/*************** -*- Mode: MACSYMA; Package: MAXIMA -*-  ******************/
/***************************************************************************
***                                                                    *****
***     Copyright (c) 1984 by William Schelter,University of Texas     *****
***     All rights reserved                                            *****
***************************************************************************/


/* rtest3 */						
kill(all);
done;
for a from -3 step 7 thru 26 do ldisplay(a);
done$
s:0;
0$
for i while i <= 10 do s:s+i;
done$
s;
55$
series:1;
1$
term:exp(sin(x));
%e^sin(x)$
for p unless p > 7 do
    (term:diff(term,x)/p,series:series+subst(x = 0,term)*x^p);
done$
series;
x^7/90-x^6/240-x^5/15-x^4/8+x^2/2+x+1$
poly:0;
0$
for i thru 5 do (for j from i step -1 thru 1 do poly:poly+i*x^j);
done$
poly;
5*x^5+9*x^4+12*x^3+14*x^2+15*x$
guess:-3.0;
-3.0$
for i thru 10 do
    (guess:subst(guess,x,0.5*(x+10/x)),
     if abs(guess^2-10) < 5.0e-5 then return(guess));
-3.162280701754386;
/* -3.1622806$ */
for count from 2 next 3*count thru 20 do ldisplay(count);
done$
x:1000;
1000$
thru 10 while x # 0 do x:0.5*(x+5/x);
done$
x;
2.282429035887867$
remvalue(x);
[x]$
newton(f,guess):=block([numer,y],local(f,df,x,guess),numer:true,
       define(df(x),diff(f(x),x)),
       do (y:df(guess),if y = 0 then error("derivative at",guess,"is zero"),
	   guess:guess-f(guess)/y,
	   if abs(f(guess)) < 5.0e-6 then return(guess)));
newton(f,guess):=block([numer,y],local(f,df,x,guess),numer:true,
       define(df(x),diff(f(x),x)),
       do (y:df(guess),if y = 0 then error("derivative at",guess,"is zero"),
	   guess:guess-f(guess)/y,
	   if abs(f(guess)) < 5.0e-6 then return(guess)))$
sqr(x):=x^2-5.0;
sqr(x):=x^2-5.0$
newton(sqr,1000);
2.236068027062195; 
for f in [log,rho,atan] do ldisp(f(1.0));
done$
ev(concat(e,linenum-1),numer);
e10$
kill(functions,values,arrays);
done$
done;
done$
exp:diff(x*f(x),x);
x*'diff(f(x),x,1)+f(x)$
f(x):=sin(x);
f(x):=sin(x)$
ev(exp,diff);
sin(x)+x*cos(x)$
x;
x$
x:3;
3$
x;
3$
'x;
x$
f(x):=x^2;
f(x):=x^2$
'f(2);
'f(2)$
ev(%,f);
4$
'(f(2));
f(2)$
f(2);
4$
sum(i!,i,1,4);
33$
'sum(i!,i,1,4);
'sum(i!,i,1,4)$
remvalue(x);
[x]$
'integrate(f(x),x,a,b);
'integrate(x^2,x,a,b)$
for i thru 5 do s:s+i^2;
done$
exp:s;
s+55$
ev(%,s:0);
55$
ev(exp);
s+110$
exp:'sum(g(i),i,0,n);
'sum(g(i),i,0,n)$
z*%e^z;
z*%e^z$
ev(%,z:x^2);
x^2*%e^x^2$
subst(x^2,z,exp);
'sum(g(i),i,0,n)$
a:%;
'sum(g(i),i,0,n)$
a+1;
'sum(g(i),i,0,n)+1$
kill(a,y);
done$
a;
a$
declare(integrate,noun);
done$
integrate(y^2,y);
integrate(y^2,y)$
''integrate(y^2,y);
y^3/3$
f(y):=diff(y*log(y),y,2);
f(y):=diff(y*log(y),y,2)$
f(y):=1/y;
f(y):=1/y$
c10;
c10$
(x+y)^3;
(y+x)^3$
diff(%,x);
3*(y+x)^2$
y:x^2+1;
x^2+1$

