2 - 0.5);
0.0$

polarform((a+1)/2 - a/2 - 0.5b0);
0.0b0$


/* #2531 Integration with inf */
errcatch(integrate((1+1/x)^(1/2),x,1,inf));
[];

error;
["defint: integral is divergent."];



/*
 * ID: 3440046: elliptic_f(0.5,1) signals error
 *
 * Add a few more tests for invalid values.
 */
closeto(elliptic_f(0.5,1)-elliptic_f(1/2,1), 1e-15);
true;

closeto(elliptic_f(0.5b0,1) - bfloat(elliptic_f(1/2,1)), 1b-16);
true;

errcatch(elliptic_f(2.0,1));
[];

errcatch(elliptic_f(2b0,1));
[];

/*
 * Bug 3428734:  integrate(bessel_y(1,z),z)  with ?z : 1
 */

(?z:1, 0);
0;

integrate(bessel_y(1,z),z);
-bessel_y(0,z);

integrate(bessel_j(1,z),z);
-bessel_j(0,z);

integrate(bessel_k(1,z),z);
-bessel_k(0,z);

integrate(bessel_i(1,z),z);
bessel_i(0,z);

/*
 * Bug 3381301: log(-1.0b0) has small realpart
 */
realpart(log(-1b0));
0;

/*
 * Bug 3559064: elliptic_f(2,1) is wrong.
 */
errcatch(elliptic_f(2,1));
[];

/*
 * Bug 2528: A variable should be real if it is both real and complex
 */
(declare(foo, real), declare(foo, complex), 0);
0$

[realpart(foo), imagpart(foo)];
[foo, 0]$

(kill(foo), 0);
0$

map(lambda([x], featurep(x, 'irrational)),[42,%pi,%phi,%e]);
[false, true, true, true]$

/* #2501 %pi/8 is definitely not an integer */
integrate(log(cot(x)-1),x,0,%pi/4);
(%i*li[2]((%i+1)/2)-%i*li[2](-((%i-1)/2)))/2 -(%i*(2*li[2](%i+1)-2*li[2](1-%i))+%pi*log(2))/4$

integrate(log(cos(x)),x,0,%pi/2);
%i*%pi^2/24-(6*%pi*log(4)+%i*%pi^2)/24$

map(lambda([x], featurep(x, noninteger)),[sqrt(5),%pi,%pi/3, %pi/8,log(42),99/2013]);
[true,true,true,true,true,true]$

map(lambda([x], featurep(x, noninteger)),[0,1,2013]);
[false,false,false]$

/* #2583 sign error for integrate(x^(8*%i-1),x) */
block([domain : 'real], integrate(x^(8*%i-1),x));
-%i*x^(8*%i)/8$

/* #2602: some-bfloatp and some-floatp recursed wrongly on rat expressions */
?some\-bfloatp(rat(1/2));
false$

/* #2594: Error in trigreduce for complicated expressions */
subst(0, x, trigreduce(product(cos(k*x), k, 1, 8)));
1$

/* SF bug #2818: Problem with trigreduce */
trigreduce(sin(1/8*%pi)*sin(3/8*%pi)*sin(5/8*%pi)*sin(7/8*%pi));
1/8;

/* #2591: Risch gives lisp error */
(risch(asinh((z^2-1)/z)/z,z), 0);
0$

/* #2682: Function zeta fails numerically for large numbers */
closeto(zeta(40.0) - 1, 1e-12);
true$

/* #2675 (1/3): Integration yields noun form with subscripted variable */
integrate(exp(-(1+%i)*x[1]),x[1],0,inf);
1/2-%i/2$

/* #2688: %e^^A returns element by element exponent */
is(%e^^matrix([1,2],[3,4]) = %e^matrix([1,2],[3,4]));
false$

/* #2676: Integral incorrect when variable is subscripted */
integrate(x[1]*exp(x[1]), x[1]);
exp(x[1])*(x[1]-1)$

/* #2726: Integrate produces wrong answer for Gaussian Moments */
(declare(m2726, even),
 block([tmp: integrate(exp(-x^2/2)/sqrt(2*%pi) * x^m2726, x, -1/4, 1/4)],
   sign (subst(m2726 = 4, tmp))));
pos$

