Association for Computing Machinery (ACM), ACM Transactions on Mathematical Software, 4(39), p. 1-19, 2013
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A very simple test, introduced by Ziv, allows one to determine if an approximation to the value f(x) of an elementary function at a given point x suffices to return the floating-point number nearest f(x) . The same test may be used when implementing floating-point operations with input and output operands of different formats, using arithmetic operators tailored for manipulating operands of the same format. That test depends on a “magic constant” e . We show how to choose that constant e to make the test reliable and efficient. Various cases are considered, depending on the availability of an fma instruction, and on the range of f(x) .