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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % Tests of the COMPACT package. % Author: Anthony C. Hearn. % First some simple examples. aa := {cos(x)^2+sin(x)^2-1}; 2 2 aa := {cos(x) + sin(x) - 1} xx := 2*cos(x)^2+2*sin(x)^2-2; 2 2 xx := 2*(cos(x) + sin(x) - 1) compact(xx,aa); 0 xx := (1-cos(x)^2)^4; 8 6 4 2 xx := cos(x) - 4*cos(x) + 6*cos(x) - 4*cos(x) + 1 compact(xx,aa); 8 sin(x) % These examples are from Lars Hornfeldt. % This should be cos x^10*sin x^10. compact(((1-(sin x)**2)**5)*((1-(cos x)**2)**5) *(((sin x)**2+(cos x)**2)**5), {cos x^2+sin x^2=1}); 10 2 4 10 8 2 sin(x) *(10*cos(x) *sin(x) - sin(x) + 5*sin(x) - 5*sin(x) + 1) % This example illustrates the problem in the above. It is cos(x)^6. compact(-3cos(x)^2*sin(x)^2-sin(x)^6+1,{cos x^2+sin x^2-1}); 2 2 6 - 3*cos(x) *sin(x) - sin(x) + 1 compact(s*(1-(sin x**2))+c*(1-(cos x)**2)+(sin x)**2+(cos x)**2, {cos x^2+sin x^2=1}); 2 2 cos(x) *s + sin(x) *c + 1 xx := s*(1-(sin x**2))+c*(1-(cos x)**2)+(sin x)**2+(cos x)**2 *((sin x)**2+(cos x)**2)*(sin x)**499*(cos x)**499; 503 499 501 501 2 2 xx := cos(x) *sin(x) + cos(x) *sin(x) - cos(x) *c - sin(x) *s 2 + sin(x) + c + s compact(xx,{cos(x)^2+sin(x)^2=1}); 501 499 2 2 2 cos(x) *sin(x) + cos(x) *s + sin(x) *c + sin(x) compact((s*(1-(sin x**2))+c*(1-(cos x)**2)+(sin x)**2+(cos x)**2) *((sin x)**2+(cos x)**2)*(sin x)**499*(cos x)**499, {cos x^2+sin x^2=1}); 499 499 2 2 cos(x) *sin(x) *(cos(x) *s + sin(x) *c + 1) compact(df((1-(sin x)**2)**4,x),{cos x^2+sin x^2=1}); 2 2 6 8*cos(x)*sin(x)*(3*cos(x) *sin(x) + sin(x) - 1) % End of Lars Hornfeld examples. xx := a*(cos(x)+2*sin(x))^3-w*(cos(x)-sin(x))^2; 3 2 2 2 xx := cos(x) *a + 6*cos(x) *sin(x)*a - cos(x) *w + 12*cos(x)*sin(x) *a 3 2 + 2*cos(x)*sin(x)*w + 8*sin(x) *a - sin(x) *w compact(xx,aa); 2 3 11*cos(x)*sin(x) *a + 2*cos(x)*sin(x)*w + cos(x)*a + 2*sin(x) *a + 6*sin(x)*a - w xx := (1-cos(x)^2)^2+(1-sin(x)^2)^2; 4 2 4 2 xx := cos(x) - 2*cos(x) + sin(x) - 2*sin(x) + 2 compact(xx,aa); 2 2 - 2*cos(x) *sin(x) + 1 xx := (c^2-1)^6+7(s-1)^4+23(c+s)^5; 12 10 8 6 5 4 4 3 2 xx := c - 6*c + 15*c - 20*c + 23*c + 115*c *s + 15*c + 230*c *s 2 3 2 4 5 4 3 2 + 230*c *s - 6*c + 115*c*s + 23*s + 7*s - 28*s + 42*s - 28*s + 8 compact(xx,{c+s=1}); 12 10 8 6 4 2 c - 6*c + 15*c - 20*c + 22*c - 6*c + 24 yy := (c+1)^6*s^6+7c^4+23; 6 6 5 6 4 6 4 3 6 2 6 6 6 yy := c *s + 6*c *s + 15*c *s + 7*c + 20*c *s + 15*c *s + 6*c*s + s + 23 compact(yy,{c+s=1}); 6 6 5 6 4 6 4 3 6 2 6 6 6 c *s + 6*c *s + 15*c *s + 7*c + 20*c *s + 15*c *s + 6*c*s + s + 23 zz := xx^3+c^6*s^6$ compact(zz,{c+s=1}); 36 34 32 30 28 26 24 22 c - 18*c + 153*c - 816*c + 3081*c - 8820*c + 20019*c - 37272*c 20 18 16 14 12 11 + 58854*c - 81314*c + 100488*c - 111840*c + 111341*c - 6*c 10 9 8 7 6 4 2 - 97545*c - 20*c + 80439*c - 6*c - 53783*c + 40608*c - 10368*c + 13824 xx := (c+s)^5 - 55(1-s)^2 + 77(1-c)^3 + (c+2s)^8; 8 7 6 2 5 3 5 4 4 4 xx := c + 16*c *s + 112*c *s + 448*c *s + c + 1120*c *s + 5*c *s 3 5 3 2 3 2 6 2 3 2 + 1792*c *s + 10*c *s - 77*c + 1792*c *s + 10*c *s + 231*c 7 4 8 5 2 + 1024*c*s + 5*c*s - 231*c + 256*s + s - 55*s + 110*s + 22 % This should reduce to something like: yy := 1 - 55c^2 + 77s^3 + (1+s)^8; 2 8 7 6 5 4 3 2 yy := - 55*c + s + 8*s + 28*s + 56*s + 70*s + 133*s + 28*s + 8*s + 2 % The result contains the same number but different terms. compact(xx,{c+s=1}); 8 7 6 5 4 3 2 s + 8*s + 28*s + 56*s + 70*s + 133*s - 27*s + 118*s - 53 compact(yy,{c+s=1}); 8 7 6 5 4 3 2 s + 8*s + 28*s + 56*s + 70*s + 133*s - 27*s + 118*s - 53 % Test showing order of expressions is important. d2:= - 4*r3a**2 - 4*r3b**2 - 4*r3c**2 + 3*r3**2$ d1:= 4 * r3a**2 * r3 + 4 * r3b**2 * r3 + 4 * r3c**2 * r3 + 16 * r3a * r3b * r3c - r3**3$ d0:= 16 * r3a**4 + 16 * r3b**4 + 16 * r3c**4 + r3**4 - 32 * r3a**2 * r3b**2 - 32 * r3a**2 * r3c**2 - 32 * r3b**2 * r3c**2 - 8 * r3a**2 * r3**2 - 8 * r3b**2 * r3**2 - 8 * r3c**2 * r3**2 - 64 * r3a * r3b * r3c * r3$ alist := { c0 = d0, c1 = d1, c2 = d2}$ blist := { c2 = d2, c1 = d1, c0 = d0}$ d:= d2 * l*l + d1 * l + d0; 2 2 2 2 2 2 2 2 3 2 d := 3*l *r3 - 4*l *r3a - 4*l *r3b - 4*l *r3c - l*r3 + 4*l*r3*r3a 2 2 4 2 2 + 4*l*r3*r3b + 4*l*r3*r3c + 16*l*r3a*r3b*r3c + r3 - 8*r3 *r3a 2 2 2 2 4 2 2 - 8*r3 *r3b - 8*r3 *r3c - 64*r3*r3a*r3b*r3c + 16*r3a - 32*r3a *r3b 2 2 4 2 2 4 - 32*r3a *r3c + 16*r3b - 32*r3b *r3c + 16*r3c compact(d,alist); 2 c0 + c1*l + c2*l % Works fine. compact(d,blist); 2 2 2 3 4 c2*l - c2*l*r3 + 2*c2*r3 + 8*c2*r3a + 2*l*r3 + 16*l*r3a*r3b*r3c - 5*r3 2 2 4 4 2 2 4 - 24*r3 *r3a - 64*r3*r3a*r3b*r3c + 48*r3a + 16*r3b - 32*r3b *r3c + 16*r3c % Only c2=d2 is applied. % This example illustrates why parallel application of the individual % side relations is necessary. lst:={x1=a+b+c, x2=a-b-c, x3=-a+b-c, x4=-a-b+c}; lst := {x1=a + b + c, x2=a - b - c, x3= - a + b - c, x4= - a - b + c} z1:=(a+b+c)*(a-b-c)*(-a+b-c); 3 2 2 2 2 3 2 2 3 z1 := - a + a *b - a *c + a*b + 2*a*b*c + a*c - b - b *c + b*c + c % This is x1*x2*x3. z2:=(a+b+c)*(a-b-c)*(-a+b-c)*(-a-b+c); 4 2 2 2 2 4 2 2 4 z2 := a - 2*a *b - 2*a *c + b - 2*b *c + c % This is x1*x2*x3*x4. compact(z1,lst); 2 x1*(4*a*b + 2*c*x1 - x1 ) % Not the best solution but better than nothing. compact(z2,lst); 4 2 2 2 2 4 2 2 4 a - 2*a *b - 2*a *c + b - 2*b *c + c % Does nothing. end; (TIME: compact 710 710)