/* # 2697: Inconsistent handling of Greek symbols */
integrate(y(%theta)=sin(%theta),%theta,%theta[0], %theta[1]);
'integrate(y(%theta),%theta,%theta[0],%theta[1]) = cos(%theta[0])-cos(%theta[1])$

integrate(y(t)=sin(t),t,t[0], t[1]);
'integrate(y(t),t,t[0],t[1]) = cos(t[0])-cos(t[1])$

integrate(y(tau)=sin(tau),tau,tau[0], tau[1]);
'integrate(y(tau),tau,tau[0],tau[1]) = cos(tau[0])-cos(tau[1])$

integrate ([foo(x), bar(x)], x, x[1], x[2]);
['integrate (foo(x), x, x[1], x[2]), 'integrate (bar(x), x, x[1], x[2])];

/* # 2738: Integrate encountered a Lisp error: The value 2 is not of type LIST. */

(kill (x, y, I, J),
 x(t):=2*cos(t),
 y(t):=2*sin(t),
 I : (x(t)+y(t)^2)*sqrt(diff(x(t),t)^2+diff(y(t),t)^2),
 J : integrate (I, t));
4*(t-tan(t)/(tan(t)^2+1))+4*sin(t)$

trigsimp (diff (J, t) - I);
0;
/* bug #2980: Infinite recursion with (e: log(e), rectform(e)) */
block ([e: log(e)], rectform(e));
log(abs(e)) + %i*atan2(0, e)$

/* bug #2159: integration_with_logabs ("integrate(tan(x),x);" etc. do not take "logabs" flag into account) */
integrate([tan(x),csc(x),sec(x),cot(x),tanh(x),coth(x),csch(x)],x);
[log(sec(x)),-log(csc(x)+cot(x)),log(sec(x)+tan(x)),log(sin(x)),log(cosh(x)),log(sinh(x)),log(tanh(x/2))]$
integrate([tan(x),csc(x),sec(x),cot(x),tanh(x),coth(x),csch(x)],x),logabs;
[log(abs(sec(x))),-log(abs(csc(x)+cot(x))),log(abs(sec(x)+tan(x))),log(abs(sin(x))),log(cosh(x)),log(abs(sinh(x))),log(abs(tanh(x/2)))]$

/* Bug #3075: #3075 answer "3*false" from "integrate(3*asinh(x),x,-inf,inf)" */
/* We can't check for the correct answer zero (yet), because Maxima can't solve integrate(asinh(x),x,-inf,inf). */
is(integrate(3*asinh(x),x,-inf,inf)#3*false);
true$

/*
 * Bug #3056: exp(1b19) signals error that 1b19 doesn't have enough
 * precision to compute its integer part.  Add test for this and also
 * the original test from commit 576c7508.)
 */

/* Don't really care what the answer is as long as we don't signal an error */
is(errcatch(exp(1b19)) # []);
true;

/* Test from the commit 576c7508 */
ceiling((207300647060*%e^-563501581931)/(403978495031*%e^-1098127402131));
ceiling((207300647060*%e^534625820200)/403978495031);

/* Bug #3098: numerical evaluation of li[3] */
li[3](0.0);
0.0;

closeto(li[3](1.0) - zeta(3.0), 1e-15);
true;

closeto(li[3](0.5) - li[3](1/2), 1e-15);
true;

closeto(li[3](-2.0) - float(subst(z=-2.0, li[3](1/z) - log(-z)/6*(log(-z)^2+%pi^2))), 1e-15);
true;

closeto(abs(li[3](2.0) - expand(float(subst(z=2.0, li[3](1/z) - log(-z)/6*(log(-z)^2+%pi^2))))), 1e-15);
true;

li[2](-1);
-%pi^2/12;

li[2](1);
%pi^2/6;

li[2](1/2);
%pi^2/12 - log(2)^2/2;

closeto(li[2](0.5) - (%pi^2/12 - log(2)^2/2), 1.1103e-16), numer;
true;

closeto(li[2](2.0) - (%pi^2/4 - %i*%pi*log(2)), 2.513e-15), numer;
true;

closeto(li[2]((1-sqrt(5))/2) - (log((sqrt(5)-1)/2)^2/2-%pi^2/15), 1.1103e-16), numer;
true;

/* Catalan's constant: 0.915965594... */
closeto(li[2](1.0*%i) - (-%pi^2/48 + %i*0.915965594177219015054603514932384110774149374281672134266), 1.3878e-15), numer;
true;

closeto(li[2](1.0-%i) - (%pi^2/16-%i*0.915965594177219015054603514932384110774149374281672134266 - %pi*%i*log(2)/4), 4.5e-16), numer;
true;

/* Make sure li[3](1/7),numer returns a float and not a bfloat */
?floatp(li[3](float(1/7)));
true;

