Hello.
I would like to know precisely why 0.1 + 0.2 = 0.30000000000000004. I know
that the floating arithmetic needs binary truncations.
I have do this first : keeping power of 2 until 2**{-52}.
*================================*
*k = var('k')*
*s_1 = sum(2**(-2-4*k), k, 0, 12)*
*s_2 = sum(2**(-5-4*k), k, 0, 11)*
*s = s_1 + s_2*
*print "s =", s.n(digits = 17)*
*--------------------------------*
*s = 0.29999999999999982*
*================================*
We understand the approximation but we do not have the one expected. So I
look at the binary approximations from Sage point of view.
*======================================================================*
*approx_1 = 0.1.str(base=2)*
*approx_2 = 0.2.str(base=2)*
*approx_sum = (0.1+0.2).str(base=2)*
*approx_3 = 0.3.str(base=2)*
*print " 0.1 =", approx_1*
*print " 0.2 =", approx_2*
*print "0.1 + 0.2 =", approx_sum*
*print " 0.3 =", approx_3*
*----------------------------------------------------------------------*
* 0.1 = 0.00011001100110011001100110011001100110011001100110011010*
* 0.2 = 0.0011001100110011001100110011001100110011001100110011010*
*0.1 + 0.2 = 0.010011001100110011001100110011001100110011001100110100*
* 0.3 = 0.010011001100110011001100110011001100110011001100110011*
*======================================================================*
The approximations for 0.1 and 0.2 have more bits that the two other ones,
and this output do not allow to do by hand the evaluation of the
approximation of 0.1 + 0.2.
What I'm doing wrong in my investigations ?
Christophe BAL
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