see http://www.math.utexas.edu/pipermail/maxima/2013/031454.html
Third root of of log(x^3-x*78+90)+1000*x is x3+epsilon, where x3 third
root of x^3-x*78+90.
How I compute diference epsilon?
(%i1) set_display('ascii)$
(%i2) fpprec:4000$
(%i3) eq3:x^3-78*x+90$
(%i4) allroots(eq3);
(%o4) [x = 1.174624074847539, x = 8.185668853475939, x = -
9.360292928323476]
(%i5) x1:bf_find_root(eq3,x,-10,-9,relerr=10^-5000)$
x2:bf_find_root(eq3,x,1,2,relerr=10^-5000)$
x3:bf_find_root(eq3,x,8,9,relerr=10^-5000)$
(%i8) eq:log((x-x1)*(x-x2)*(x-x3))+1000*x$
(%i9) eq1:%,logexpand=super$
(%i10) eq2:subst(x=x3+10^(-t),eq1)$
(%i11) t1:bf_find_root(eq2,t,3500,3600,relerr=10^-5000)$
(%i12) epsilon:bfloat(10^(-t1))$
(%i13) fpprec:442$
(%i14) bfloat(epsilon);
(%o14)
8.302833484588825497474457774037010207258614403421202647496804762002746\
791385011125050393723851741501668709043262400716333777722098240692435332419533\
101896112466951287114761320993507344883585599759639161435076848887464021132424\
229992098204901162925175858992701463272558057631490094106001800921948537005874\
462638104923848174964988193816324205217476658073360951233886266423605955115326\
944541545588173193117621954861313397249231152514832988079791b-3558
(%i15) elapsed_run_time ();
(%o15) 101.44
Run time is ~~100 seconds on AMD Athlon(tm) 64 X2 Dual Core Processor
5000+ 2.60 GHz
best
Aleksas D