?floatp(realpart(li[2](1.0+1.0*%i)));
true;

closeto(li[3](0.5) - float((7*zeta(3))/8+log(2)^3/6-(%pi^2*log(2))/12), 1.1103e-16);
true;

closeto(float(li[3](exp(%pi*%i/3))) - float(zeta(3)/3 + %i*5*%pi^3/162), 5.6611e-16);
true;

/*
 * li[3] should not return a small imaginary part for this value.
 * (There are others, but the solution fixes those as well.
 */
imagpart(li[3](-.862));
0;

/*
 * Likewise li[s](-1.5) was returning small imaginary parts.  In fact,
 * this was true for -2 < z < -1 because the log series was used to
 * compute the value.  Check that we return exactly zero now for
 * select values of s
 */

makelist(imagpart(li[s](-1.5)), s, 4, 10);
[0, 0, 0, 0, 0, 0, 0];  

/* Bug 3582: numerical evaluation of li[2] */
/* this gave an infinite loop */
closeto(li[2]( 0.5 + %i* 0.6) - ( 0.7535498331267791 * %i + 0.4081870726952078 ), 1e-15);
true;

/* Bug 3112: zeta(n) for negative even n is inaccurate */
zeta(-4.0);
0.0;

zeta(-6b0);
0b0;

/* Bug 3105: li[s](1.0) doesn't simplify */
closeto(li[4](1.0)-li[4](1), 1.11022e-16);
true;

closeto(li[4](-1.0)-li[4](-1), 1.11022e-16);
true;

closeto(li[4](1b0)-li[4](1), 10^-fpprec);
true;

closeto(li[4](-1b0)-li[4](-1), 10^-fpprec);
true;


/*
 * More numeric test for polylog function li[s](z).
 *
 * The reference values were obtained from
 * http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=PolyLog
 */
fpprec:24;
24;

closeto(li[4](0.25) - 0.25411619074634353405967371315352,
  5.5512e-17);
true;

closeto(li[4](0.25b0) - 0.25411619074634353405967371315352b0,
  2.06796b-25);
true;

closeto(li[4](0.75) - 0.79222102797282777952948578955736,
  2.2205e-16);
true;

closeto(li[4](0.75b0) - 0.79222102797282777952948578955736b0,
  6.20386b-25);
true;

closeto(li[4](1.5) - (1.7347570807760620737768805117515 - 0.0349027048283367002627421237287*%i),
  2.2205e-16);
true;

closeto(li[4](1.5b0) - (1.7347570807760620737768805117515b0 - 0.0349027048283367002627421237287b0*%i),
  4.14398b-25);
true;

closeto(li[4](10.0) - (9.6140263862742968515251940747859 - 6.3921313179656069159740055708257*%i),
  5.618e-15);
true;

closeto(li[4](10.0b0) - (9.6140263862742968515251940747859b0 - 6.3921313179656069159740055708257b0*%i),
  1.9364b-23);
true;

closeto(li[4](-5.0) - (-4.1064679790949702621073505378164),
  8.8818e-16);
true;

closeto(li[4](-5b0) - (-4.1064679790949702621073505378164b0),
  4.9631b-24);
true;

closeto(li[4](2.0*%i) - (-0.2113943747614829764326347997923 + 1.9289340331586646502356787803360*%i),
  7.22e-16);
true;

closeto(li[4](2b0*%i) - (-0.2113943747614829764326347997923b0 + 1.9289340331586646502356787803360b0*%i),
  1.9788b-24);
true;

closeto(li[5](0.25) - 0.25202158817857420100669519623555,
  5.55112e-17);
true;

closeto(li[5](0.25b0) - 0.25202158817857420100669519623555b0,
  5.16988b-25);
true;

closeto(li[5](0.75) - 0.76973541059975738097269173152535,
  1.111e-16);
true;

closeto(li[5](0.75b0) - 0.76973541059975738097269173152535b0,
  2.06796b-25);
true;

closeto(li[5](1.5) - (1.5961739456813534102689069143338 - 0.0035379572466222227823786042761*%i),
  4.33681e-19);
true;

closeto(li[5](1.5b0) - (1.5961739456813534102689069143338b0 - 0.0035379572466222227823786042761b0*%i),
  4.13594b-25);
true;

closeto(li[5](10.0) - (11.2390407376112991620107110964539 - 3.6796065713019972004384472107791*%i),
  9.93014e-15);
true;

closeto(li[5](10.0b0) - (11.2390407376112991620107110964539b0 - 3.6796065713019972004384472107791b0*%i),
  2.18382b-23);
true;

closeto(li[5](-5.0) - (-4.4800824065112010228046981292686),
  1e-16);
true;

