c++ - Computer precision: when should I have to worry about it? -

in c++ programming, when need worry precision issue? take small example (it might not perfect 1 though),

std::vector<double> first (50000, 0.0); std::vector<double> second (first); 

could possible second[619] = 0.00000000000000000000000000001234 (i mean small value). or sum = second[0]+second[1]+...+second[49999] => 1e-31? or sum = second[0]-second[1]-...-second[49999] => -7.987654321e-12?

my questions:

1. could small disturbances in working double type numbers?
2. what may cause these kind of small disturbances? i.e. rounding errors become large? please list them? how take precautions?
3. if there small disturbance in operations, mean after these operations, using if (sum == 0) dangerous? 1 should use if (sum < small) instead, small defined small value, such 1e-30?
4. lastly, small disturbances result negative value? because if possible, should better use if (abs(sum) < small) instead.

any experiences?

this reference document floating point precision: what every computer scientist should know floating-point arithmetic

one of more important parts catastrophic cancellation

catastrophic cancellation occurs when operands subject rounding errors. example in quadratic formula, expression b2 - 4ac occurs. quantities b2 , 4ac subject rounding errors since results of floating-point multiplications. suppose rounded nearest floating-point number, , accurate within .5 ulp. when subtracted, cancellation can cause many of accurate digits disappear, leaving behind digits contaminated rounding error. hence difference might have error of many ulps. example, consider b = 3.34, = 1.22, , c = 2.28. exact value of b2 - 4ac .0292. b2 rounds 11.2 , 4ac rounds 11.1, hence final answer .1 error 70 ulps, though 11.2 - 11.1 equal .16. subtraction did not introduce error, rather exposed error introduced in earlier multiplications.

benign cancellation occurs when subtracting known quantities. if x , y have no rounding error, theorem 2 if subtraction done guard digit, difference x-y has small relative error (less 2).

a formula exhibits catastrophic cancellation can rearranged eliminate problem. again consider quadratic formula