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diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 4dc9095..a1d09e2 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1175,6 +1175,35 @@ A relevant reference is \cite{BrDiJeLeMeMuReStTo09}, especially Section 4.5 which discusses complex floating-point numbers, and gives error bounds for multiplication, division and square root. +\paragraph{Sign of zeroes.} +When the output value has a zero real or imaginary part, its sign should be +decided. When the inputs also have a zero real or imaginary part, we +consider all possible limits, and if all those limits give the same sign, +we take this as the sign of the zero part. +Otherwise, we round to $+0$, except for rounding toward $-\infty$, where we +round to $-0$. + +Example~: consider $x = 1 + 0 i$ and $y = -0 + 0 i$, we consider that $x$ +is the limit of $1 + \epsilon i$ for $\epsilon > 0$ tending to zero. +Similarly, $y = -\delta + \gamma i$ with $\delta, \gamma > 0$. +Now $\log x \approx \epsilon^2/2 + \epsilon i$, thus +$y \log x \approx (-\epsilon^2 \delta/2 - \epsilon \gamma) ++ i (\epsilon^2 \gamma/2 - epsilon \delta)$. +Thus if we neglect terms of order $3$ or more, +whatever the relative growth of $\epsilon, \delta, \gamma$, the real and +imaginary parts of $y \log x$ are negative, thus we decide $x^y$ is rounded +to $1 - 0 i$. + +For $x = 1 - 0 i$ and $y = 0 + 0 i$, we write +$x = 1 - \epsilon i$ and $y = \delta + \gamma i$, +which gives $\log x \approx \epsilon^2/2 - \epsilon i$ +and $y \log x \approx \epsilon \gamma - i \epsilon \delta + O(\epsilon^2)$, +thus we also round $x^y$ to $1 - 0 i$. + +% (1 -0)^(-0 -0): +% x = 1 - epsilon i, y = -delta-gamma*i +% y log(x) = -epsilon gamma + i epsilon delta + O(epsilon^2) + \bibliographystyle{acm} \bibliography{algorithms} |