closeto(li[5](-5b0) - (-4.4800824065112010228046981292686b0),
  6.61745b-24);
true;

closeto(li[5](2.0*%i) - (-0.1139660114783041974283299769126 + 1.9734617121122917650272273558071*%i),
  6.98e-16);
true;

closeto(li[5](2b0*%i) - (-0.1139660114783041974283299769126b0 + 1.9734617121122917650272273558071b0*%i),
  6.42086b-25);
true;

closeto(li[20](5.0) - (5.0000238783147176199891129814336 - 2.182054400942151422e-13*%i),
  2.41797e-15);
true;

closeto(li[20](5b0) - (5.0000238783147176199891129814336b0 - 2.182054400942151422b-13*%i),
  6.76366b-24);
true;

/*
 * Bug 3966: li[s](1) = zeta(s)
 */
makelist(li[s+1/2](1), s, 1, 5);
[zeta(3/2), zeta(5/2), zeta(7/2), zeta(9/2), zeta(11/2)];

li[7/3](1);
zeta(7/3);

li[-5/2](1);
li[-5/2](1);

li[-5/2](1.0);
li[-5/2](1.0);

/* Bug #3194: No simplification of "tan(x+n*%pi)" and "cot(x+n*%pi)" with "n" being a declared integer */
/* Also test that the "modulus 1/2" check in "%piargs-tan/cot" works correctly. */
declare(i, integer, ei, even, oi, odd);
done$
[tan(x+i*%pi),cot(x+i*%pi)];
[tan(x),cot(x)]$
[tan(x+ei*%pi),cot(x+ei*%pi)];
[tan(x),cot(x)]$
[tan(x+oi*%pi),cot(x+oi*%pi)];
[tan(x),cot(x)]$
[tan(x-3/2*%pi),tan(x-1/2*%pi),tan(x+1/2*%pi),tan(x+3/2*%pi)];
[-cot(x),-cot(x),-cot(x),-cot(x)]$
[cot(x-3/2*%pi),cot(x-1/2*%pi),cot(x+1/2*%pi),cot(x+3/2*%pi)];
[-tan(x),-tan(x),-tan(x),-tan(x)]$
kill(i, ei, oi);
done$

/* Bug #3148: sign can't figure out sign(a - b) but it knows sign(b - a) where a and b are exponentials */
assume(t>0);
[t>0]$
sign(2^(500000*t)-2^(500007*t));
neg$
sign(2^(500007*t)-2^(500000*t));
pos$
forget(t>0);
[t>0]$

/* Bug #3246: Integrating u'(x) * f(u(x) + c) fails for f any inverse trigonometric/hyperbolic function */
/* Also make sure that all the antiderivatives of (inverse) trigonometric/hyperbolic functions are correct. */
integrate(map(lambda([f], diff(u(x), x) * f(u(x) + c)), [asin,acos,atan,acsc,asec,acot,asinh,acosh,atanh,acsch,asech,acoth]), x);
[(u(x)+c)*asin(u(x)+c)+sqrt(1-(u(x)+c)^2),(u(x)+c)*acos(u(x)+c)-sqrt(1-(u(x)+c)^2),(u(x)+c)*atan(u(x)+c)-log((u(x)+c)^2+1)/2,log(sqrt(1-1/(u(x)+c)^2)+1)/2-log(1-sqrt(1-1/(u(x)+c)^2))/2+(u(x)+c)*acsc(u(x)+c),-log(sqrt(1-1/(u(x)+c)^2)+1)/2+log(1-sqrt(1-1/(u(x)+c)^2))/2+(u(x)+c)*asec(u(x)+c),log((u(x)+c)^2+1)/2+(u(x)+c)*acot(u(x)+c),(u(x)+c)*asinh(u(x)+c)-sqrt((u(x)+c)^2+1),(u(x)+c)*acosh(u(x)+c)-sqrt((u(x)+c)^2-1),log(1-(u(x)+c)^2)/2+(u(x)+c)*atanh(u(x)+c),log(sqrt(1/(u(x)+c)^2+1)+1)/2-log(sqrt(1/(u(x)+c)^2+1)-1)/2+(u(x)+c)*acsch(u(x)+c),(u(x)+c)*asech(u(x)+c)-atan(sqrt(1/(u(x)+c)^2-1)),log(1-(u(x)+c)^2)/2+(u(x)+c)*acoth(u(x)+c)]$
radcan(exponentialize(map(lambda([f], f(x) - diff(integrate(f(x), x), x)), [sin,cos,tan,csc,sec,cot,sinh,cosh,tanh,csch,sech,coth,asin,acos,atan,acsc,asec,acot,asinh,acosh,atanh,acsch,asech,acoth]))), domain:complex;
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$

