how does arithmetic exactly work in Calc?

Michel Onoff michel.onoff at
Thu Dec 3 12:48:25 UTC 2020


I would like to understand how simple arithmetic (+ - * /) works in Calc.
I kind of guess that the underlying internal number representation is
IEEE 754 double (64 bit).
I also guess that when a double is shown on the sheet it is approximated
by a decimal with at most 15 significand digits.

To reproduce the behavior below, use scientific notation with 20 digits
or more.

I have two slightly different number x and x'.
x is 2^-49 (a formula) while x' is 1.77635683940025E-15 (a literal).
Their decimal representation appear equal on the sheet, but they are,
indeed, slightly different internally. You can set them apart as follows:

y = x - 1.7763568394002E-15
y' = x' - 1.7763568394002E-15

that is, by subtracting the same number from x and x'.
y and y' appear differently on the sheet, meaning that x and x' are
different internally to start with.

However, x - x' is exactly 0. In IEEE 754 arithmetic, two numbers are
equal if and only if their difference is 0. That would mean that x and
x' are equal, which they are not from the above.

So, what is the exact model for Calc's arithmetic?
Which calculations are performed exactly on fundamental arithmetic


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