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Error in Ulps of the Multiplication or Division by a Correctly-Rounded Function or Constant in Binary Floating-Point Arithmetic | IEEE Journals & Magazine | IEEE Xplore

Error in Ulps of the Multiplication or Division by a Correctly-Rounded Function or Constant in Binary Floating-Point Arithmetic


Abstract:

Assume we use a binary floating-point arithmetic and that \operatorname{RN} is the round-to-nearest function. Also assume that c is a constant or a real function of o...Show More

Abstract:

Assume we use a binary floating-point arithmetic and that \operatorname{RN} is the round-to-nearest function. Also assume that c is a constant or a real function of one or more variables, and that we have at our disposal a correctly rounded implementation of c, say \hat{c}= \operatorname{RN}(c). For evaluating x \cdot c (resp. x / c or c / x), the natural way is to replace it by \operatorname{RN}(x \cdot \hat{c}) (resp. \operatorname{RN}(x / \hat{c}) or \operatorname{RN}(\hat{c}/ x)), that is, to call function \hat{c} and to perform a floating-point multiplication or division. This can be generalized to the approximation of n/d by \operatorname{RN}(\hat{n}/\hat{d}) and the approximation of n \cdot d by \operatorname{RN}(\hat{n} \cdot \hat{d}), where \hat{n} = \operatorname{RN}(n) and \hat{d} = \operatorname{RN}(d), and n and d are functions for which we have at our disposal a correctly rounded implementation. We discuss tight error bounds in ulps of such approximations. From our results, one immediately obtains tight error bounds for calculations such as \mathtt {x * pi}, \mathtt {ln(2)/x}, \mathtt {x/(y+z)}, \mathtt {(x+y)*z}, \mathtt {x/sqrt(y)}, \mathtt {sqrt(x)/{y}}, \mathtt {(x+y)(z+t)}, \mathtt {(x+y)/(z+t)}, \mathtt {(x+y)/(zt)}, etc. in floating-point arithmetic.
Published in: IEEE Transactions on Emerging Topics in Computing ( Volume: 12, Issue: 2, April-June 2024)
Page(s): 656 - 666
Date of Publication: 18 July 2023

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