/* Bug #3258: integrate's answer contains diff operator with negative order */
freeof(diff, integrate(%e^(c*x-r^2/(4*x))/x^(3/2),x));
true$

/* Bug #3220: create_list doesn't bind variables properly */
errcatch (create_list (%i, %i, [1, 2, 3]));
[]$

create_list (bfloat (1 - 10^-50) - 1, fpprec, [16, 100]);
[0.0b0, -1.0b-50]$

create_list (bfloat (3.14), fpprec, 1, 3);
[3.0b0, 3.1b0, 3.14b0]$

/* Bug #3356: sign(nz * nz) = nz */
assume(apos > 0, aneg < 0, apz >= 0, anz <= 0, notequal(apn, 0), bpos > 0, bneg < 0, bpz >= 0, bnz <= 0, notequal(bpn, 0));
[apos > 0, aneg < 0, apz >= 0, anz <= 0, notequal(apn, 0), bpos > 0, bneg < 0, bpz >= 0, bnz <= 0, notequal(bpn, 0)]$
map('sign,
        [
                apos * bpos, apos * bneg, apos * bpz, apos * bnz, apos * bpn, apos * bpnz,
                aneg * bpos, aneg * bneg, aneg * bpz, aneg * bnz, aneg * bpn, aneg * bpnz,
                apz * bpos, apz * bneg, apz * bpz, apz * bnz, apz * bpn, apz * bpnz,
                anz * bpos, anz * bneg, anz * bpz, anz * bnz, anz * bpn, anz * bpnz,
                apn * bpos, apn * bneg, apn * bpz, apn * bnz, apn * bpn, apn * bpnz,
                apnz * bpos, apnz * bneg, apnz * bpz, apnz * bnz, apnz * bpn, apnz * bpnz
        ]
);
[
        'pos, 'neg, 'pz, 'nz, 'pn, 'pnz,
        'neg, 'pos, 'nz, 'pz, 'pn, 'pnz,
        'pz, 'nz, 'pz, 'nz, 'pnz, 'pnz,
        'nz, 'pz, 'nz, 'pz, 'pnz, 'pnz,
        'pn, 'pn, 'pnz, 'pnz, 'pn, 'pnz,
        'pnz, 'pnz, 'pnz, 'pnz, 'pnz, 'pnz
];
forget(apos > 0, aneg < 0, apz >= 0, anz <= 0, notequal(apn, 0), bpos > 0, bneg < 0, bpz >= 0, bnz <= 0, notequal(bpn, 0));
[apos > 0, aneg < 0, apz >= 0, anz <= 0, notequal(apn, 0), bpos > 0, bneg < 0, bpz >= 0, bnz <= 0, notequal(bpn, 0)]$

/* test cases mentioned in bug report #3356 */

(kill (a, b), assume(a<=0, b<=0), sign(a*b));
pz;

(kill (q, r), assume(q<0, r<0), sign(q*r));
pos;

sign(a*r);
pz;

sign(a*-2);
pz;

sign(a*q*r);
nz;

sign(a*b*q);
nz;

(forget (a <= 0, b <= 0, q < 0, r < 0), 0);
0;

/* Bug #3403: Function/lambda parameters declared constant cause error */
declare(c, constant);
done$
emptyp(errcatch(f(c) := c));
false$
emptyp(errcatch(lambda([c], c)));
false$
emptyp(errcatch(f(%e) := %e));
true$
emptyp(errcatch(lambda([%e], %e)));
true$
kill(c, f);
done$

/*        Bug #3009: factoring exponentials causes MAKE-ARRAY error */
/* a.k.a. Bug #3146: max() runs out of memory when comparing exponential functions */
sign(((-1)-%e^-(4.075321792706671E-4*d_1))*%e^-(4.075321792706671E-4*d_2)+%e^-(8.150643585413342E-4*d_2)-%e^-(4.075321792706671E-4*d_1)+%e^-(8.150643585413342E-4*d_1)+1);
pnz$
factor(a^1000000+a+1);
a^1000000+a+1$
max(2.0325-6.9825*%e^-(492380.0*t),2.103-7.053*%e^-(406810.0*t));
'max(2.0325-6.9825*%e^-(492380.0*t),2.103-7.053*%e^-(406810.0*t))$
sign(exp(-492380*t)-2793);
pnz$

/* Bug #3147: sign of expressions with exponents crashes */
assume(t>0);
[t>0]$
sign(2^(500005*t)-2^(500001*t));
pos$
kill(t);
done$

