How do I check and handle numbers very close to zero

Question

I have some math (in C++) which seems to be generating some very small, near zero, numbers (I suspect the trig function calls are my real problem), but I'd like to detect these cases so that I can study them in more detail.

I'm currently trying out the following, is it correct?

if ( std::abs(x) < DBL_MIN ) {
     log_debug("detected small num, %Le, %Le", x, y);
}

Second, the nature of the mathematics is trigonometric in nature (aka using a lot of radian/degree conversions and sin/cos/tan calls, etc), what sort of transformations can I do to avoid mathematical errors?

Obviously for multiplications I can use a log transform - what else?

Answer 1

Contrary to widespread belief, DBL_MIN is not the smallest positive double value but the smallest positive normalized double value. Typically - for 64-bit ieee754 doubles - it's 2^-1022, while the smallest positive double value is 2^-1074. Therefore

I'm currently trying out the following, is it correct?

if ( std::abs(x) < DBL_MIN ) {
     log_debug("detected small num, %Le, %Le", x, y);
}

may have an affirmative answer. The condition checks whether x is a denormalized (also called subnormal) number or ±0.0. Without knowing more about your specific situation, I cannot tell if that test is appropriate. Denormalized numbers can be legitimate results of calculations or the consequence of rounding where the correct result would be 0. It is also possible that rounding produces numbers of far greater magnitude than DBL_MIN when the mathematically correct result would be 0, so a much larger threshold could be sensible.