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First, when you read 6.00000000000000, you should understand that this not the value of the number (which is a floating point number having 53 bits of precision), but a string representation of it, as a decimal expansion (not binary). In your third case, the value of N(log(64)/log(2)) is less that 6, as you can check by typing:

sage: a = N(log(64)/log(2))
sage: a < 6 
True
sage: a.exact_rational()
6755399441055743/1125899906842624
sage: a.sign_mantissa_exponent()
(1, 6755399441055743, -50)

In the first two cases, you can check that the first two values are equal to 6.

Now, where could this difference come from ? The first two values are the same since N(64) is just the conversion of the integer 64 into an element of RealField(53), which is the same as 64.0. What about the third ? log(64)/log(2) is not a floating point number but an element of the Symbolic Ring, as you can check by typing:

sage: (log(64)/log(2)).parent()     
Symbolic Ring

Now, when you type N(log(64)/log(2)), you ask Sage to convert this element of the Symbolic Ring to an element of the RealField(53). Hence, it seems that the rounding from the Symbolic Ring to RealField(53) is not done to the nearest floating point number, but toward zero:

sage: N(log(64)/log(2)) < 6 
True
sage: N(-log(64)/log(2)) > -6
True

It looks like to be the same problem that appeared for the conversion from the Rational Field to the Real Double Field, see trac ticket 14416 and ask question 2444.

First, when you read 6.00000000000000, you should understand that this not the value of the number (which is a floating point number having 53 bits of precision), but a string representation of it, as a decimal expansion (not binary). In your third case, the value of N(log(64)/log(2)) is less that 6, 6 (though very close, hence the decimal representation), as you can check by typing:

sage: a = N(log(64)/log(2))
sage: a < 6 
True
sage: a.exact_rational()
6755399441055743/1125899906842624
sage: a.sign_mantissa_exponent()
(1, 6755399441055743, -50)

In the first two cases, you can check that the first two values are equal to 6.

Now, where could this difference come from ? The first two values are the same since N(64) is just the conversion of the integer 64 into an element of RealField(53), which is the same as 64.0. What about the third ? log(64)/log(2) is not a floating point number but an element of the Symbolic Ring, as you can check by typing:

sage: (log(64)/log(2)).parent()     
Symbolic Ring

Now, when you type N(log(64)/log(2)), you ask Sage to convert this element of the Symbolic Ring to an element of the RealField(53). Hence, it seems that the rounding from the Symbolic Ring to RealField(53) is not done to the nearest floating point number, but toward zero:

sage: N(log(64)/log(2)) < 6 
True
sage: N(-log(64)/log(2)) > -6
True

It looks like to be the same a similar problem that the one that appeared for the conversion from the Rational Field to the Real Double Field, see trac ticket 14416 and ask question 2444.

First, when you read 6.00000000000000, you should understand that this not the value of the number (which is a floating point number having 53 bits of precision), but a string representation of it, as a decimal expansion (not binary). In your third case, the value of N(log(64)/log(2)) is less that 6 (though very close, hence the decimal representation), as you can check by typing:

sage: a = N(log(64)/log(2))
sage: a < 6 
True
sage: a.exact_rational()
6755399441055743/1125899906842624
sage: a.sign_mantissa_exponent()
(1, 6755399441055743, -50)

In the first two cases, you can check that the first two values are equal to 6.

Now, where could this difference come from ? The first two values are the same since N(64) is just the conversion of the integer 64 into an element of RealField(53), which is the same as 64.0. What about the third ? log(64)/log(2) is not a floating point number but an element of the Symbolic Ring, as you can check by typing:

sage: (log(64)/log(2)).parent()     
Symbolic Ring

Now, when you type N(log(64)/log(2)), you ask Sage to convert this element of the Symbolic Ring to an element of RealField(53). Hence, it seems that the rounding from the Symbolic Ring to RealField(53) is not done to the nearest floating point number, but toward zero:

sage: N(log(64)/log(2)) < 6 
True
sage: N(-log(64)/log(2)) > -6
True

It looks like to be a similar problem that the one that appeared for the conversion from the Rational Field to the Real Double Field, see trac ticket 14416 and ask question 2444.

sage: a = log(64)/log(2)
sage: a.full_simplify()
6
sage: N((log(64)/log(2)).full_simplify()) == 6.0
True