/* Test factor_max_degree. */
block([factor_max_degree : 0], [factor(x^2+2*x+1), factor(x^2-1), factor(x^2+3*x+2), factor(x^2000+x^1999)]);
[(x+1)^2, (x-1)*(x+1), (x+1)*(x+2), x^1999*(x+1)]$
block([factor_max_degree : 1, factor_max_degree_print_warning : false], [factor(x^2+2*x+1), factor(x^2-1), factor(x^2+3*x+2), factor(x^2000+x^1999)]);
[(x+1)^2, x^2-1, x^2+3*x+2, x^1999*(x+1)]$
block([factor_max_degree : 2], [factor(x^2+2*x+1), factor(x^2-1), factor(x^2+3*x+2), factor(x^2000+x^1999)]);
[(x+1)^2, (x-1)*(x+1), (x+1)*(x+2), x^1999*(x+1)]$

/*        Bug #2928: Loops forever: ev(ratexpand(1/(x^(2/3)+1)), algebraic:true); */
/* a.k.a. Bug #2994: Infinite loop: ratsimp(x/(sqrt(x^(2/3)+1)),x),algebraic; */
/* a.k.a. Bug #3419: Endless loop in BPROG (rat(1/(x^(2/3)+1)), algebraic) */

/* examples present in previous version of rtest16 */

string (ev (rat (1/(x^(1/3)+1)), algebraic));
"((x^(1/3))^2-x^(1/3)+1)/(x+1)";

string (ev (rat (1/(x^(2/3)+1)), algebraic));
"(x^(1/3)*x-(x^(1/3))^2+1)/(x^2+1)";

string (ev (rat(1/((x-1)^(2/3)+1)), algebraic));
"((x-1)^(1/3)*x-((x-1)^(1/3))^2-(x-1)^(1/3)+1)/(x^2-2*x+2)";

string (ev (rat(x/(sqrt(x^(2/3)+1))), algebraic));
"(sqrt(x^(2/3)+1)*x^(1/3)*x^2+((-sqrt(x^(2/3)+1)*(x^(1/3))^2)+sqrt(x^(2/3)+1))*x)/(x^2+1)";

/* additional examples courtesy of M. Talon */

string (ev (rat(1/(x^(2/5)+1)), algebraic));
"(((x^(1/5))^3-x^(1/5))*x+(x^(1/5))^4-(x^(1/5))^2+1)/(x^2+1)";

/* Maxima cannot yet handle this next one (incomplete factorization, gets one factor but misses the other)
 *
rat(1/(x^(3/5)+x^(1/3))),algebraic;
 */

string (ev (rat(1/(x^(4/5)+1)), algebraic));
"(x^(1/5)*x^3-(x^(1/5))^2*x^2+(x^(1/5))^3*x-(x^(1/5))^4+1)/(x^4+1)";

/* Bug #3431: error system variable holds unsimplified list, causing errors to be repeated when trying to access it */
errcatch(0^0);
[]$
errcatch(print(error), true);
[true]$

/* SF bug #3458: "does not interpret addcol as it should but instead interprets same as addrow" */

addcol (matrix (), [11, 22, 33]);
matrix ([11], [22], [33]);

/* additional tests for addcol and addrow */

addcol (matrix (), [11, 22, 33], [111, 222, 333]);
matrix ([11, 111], [22, 222], [33, 333]);

addcol (matrix (), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([11, 111, 1111], [22, 222, 2222], [33, 333, 3333]);

addcol (matrix ([1, 2, 3]), [11, 22, 33]);
matrix ([1, 2, 3, 11, 22, 33]);

addcol (matrix ([1, 2, 3]), [11, 22, 33], [111, 222, 333]);
matrix ([1, 2, 3, 11, 22, 33, 111, 222, 333]);

addcol (matrix ([1, 2, 3]), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([1, 2, 3, 11, 22, 33, 111, 222, 333, 1111, 2222, 3333]);

addcol (matrix ([1], [2], [3]), [11, 22, 33]);
matrix ([1, 11], [2, 22], [3, 33]);

addcol (matrix ([1], [2], [3]), [11, 22, 33], [111, 222, 333]);
matrix ([1, 11, 111], [2, 22, 222], [3, 33, 333]);

addcol (matrix ([1], [2], [3]), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([1, 11, 111, 1111], [2, 22, 222, 2222], [3, 33, 333, 3333]);

addrow (matrix (), [11, 22, 33]);
matrix ([11, 22, 33]);

addrow (matrix (), [11, 22, 33], [111, 222, 333]);
matrix ([11, 22, 33], [111, 222, 333]);

addrow (matrix (), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);

addrow (matrix ([1, 2, 3]), [11, 22, 33]);
matrix ([1, 2, 3], [11, 22, 33]);

addrow (matrix ([1, 2, 3]), [11, 22, 33], [111, 222, 333]);
matrix ([1, 2, 3], [11, 22, 33], [111, 222, 333]);

addrow (matrix ([1, 2, 3]), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([1, 2, 3], [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);

