Every book on numerical methods studies the precision of the algorithms as if they will be executed on a machine with infinite precision. Apparently, the effects of using floating point arithmetic (with finite representation) are not very important since they are neglected in the textbooks. However, I guess that these effects must have been studied somewhere. Could you please provide me with a reference?
Updated. Changed the wording of the question as suggested by NoChance.
On
Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations by Forman S. Acton includes a lot of practical advice about preserving significant figures in calculations.
However, I don't think there is much in Acton about formal error analysis in the sense of propagation of errors. For that, you might consider Rounding Errors in Algebraic Processes by J.H. Wilkinson.
Numerical recipes by Press, Teukolsky and two others does do the analysis including the effects of machine-epsilon rounding.