Trigonometric Functions of Real Radian Angles

[David Hough]

Trigonometric Functions of Real Radian Angles is the name of a section
in the HP-15C Advanced Functions Handbook, page 184, August 1982,
part number 00015-90011.  
This booklet is really an outstandingly readable graduate
course in practical error analysis.  
Some will recognize the style as that of W. Kahan.

The HP-15C uses an internal
value p.13 consisting of a 13-digit approximation to pi that can't
be entered on the calculator, which is limited to 10 significant digits
of input.   Two excerpts from page 186:

 ...
In this calculation the phase shift introduced by p.13 != pi has less
effect than changing the distance between Earth and moon by as little
as the thickness of this page.
 ...
It all seems like much ado about very little.  After a blizzard of
formulas and examples, we conclude that the error caused by p.13 != pi
is negligible for engineering purposes, so we need not have bothered
to know about it.  That is the burden that conscientious error analysts
must bear; if they merely took for granted that small errors are
negligible, they might be wrong.