/* begin fix */
kill(all);
done;
 ev(%e^x*sin(x)^2,exponentialize);
 -%e^x*(%e^(%i*x)-%e^-(%i*x))^2/4;
  integrate(%,x);
-((%e^((2*%i+1)*x)/(2*%i+1)+%e^((1-2*%i)*x)/(1-2*%i)-2*%e^x)/4); 
 ev(%,demoivre);
 -((%e^x*(%i*sin(2*x)+cos(2*x))/(2*%i+1)
      +%e^x*(cos(2*x)-%i*sin(2*x))/(1-2*%i)-2*%e^x)
      /4);
 ans:ev(%,ratexpand);
 -%e^x*sin(2*x)/5-%e^x*cos(2*x)/10+%e^x/2;
 ev(ans,x:1,numer)-ev(ans,x:0,numer);
 0.5779160182042402;
 (fpprec : 35, 0);
 0;
 ev(ans,x:1,bfloat)-ev(ans,x:0,bfloat);
 5.7791601820424019599988308251707781b-1;
 integrate(%e^x*sin(x)^2,x);
 -(((2*%e^x*sin(2*x)+%e^x*cos(2*x)-5*%e^x)/10));
 trigreduce(%);
 -((2*%e^x*sin(2*x)+%e^x*cos(2*x)-5*%e^x)/10);
 % - ans,ratsimp;
 0 ;
 reset (fpprec);
 [fpprec];

/* end fix*/

ev(sin(x),%emode);
sin(x)$
sin(%pi/12)+tan(%pi/6);
sin(%pi/12)+1/sqrt(3)$
ev(%,numer);
0.8361693142921465;
/* tops 20 : 0.83616931$ */
sin(1);
sin(1)$
ev(sin(1),numer);
0.8414709848078965$
beta(1/2,2/5);
beta(1/2,2/5)$
ev(%,numer);
3.679093980405881;
/* tops 20: 3.67909265$ */
diff(atanh(sqrt(x)),x);
1/(2*(1-x)*sqrt(x))$
fpprec:25;
25$
sin(5.0b-1);
4.794255386042030002732879b-1$
(reset (fpprec), 0);
0;
/*begin fix */
 exp:cos(x)^2-sin(x)^2;
 cos(x)^2-sin(x)^2$
 ev(%,x:%pi/3);
 -1/2$
 diff(exp,x);
 -4*cos(x)*sin(x)$
 integrate(exp,x);
 (sin(2*x)/2+x)/2-(x-sin(2*x)/2)/2$
 expand(%);
 sin(2*x)/2$
 trigexpand(%);
 cos(x)*sin(x)$
 trigreduce(%);
 sin(2*x)/2$
 diff(%,x);
 cos(2*x)$
 %-exp,trigreduce,ratsimp;
  0;
/*end fix*/
sech(x)^2*sinh(x)*tanh(x)/coth(x)^2+cosh(x)^2*sech(x)^2*tanh(x)/coth(x)^2
				   +sech(x)^2*tanh(x)/coth(x)^2;
sech(x)^2*sinh(x)*tanh(x)/coth(x)^2+cosh(x)^2*sech(x)^2*tanh(x)/coth(x)^2
				   +sech(x)^2*tanh(x)/coth(x)^2$
trigsimp(%);
(sinh(x)^5+sinh(x)^4+2*sinh(x)^3)/cosh(x)^5$
/* These are from the trgsmp.dem file.  
 * I (rtoy) hand-verified these results (using maxima, of course)
 */
(1-sin(x)^2)*cos(x)/cos(x)^2+tan(x)*sec(x)^2;
(1-sin(x)^2)*cos(x)/cos(x)^2+tan(x)*sec(x)^2$
trigsimp(%);
(sin(x)+cos(x)^4)/cos(x)^3$

tan(x)^2+sec(x)^2/(1-tan(x)*sec(x));
tan(x)^2+sec(x)^2/(1-tan(x)*sec(x))$
trigsimp(%);
(sin(x)^4+sin(x)^3-1)/(cos(x)^2*sin(x)-cos(x)^4)$

