Search results
Results From The WOW.Com Content Network
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions.
While the real numbers are infinite and continuous, a floating-point number system is finite and discrete. Thus, representation error, which leads to roundoff error, occurs under the floating-point number system.
Missing values in a lookup table used by the FPU's floating-point division algorithm led to calculations acquiring small errors. While these errors would in most use-cases only occur rarely and result in small deviations from the correct output values, in certain circumstances the errors can occur frequently and lead to more significant deviations.
Even in this case, there is no guarantee that it would converge to the correct solution, because the floating-point round-off or truncation errors can be magnified, instead of damped, causing the deviation from the exact solution to grow exponentially.
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.
Arithmetic underflow can occur when the true result of a floating-point operation is smaller in magnitude (that is, closer to zero) than the smallest value representable as a normal floating-point number in the target datatype. [1] Underflow can in part be regarded as negative overflow of the exponent of the floating-point value. For example ...
The actual errors of machine arithmetic are far too complicated to be studied directly, so instead, the following simple model is used. The IEEE arithmetic standard says all floating-point operations are done as if it were possible to perform the infinite-precision operation, and then, the result is rounded to a floating-point number.
Floating-point operations other than ordered comparisons normally propagate a quiet NaN (qNaN). Most floating-point operations on a signaling NaN ( sNaN ) signal the invalid-operation exception ; the default exception action is then the same as for qNaN operands and they produce a qNaN if producing a floating-point result.