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- W2056837298 abstract "The hypotenuse function, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=z equals StartRoot x squared plus y squared EndRoot> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msqrt> </mml:mrow> <mml:annotation encoding=application/x-tex>z=sqrt {x^2+y^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is sometimes included in math library packages. Assuming that it is being computed by a straightforward algorithm, in a binary floating point environment, with round to nearest rounding mode, a sharp roundoff error bound is derived, for arbitrary precision. For IEEE single precision, or higher, the bound implies that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue z overbar minus z EndAbsoluteValue greater-than 1.222 u l p left-parenthesis z right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mover> <mml:mi>z</mml:mi> <mml:mo accent=false>¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo>−<!-- − --></mml:mo> <mml:mi>z</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1.222</mml:mn> <mml:mspace width=thinmathspace /> <mml:mi>u</mml:mi> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|overline z-z|>1.222, ulp(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue z overbar minus z EndAbsoluteValue greater-than 1.222 u l p left-parenthesis z overbar right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mover> <mml:mi>z</mml:mi> <mml:mo accent=false>¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo>−<!-- − --></mml:mo> <mml:mi>z</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1.222</mml:mn> <mml:mspace width=thinmathspace /> <mml:mi>u</mml:mi> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mover> <mml:mi>z</mml:mi> <mml:mo accent=false>¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|overline z-z|>1.222, ulp(overline z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Numerical experiments indicate that this bound is sharp and cannot be improved." @default.
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- W2056837298 date "1999-02-13" @default.
- W2056837298 modified "2023-09-25" @default.
- W2056837298 title "Sharp ULP rounding error bound for the hypotenuse function" @default.
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- W2056837298 doi "https://doi.org/10.1090/s0025-5718-99-01103-5" @default.
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