HP -15C Advanced Functions Handbook page 76

74 Section 3: Calculating in Complex Mode
Y
To require approximations with accurate components is to demand
more than keeping relative errors small. For example, consider this
problem in which the calculations are done with four significant
digits. It illustrates the limitations imposed on a complex
calculation by finite accuracy.
2
2
= Z
2
= 37.5+ 37.3*
and
"1 X ^2
= (37.10 X 37.50 - 37.30 X 37.30) + z(37.10 X 37.30 + 37.30 X 37.50)
= (1391. - 1391.) +i(1384. + 1399.)
= 0 + /(2783.)
The true value z^z
2
= -0.04 + 2782.58*. Even though Z
l
and Z
2
have
no error, the real part of their four-digit product has no correct
significant decimals, although the relative error of the complex
product is less than 2 X 10~
4
.
The example illustrates that complex multiplication doesn't
propogate its errors componentwise. But even if complex
multiplication produced accurate components, the rounding errors
of a chain computation could quickly produce inaccurate
components. On the other hand, the relative error, which
corresponds to the precision of the calculation, grows only slowly.