Comment #1 on issue 2878 by [email protected]: numerical evaluation problem
http://code.google.com/p/sympy/issues/detail?id=2878
I think that's what happened. When you did S("0.6"), it created a Float
with the default precision (15 digits). evalf'ing it doesn't make a
difference, because once you make it as a Float, it only has that 15 digits
of precision in there. The digits beyond that are just floating point
junk. The S(6)/10 is different because this is exact, so it can be
evalf'ed to any precision, without any loss of the original information.
Here's what I get if I set the precision to 100 from the outset:
In [27]: a.subs(x, Float("0.6", 100))
Out[27]:
-3.571835597757109319424323588454190498287218733821245816297268260086351987208556861485775068313037593e-102
So the solution is to either use exact numbers up until the point where you
want to get floating point numbers, or else to always put high enough
precision on your floating point numbers so that the final result will be
correct to the precision you desire.
P.S., here's a trick to symbolically simplify this:
In [33]: a.replace(sin, lambda x:sqrt(1 - cos(x)**2)).expand()
Out[33]: 0
Unfortunately, trigsimp() isn't smart enough to do this automatically yet.
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