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Fungrim entry: fa30c7

∣atan⁡ ⁣(x+y)−atan⁡(x)∣≤∣y∣1+(max⁡ ⁣(0,∣x∣−∣y∣))2\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}(x)\right| \le \frac{\left|y\right|}{1 + {\left(\max\!\left(0, \left|x\right| - \left|y\right|\right)\right)}^{2}}
Assumptions:x∈R  and⁡  y∈Rx \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}
TeX:
\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}(x)\right| \le \frac{\left|y\right|}{1 + {\left(\max\!\left(0, \left|x\right| - \left|y\right|\right)\right)}^{2}}

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}
Definitions:
Fungrim symbol Notation Short description
Abs∣z∣\left|z\right| Absolute value
Atanatan⁡(z)\operatorname{atan}(z) Inverse tangent
Powab{a}^{b} Power
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("fa30c7"),
    Formula(LessEqual(Abs(Sub(Atan(Add(x, y)), Atan(x))), Div(Abs(y), Add(1, Pow(Max(0, Sub(Abs(x), Abs(y))), 2))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR))))

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2021-03-15 19:12:00.328586 UTC