precision - Why is Sin[2.0 Pi] = -2.44929 x 10^-16 and not 0.0?

Why does Sin[2.0 Pi] evaluate to -2.44929 x 10^-16 and not 0.0?

Answer

If you use inexact arithmetic, you must accept that you will get inexact results, that is, an approximation to the exact result. 2.0 can be represent exactly as a machine float, but 2.0 Pi cannot. Therefore, there is a small error in the argument given to Sin which propagates during evaluation and produces the result you see. High precision numerics is something of an art. Mathematica supplies tools to make mastering that art easier than used to be, but you still have apply yourself to the mastery.

First you must understand the difference between machine precision arithmetic (CPU floating point) and Mathematica's own arbitrary precision arithmetic. You get the first by calling N with one argument and the second by calling N with two arguments, the second being the precision you want to maintain.

You can also activate arbitrary precision arithmetic by indicating the precision of all the numerical quantities appearing in a calculation. If no precision is explicitly indicated, you get machine CPU precision. Applied to the computation given in your example, these remarks play out as follows;

Sin[2.0 Pi] (* default machine precision *)

-2.44929*10^-16

N[Sin[2 Pi]] (* exact computation reduced to machine precision *)

0.

Sin[2.0`10 Pi] (* computation done with 10-digits as the precision goal *)

0.*10^-10

N[Sin[2.0`10 Pi]]  (* the above reduced to machine precision *)

0.

The output from the 10-digit precision computation may look more precise than the default machine precision result, but it's not.

 Abs[Sin[2.0`10 Pi]] < Abs[Sin[2.0 Pi]]

False

However, if 30-digit precision is requested

 Abs[Sin[2.0`30 Pi]] < Abs[Sin[2.0 Pi]]

True