(sin(x)^4-6*cos(x)^2*sin(x)^2+4*(cos(x)^2-sin(x)^2)+8*sin(x)+cos(x)^4+3)/(8*cos(x)^3);
(sin(x)^4-6*cos(x)^2*sin(x)^2+4*(cos(x)^2-sin(x)^2)+8*sin(x)+cos(x)^4+3)/(8*cos(x)^3)$
trigsimp(%);
(sin(x)+cos(x)^4)/cos(x)^3$


sech(x)^2*sinh(x)*tanh(x)/coth(x)^2+cosh(x)^2*sech(x)^2*tanh(x)/coth(x)^2+sech(x)^2*tanh(x)/coth(x)^2;
sech(x)^2*sinh(x)*tanh(x)/coth(x)^2+cosh(x)^2*sech(x)^2*tanh(x)/coth(x)^2+sech(x)^2*tanh(x)/coth(x)^2$
trigsimp(%);
(sinh(x)^5+sinh(x)^4+2*sinh(x)^3)/cosh(x)^5$

-sech(x)^5*(sinh(x)^5+2*(sinh(x)^4+6*cosh(x)^2*sinh(x)^2+cosh(x)^4)-13*(sinh(x)^3+3*cosh(x)^2*sinh(x))+10*cosh(x)^2*sinh(x)^3-8*(sinh(x)^2+cosh(x)^2)+5*cosh(x)^4*sinh(x)+34*sinh(x)+6)/16;
-sech(x)^5*(sinh(x)^5+2*(sinh(x)^4+6*cosh(x)^2*sinh(x)^2+cosh(x)^4)-13*(sinh(x)^3+3*cosh(x)^2*sinh(x))+10*cosh(x)^2*sinh(x)^3-8*(sinh(x)^2+cosh(x)^2)+5*cosh(x)^4*sinh(x)+34*sinh(x)+6)/16$
trigsimp(%);
-((sinh(x)^5+sinh(x)^4-2*sinh(x)^3)/cosh(x)^5)$

cos(x)*(sec(x)^2*tan(x)+1)-sec(x)^2*sin(x)-cos(x);
cos(x)*(sec(x)^2*tan(x)+1)-sec(x)^2*sin(x)-cos(x)$
trigsimp(%);
0$

v*cos(x)*sec(x)^2*tan(x)+(-v*sec(x)^2-2*'diff(v,x))*sin(x)+'diff(v,x)*cos(x)*sec(x)+'diff(v,x,2)*cos(x);
v*cos(x)*sec(x)^2*tan(x)+(-v*sec(x)^2-2*'diff(v,x))*sin(x)+'diff(v,x)*cos(x)*sec(x)+'diff(v,x,2)*cos(x)$
trigsimp(%);
-2*'diff(v,x,1)*sin(x)+'diff(v,x,2)*cos(x)+'diff(v,x,1)$

triginverses : all;
all;

sinh(acosh(x));
sqrt(x-1)*sqrt(x+1);

sinh(atanh(x));
x/(sqrt(1-x)*sqrt(x+1));

cosh(asinh(x));
sqrt(x^2+1);

cosh(atanh(x));
1/(sqrt(1-x)*sqrt(x+1));

tanh(asinh(x));
x/sqrt(x^2+1);

tanh(acosh(x));
sqrt(x-1)*sqrt(x+1)/x;

/* A few checks to see that triginverses false disables the above transformations */
triginverses: false;
false;

cos(acosh(x));
cos(acosh(x));

triginverses : all;
all;

