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