Sun Nov  1 17:27:20 MST 1998
aquarius% reduce
REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ...

1: % ----------[ R e d u c e ]----------
% ---------- Initialization ----------
on time;


Time: 0 ms

% ---------- Determining Zero Equivalence ----------
% The following expressions are all equal to zero
sqrt(997) - (997^3)^(1/6);


            1/6
 - 991026973    + sqrt(997)

Time: 10 ms

on ifactor;


Time: 0 ms

ws;


0

Time: 0 ms

sqrt(999983) - (999983^3)^(1/6);


0

Time: 480 ms

off ifactor;


Time: 0 ms

(2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6;


0

Time: 0 ms

cos(x)^3 + cos(x)*sin(x)^2 - cos(x);


              2         2
cos(x)*(cos(x)  + sin(x)  - 1)

Time: 10 ms

% See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'',
% _Communications of the Association of Computing Machinery_, Volume 14,
% Number 8, August 1971, 527--537.  This expression is zero if Re(x) is
% contained in the interval ((4 n - 1)/2 pi, (4 n + 1)/2 pi) for n an integer:
% ..., (-5/2 pi, -3/2 pi), (-pi/2, pi/2), (3/2 pi, 5/2 pi), ...
load_package(trigsimp)$

Time: 280 ms

xpr:= log(tan(1/2*x + pi/4)) - asinh(tan(x));


                                   pi + 2*x
xpr :=  - asinh(tan(x)) + log(tan(----------))
                                      4

Time: 10 ms

trigsimp(ws);


                               x          x
                          cos(---) + sin(---)
          sin(x)               2          2
 - asinh(--------) + log(---------------------)
          cos(x)               x          x
                          cos(---) - sin(---)
                               2          2

Time: 170 ms

% Use a roundabout method---show that expr is a constant equal to zero
df(xpr, x);


                 2           pi + 2*x          pi + 2*x  2
  - 2*sqrt(tan(x)  + 1)*tan(----------) + tan(----------)  + 1
                                4                 4
---------------------------------------------------------------
                              pi + 2*x
                       2*tan(----------)
                                 4

Time: 20 ms

trigsimp(ws);


               2               x  2
  - sqrt(sin(x)  - 1) - 2*sin(---) *i + i
                               2
------------------------------------------
              2              x  2
   sqrt(sin(x)  - 1)*(2*sin(---)  - 1)
                             2

Time: 240 ms  plus GC time: 40 ms
sub(x = 0, xpr);


0

Time: 0 ms

clear xpr;


Time: 0 ms

log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1));


0

Time: 0 ms

(4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1))
   * (2*sqrt(r) + 1)^(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1;


               1/(2*sqrt(r) + 1)                      sqrt(r)/(2*sqrt(r) + 1)
(2*sqrt(r) + 1)                 *(4*sqrt(r) + 4*r + 1)

 - 2*sqrt(r) - 1

Time: 30 ms

% [Gradshteyn and Ryzhik 9.535(3)]
load_package(specfn)$


Time: 1570 ms  plus GC time: 70 ms
2^(1 - z)*Gamma(z)*Zeta(z)*cos(z*pi/2) - pi^z*Zeta(1 - z);


        pi*z                        z  z
 2*cos(------)*gamma(z)*zeta(z) - pi *2 *zeta( - z + 1)
         2
--------------------------------------------------------
                            z
                           2

Time: 40 ms

% ---------- Quit ----------
quit;

Quitting

real 13.98
user 3.08
sys 1.62