/*
 * Seems like these should succeed, but they fail
 * with error about "incompatible structure", although similar calls succeed.
 * Just omit these tests unless addrow/addcol are ever revised for greater consistency.
 *
addrow (matrix ([1], [2], [3]), [11, 22, 33]);
matrix ([1], [2], [3], [11], [22], [33]);

addrow (matrix ([1], [2], [3]), [11, 22, 33], [111, 222, 333]);
matrix ([1], [2], [3], [11], [22], [33], [111], [222], [333]);

addrow (matrix ([1], [2], [3]), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([1], [2], [3], [11], [22], [33], [111], [222], [333], [1111], [2222], [3333]);
 *
 */

addcol (matrix (), matrix ([11], [22], [33]));
matrix ([11], [22], [33]);

addcol (matrix (), matrix ([11, 22, 33]));
matrix ([11, 22, 33]);

addcol (matrix (), matrix ([11], [22], [33]), matrix ([111], [222], [333]));
matrix ([11, 111], [22, 222], [33, 333]);

addcol (matrix (), matrix ([11, 22, 33]), matrix ([111, 222, 333]));
matrix ([11, 22, 33, 111, 222, 333]);

addcol (matrix (), matrix ([11], [22], [33]), matrix ([111], [222], [333]), matrix ([1111], [2222], [3333]));
matrix ([11, 111, 1111], [22, 222, 2222], [33, 333, 3333]);

addcol (matrix (), matrix ([11, 22, 33]), matrix ([111, 222, 333]), matrix ([1111, 2222, 3333]));
matrix ([11, 22, 33, 111, 222, 333, 1111, 2222, 3333]);

addcol (matrix ([1, 2, 3]), matrix ([11, 22, 33]));
matrix ([1, 2, 3, 11, 22, 33]);

addcol (matrix ([1, 2, 3]), matrix ([11, 22, 33]), matrix ([111, 222, 333]));
matrix ([1, 2, 3, 11, 22, 33, 111, 222, 333]);

addcol (matrix ([1, 2, 3]), matrix ([11, 22, 33]), matrix ([111, 222, 333]), matrix ([1111, 2222, 3333]));
matrix ([1, 2, 3, 11, 22, 33, 111, 222, 333, 1111, 2222, 3333]);

addrow (matrix (), matrix ([11], [22], [33]));
matrix ([11], [22], [33]);

addrow (matrix (), matrix ([11, 22, 33]));
matrix ([11, 22, 33]);

addrow (matrix (), matrix ([11], [22], [33]), matrix ([111], [222], [333]));
matrix ([11], [22], [33], [111], [222], [333]);

addrow (matrix (), matrix ([11, 22, 33]), matrix ([111, 222, 333]));
matrix ([11, 22, 33], [111, 222, 333]);

addrow (matrix (), matrix ([11], [22], [33]), matrix ([111], [222], [333]), matrix ([1111], [2222], [3333]));
matrix ([11], [22], [33], [111], [222], [333], [1111], [2222], [3333]);

addrow (matrix (), matrix ([11, 22, 33]), matrix ([111, 222, 333]), matrix ([1111, 2222, 3333]));
matrix ([11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);

addrow (matrix ([1, 2, 3]), [11, 22, 33]);
matrix ([1, 2, 3], [11, 22, 33]);

addrow (matrix ([1, 2, 3]), [11, 22, 33], [111, 222, 333]);
matrix ([1, 2, 3], [11, 22, 33], [111, 222, 333]);

addrow (matrix ([1, 2, 3]), [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);
matrix ([1, 2, 3], [11, 22, 33], [111, 222, 333], [1111, 2222, 3333]);

/* addrow/addcol that should fail */

errcatch (addrow (matrix ([1], [2], [3]), [11, 22, 33]));
[];