/* SF bug # 1981518, Calling desolve inside a "for...do" makes it loop endlessly
 * (protect against endless loop by throw--catch in case bug is triggered)
 */
catch (block ([foo:1],
 for i thru 3 do (ilt (1/s, s, t),
 if foo > 3 then throw ('i = i) else foo : foo + 1)));
done;

/* bug reported to mailing list 2009-05-09
 * unexpected behavior in for loop with variable step
 */

block ([L : []], for r:0 thru 7 step +2 do L : cons (r, L), L);
[6, 4, 2, 0];

block ([L : []], for r:7 thru 0 step -2 do L : cons (r, L), L);
[1, 3, 5, 7];

block ([L : [], r0 : 0, r1 : 7, s : +2], for r:r0 thru r1 step s do L : cons (r, L), L);
[6, 4, 2, 0];

block ([L : [], r0 : 7, r1 : 0, s : -2], for r:r0 thru r1 step s do L : cons (r, L), L);
[1, 3, 5, 7];

/* step is evaluated once at start of loop, so these loops are defined */

block ([L : [], s : +2], for i:1 thru 10 step s do L : cons (s : -s, L), L);
[-2, 2, -2, 2, -2];

block ([L : [], s : -2], for i:10 thru 1 step s do L : cons (s : -s, L), L);
[2, -2, 2, -2, 2];

/* bug reported to mailing list 2009-05-13 "reset ( radexpand,  domain )"
 *
 * display2d is a resetable 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 (), [radexpand, domain]);
[true, real];
(display2d:save, done);
done$

[radexpand, domain] : [all, complex];
[all, complex];

reset (radexpand, domain);
[radexpand, domain];

[radexpand, domain];
[true, real];

([foo, bar, baz] : [1, 2, 3],
 /* should ignore these non-defmvar's */
 reset (foo, bar, baz));
[];

/* verify that ORDFNA can handle CRE.
 */
(kill (a, b), [doallmxops, doscmxops] : [false, false], 0);
0;

b*matrix([rat(a)]);
b*matrix([''(rat(a))]);

(reset (doallmxops, doscmxops), 0);
0;

/* SF bug #2936: stack overflow in integrate */

kill (x, A, B, MU, SIGMA);
done;

trigsimp (gamma_incomplete (1, log (x)));
gamma_incomplete (1, log (x));

trigsimp ((%i*gamma_incomplete(1,(1-2*log(x))^2/4)*(1-2*log(x))^2)
           /(2*log(x)-1)^2);
%i*gamma_incomplete(1,(4*log(x)^2-4*log(x)+1)/4)$

trigsimp (integrate (%e^((-log(x)^2)-1)*log(x),x));
-(%e^-(3/4)*(2*gamma_incomplete(1,(4*log(x)^2-4*log(x)+1)/4)*abs(2*log(x)-1)
            +2*gamma_incomplete(1/2,(4*log(x)^2-4*log(x)+1)/4)*log(x)
            -gamma_incomplete(1/2,(4*log(x)^2-4*log(x)+1)/4)))
 /(4*abs(2*log(x)-1))$

/* throw away results of integrate, just make sure it runs without crashing */
block ([foo, bar, ctxt],
  foo : exp(-(log(x) - MU)*(log(x) - MU)/(2*SIGMA*SIGMA))/(x*SIGMA*sqrt(2*%pi)),
  bar : (log(B) - log(x*SIGMA) + ((x-A)*(x-A)/(2*B*B) - (log(x) -MU)*(log(x) -MU)/(2*SIGMA*SIGMA))),
  [foo, bar] : subst ([A=2, MU=3], [foo, bar]),
  ctxt : newcontext (),
  assume (SIGMA > 0, B > 0),
  integrate (expand (foo*bar), x),
  integrate (expand (foo*bar), x, 2, inf),
  killcontext (ctxt),
  0);
0;
