I am trying to understand the effects of round off error and floating point arithmetic on my research. I am reading through "What Every Computer Scientist Should Know About Floating-Point Arithmetic" and have some questions on my understanding of it and how it applies to SciPy/NumPy. My specific problem has to do with trying to find the roots of (sinh(b)+sin(b))**2-(sinh(b)-sin(b))**2 and other problematic expressions. (Acutually it is more like the determinant of mat[0,0]=(a/b)*(sinh(b)+sin(b) mat[0,1]=-a**2*d/(b**3)*(sinh(b)-sin(b)) mat[1,0]=b/d*(sinh(b)-sin(b)) mat[1,1]=-(a/b)*(sinh(b)-sin(b)) It turns out that with some symbolic manipulation, this can be rearranged to trying to find the roots of sinh(b)*sin(b) which is much better behaved. How will SciPy handle sinh(b)+ (or -) sin(b) when sinh(b) is very large and sin(b) is obviously between +/-1? As I understand it, the IEEE standard requires exact rounding for this operation which seems difficult and costly to me. What I am seeing is that when the determinant whose zeros I need to find is symbolically rearrange to sinh(b)*sin(b), the curve is smooth and has well defined places where it goes very near zero (I actually need to look for minima of the abs of this value in real problems because the results are complex valued). Note that this symbolic rearrangement might only be possible for simple systems. I am analyzing this simple system to get a handle on these errors. If I don't do the symbolic rearrangement, the determinant curve is very noisy for increasing b and the determinant might have a local minima, but that minima is not very close to zero. Is there a way to create a single precision number and do single precision math so that I can compare good and bad answers to each step in this problem (i.e. for a certain range of b, I could consider the double precision value to be "exact" and see what problems are arising because of round-off). I am trying to justify a pragmatic approach to this problem. Assuming that the expression is too complicated to look at and pick out floating-point problems and rearrange it, I want to be able to say that I can trust the roots (actually minima) if the following are true: 1. You plot the function you are trying to find minima of and it doesn't look noisy over the range of input values you are concerned with 2. The value of the expression at the minima is actually fairly small 3. Numerical searching starting at points in a region around the minima all converge to the same minima (basically there is a sort of numerical domain of attraction) Does this seem valid? I am a mechanical engineer and neither a computer scientist nor a mathematician. So, I don't know if I can or want to prove this rigidly, but I want to be convinced and convince my thesis committee at least that it is trust-worthy. My biggest fear is this: is there a case where floating-point precision problems would lead to well-behaved, stable, WRONG answers? Especially for cases involving trying to find small differences between large numbers like (a-b)**2-(a+b) for a>>>>b. Thanks for any thoughts on how to think about this, Ryan