Invalid simplification from Expression.simplify_trig()

[orlitzky]

Same issue as #12794 from the looks of it. Maybe this is just a particularly nasty function?

u,x = var('u,x')

F = -1/12*(2*(sqrt(24*sqrt(2) - 30)*(155*sqrt(2) - 228) + 2368*sqrt(2) - 3086)*u^3 + (8*(5*(3*sqrt(2) - 4)*sqrt(24*sqrt(2) - 30) - 128*sqrt(2) + 106)*u^3 + 32*(5*(3*sqrt(2) - 4)*sqrt(24*sqrt(2) - 30) - 128*sqrt(2) + 106)*u^2 + (sqrt(24*sqrt(2) - 30)*(385*sqrt(2) - 492) - 6976*sqrt(2) + 6902)*u)*x^3 + 2*(sqrt(24*sqrt(2) - 30)*(525*sqrt(2) - 764) + 6592*sqrt(2) - 8834)*u^2 + 3*(8*(5*(3*sqrt(2) - 4)*sqrt(24*sqrt(2) - 30) - 128*sqrt(2) + 106)*u^3 + 32*(5*(3*sqrt(2) - 4)*sqrt(24*sqrt(2) - 30) - 128*sqrt(2) + 106)*u^2 + (sqrt(24*sqrt(2) - 30)*(575*sqrt(2) - 788) - 1216*sqrt(2) - 118)*u)*x^2 + (5*sqrt(24*sqrt(2) - 30)*(191*sqrt(2) - 276) + 10304*sqrt(2) - 14158)*u + 3*(8*(5*(3*sqrt(2) - 4)*sqrt(24*sqrt(2) - 30) - 128*sqrt(2) + 106)*u^3 + 2*(sqrt(24*sqrt(2) - 30)*(335*sqrt(2) - 468) + 832*sqrt(2) - 1814)*u^2 + (sqrt(24*sqrt(2) - 30)*(765*sqrt(2) - 1084) + 4544*sqrt(2) - 7138)*u)*x)/(sqrt(24*sqrt(2) - 30)*(95*sqrt(2) - 148) + 2880*sqrt(2) - 3510)

b = -1/6*(128*x^3 + 384*x^2 + 6*sqrt(24*sqrt(2) - 30)*((3*sqrt(2) - 4)*x + 3*sqrt(2) - 4) + (x + 1)*sqrt(9216*sqrt(2) + 1216*x^4 + 22144*x^3 + 16*(576*sqrt(2) - 1253)*x + 13056*x^2 + 24*sqrt(24*sqrt(2) - 30)*(4*(3*sqrt(2) - 4)*x^4 - 8*(3*sqrt(2) - 4)*x^3 + 16*(3*sqrt(2) - 4)*x^2 + 25*(3*sqrt(2) - 4)*x - 9*sqrt(2) + 12) - 12176) + 204*x - 52)/(16*x^3 + 48*x^2 + (3*sqrt(2) - 4)*sqrt(24*sqrt(2) - 30) + 48*x - 14)

p  = plot(F(u=b), x, -1, 0)
p += plot(F(u=b).simplify_trig(), x, -1, 0, color='red')
p.show(filename='simplify_trig_gone_wild.png')

CC: @kcrisman

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/12795

[orlitzky]

Attachment: simplify_trig_gone_wild.sage.gz

A .sage file exhibiting the problem

[kcrisman]

comment:4

I'm inclined to believe it's a pretty nasty function. Given that the simplifications in both cases appear to be the same (and that G.expand() gives a different, but still correct, thing without numerical issues), I think the problem is just that you lose too much accuracy. See my comment on #12794.

[kcrisman]

shows solution

[kcrisman]

comment:5

Attachment: better.png

Thanks for pointing out #12795, which got me trying this out again... Pretty sure this is invalid.

sage: G = F(u=b)
sage: H = G.simplify_trig()
sage: R = RealField(1000)
sage: L = [(j,H(R(j))) for j in srange(-1,.2,.001)]
sage: M = [(a,R(b)) for a,b in L] # should have done this in one step, but oh well
sage: points(M)+plot(G,(x,-1,.2),color='red')

yields the following bee-yoo-tiful picture that every student of mathematics should have the chance to understand - why you don't just plug and chug:

[orlitzky]

comment:10

You are right about these, I should have closed them earlier. The problem is with plot() using floats. The result of simplify_rational() is still accurate.