/*
 * Seems like these should fail, but they succeed.
 * Just omit these tests unless addrow/addcol are ever revised for greater consistency.
 *
errcatch (addrow (matrix ([1], [2], [3]), matrix ([11, 22, 33])));
[];

errcatch (addrow (matrix ([1, 2, 3]), matrix ([1], [2], [3])));
[];

errcatch (addcol (matrix ([1, 2, 3]), matrix ([1], [2], [3])));
[];

errcatch (addcol (matrix ([1], [2], [3]), matrix ([11, 22, 33])));
[];
 *
 */

/* Bug #3532: Integrator doesn't use a new variable internally, causing facts on the original variable to be used for the substitution variable */
is(integrate(cos(x) * abs(cos(x)), x, 0, %pi) = %pi / 2);
false$

/* Bug found on mailing list thread "Wrong integral, but antiderivative and limits are correct" */
integrate(x^8/(5+x^9)^2, x, 0, inf);
1/45$

/* Test case for bug mentioned at https://sourceforge.net/p/maxima/mailman/message/8488105/ */
is(abs(integrate(1/(x^4+1), x, 0, 1/2) - 0.49396) < 0.00001);
true$

/* Another bug found on mailing list thread "Wrong integral, but antiderivative and limits are correct" */
freeof(?xz, residue(plog(x)/(x-ratsimp(rectform(%e^(%i*%pi/9)))), x, %e^(%i*%pi/9)));
true$

/* Bug #3562: "integrate(1/(1+tan(x)), x, 0, %pi/2) gives complex result, should be %pi/4" */
integrate(1/(1+tan(x)), x, 0, %pi/2);
%pi/4$

/* Also from bug #3562 */
limit(log((x^2+1)/x^2)-2*log((x-1)/x), x, 0, minus);
0$

/* #3787 fixnump checks in simpexpt */
(declare(n,integer),0);
0$

(-1)^((2^62-1)*n);
(-1)^n$

(-1)^((2^62+1)*n);
(-1)^n$

(-1)^((2^62+2)*n);
1$

(remove(n,integer),0);
0$

/*
 * Check sech/csch don't overflow for large numbers.  Use relative
 *  error to test for accuracy.
 */

closetorel(actual, expected, tol) := block([numer:true, relerr: abs(actual-expected)/expected], if (relerr < tol) then true else relerr);
closetorel(actual, expected, tol) := block([numer:true, relerr: abs(actual-expected)/expected], if (relerr < tol) then true else relerr);

closeto(sech(715.0), 6.033b-311);
true;
closeto(csch(715.0), 6.033b-311);
true;

/*
 *
 * Check acsch doesn't overflow for small numbers.  But some lisps
 * like clisp don't support denormals, so we have to be careful.
 *
 */

if (1e-309 = 0.0) then
    closetorel(acsch(1e-300), acsch(bfloat(1e-300)), 1.1375b-16)
  else
    closetorel(acsch(1e-309), acsch(bfloat(1b-309)), 7.1559b-17);
true;

/* SF bug #3825: "apply('forget, facts()) gives Lisp error" */

(kill (all), declare (F, increasing));
done;

facts ();
[kind (F, increasing)];

featurep (F, increasing);
true;

apply ('forget, facts ());
[kind (F, increasing)];

facts ();
[];

featurep (F, increasing);
false;

declare ([F, G], rational, [H, I], irrational);
done;

facts ();
[kind (F, rational), kind (G, rational), kind (H, irrational), kind (I, irrational)];

[featurep (F, rational),
 featurep (G, rational),
 featurep (H, irrational),
 featurep (I, irrational)];
[true, true, true, true];

forget (kind (G, rational), kind (I, irrational));
[kind (G, rational), kind (I, irrational)];

facts ();
[kind (F, rational), kind (H, irrational)];

forget (kind (F, rational), kind (H, irrational));
[kind (F, rational), kind (H, irrational)];

facts ();
[];

[featurep (F, rational),
 featurep (G, rational),
 featurep (H, irrational),
 featurep (I, irrational)];
[false, false, false, false];

/* Bug #3934: expand(1/(1+%i)^4) => (-4)^(-1) (unsimplified) */

expand((1+%i)^-2);
-%i/2;

expand(1/(1+%i)^4);
-1/4;

expand(4/(%i+1)^4)+1;
0;

expand(1/(1+%i)^8-1/16);
0;

expand((sqrt(-1/2)+sqrt(1/2))^-8);
1;

/* Bug #3956: expand(1/((sqrt(2)-1)*(sqrt(2)+1))) => 1/1 (unsimplified) */

expand(1/((sqrt(2)-1)*(sqrt(2)+1)));
1;

expand(1/(-(sqrt(3)-1)*(sqrt(3)+1)));
-1/2;

expand(1/(-(sqrt(n+1)-1)*(sqrt(n+1)+1)));
-1/n;