Catastrophic cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one  long and the other  long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be  and . These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, . However, if the approximate lengths are subtracted, the difference will be , even though the true difference between the lengths is . The difference of the approximations, , is in error by 100% of the magnitude of the difference of the 
true values, . Catastrophic cancellation isn't affected by how large the inputs are—it applies just as much to large and small inputs. It depends only on how large the difference is, and on the error of the inputs. Exactly the same error would arise by subtracting  from  as approximations to  and , or by subtracting  from  as approximations to  and . Catastrophic cancellation may happen even if the difference is computed exactly, as in the example above—it is not a property of any particular kind of arithmetic like floating-point arithmetic; rather, it is inherent to subtraction, when the inputs are approximations themselves. Indeed, in floating-point arithmetic, when the inputs are close enough, the floating-point difference is computed exactly, by the Sterbenz lemma—there is no rounding error introduced by the floating-point subtraction operation. Formal analysis 
Formally, catastrophic cancellation happens because subtraction is ill-conditioned at nearby inputs: even if approximations  and  have small relative errors  and  from true values  and , respectively, the relative error of the difference  of the approximations from the difference  of the true values is inversely proportional to the difference of the true values:

Thus, the relative error of the exact difference  of the approximations from the difference  of the true values is

which can be arbitrarily large if the true values  and  are close. In numerical algorithms 
Subtracting nearby numbers in floating-point arithmetic does not always cause catastrophic cancellation, or even any error—by the Sterbenz lemma, if the numbers are close enough the floating-point difference is exact. But cancellation may amplify errors in the inputs that arose from rounding in other floating-point arithmetic. Example: Difference of squares 
Given numbers  and , the naive attempt to compute the mathematical function

by the floating-point arithmetic

is subject to catastrophic cancellation when  and  are close in magnitude, because the subtraction can expose the rounding errors in the squaring. The alternative factoring
,
evaluated by the floating-point arithmetic
,
avoids catastrophic cancellation because it avoids introducing rounding error leading into the subtraction. For example, if

and
,
then the true value of the difference

is
. In IEEE 754 binary64 arithmetic, evaluating the alternative factoring

gives the correct result exactly (with no rounding), but evaluating the naive expression

gives the floating-point number
,
of which less than half the digits are correct and the other (underlined) digits reflect the missing terms , lost due to rounding when calculating the intermediate squared values. Example: Complex arcsine 
When computing the complex arcsine function, one may be tempted to use the logarithmic formula directly:

However, suppose  for . Then  and ; call the difference between them —a very small difference, nearly zero. If  is evaluated in floating-point arithmetic giving

with any error , where  denotes floating-point rounding, then computing the difference

of two nearby numbers, both very close to , may amplify the error  in one input by a factor of —a very large factor because  was nearly zero. For instance, if , the true value of  is approximately , but using the naive logarithmic formula in IEEE 754 binary64 arithmetic may give , with only five out of sixteen digits correct and the remainder (underlined) all garbage. In the case of  for , using the identity  avoids cancellation because  but , so the subtraction is effectively addition with the same sign which does not cancel. Example: Radix conversion 

Numerical constants in software programs are often written in decimal, such as in the C fragment double x = 1.000000000000001; to declare and initialize an IEEE 754 binary64 variable named x. However,  is not a binary64 floating-point number; the nearest one, which x will be initialized to in this fragment, is . Although the radix conversion from decimal floating-point to binary floating-point only incurs a small relative error, catastrophic cancellation may amplify it into a much larger one:

double x = 1.000000000000001;  // rounded to 1 + 5*2^{-52}
double y = 1.000000000000002;  // rounded to 1 + 9*2^{-52}
double z = y - x;              // difference is exactly 4*2^{-52}

The difference  is . The relative errors of x from  and of y from  are both below , and the floating-point subtraction y - x is computed exactly by the Sterbenz lemma. But even though the inputs are good approximations, and even though the subtraction is computed exactly, the difference of the approximations  has a relative error of over  from the difference  of the original values as written in decimal: catastrophic cancellation amplified a tiny error in radix conversion into a large error in the output. Benign cancellation 

Cancellation is sometimes useful and desirable in numerical algorithms. For example, the 2Sum and Fast2Sum algorithms both rely on such cancellation after a rounding error in order to exactly compute what the error was in a floating-point addition operation as a floating-point number itself. The function
,
if evaluated naively at points
,
will lose most of the digits of  in rounding
. However, the function

itself is well-conditioned at inputs near . Rewriting it as

exploits cancellation in

to avoid the error from

evaluated directly. This works because the cancellation in the numerator

and the cancellation in the denominator

counteract each other; the function

is well-enough conditioned near zero that

gives a good approximation to
,
and thus

gives a good approximation to
. References 

Numerical analysis