Interval Power Function

Calculation with Double Precision

The interval boundaries have to be given in decimal notation, e.g., enter 0.5 for ½, or -3.2e6 for -3200000 respectively. Infinite boundaries can be entered as “inf” or “-inf”. Empty intervals have to be entered with inversed boundaries, e.g., [1,0]=∅.

x = [, ]
y = [, ]
z = [, ]

Your Input Converted to Double Precision

x=	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]
y=	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]
z=	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]

pow1

pow1(x, y)⊆	[1.0, 4.0000000000000018]	[1.0, 4.0000000000000017763568394002504646778106689453125]	[1, 2251799813685249 × 2-49]
pow1⁻₁(]-∞, ∞[, y, z)⊆	[1.0, 2.0000000000000005]	[1.0, 2.000000000000000444089209850062616169452667236328125]	[1, 4503599627370497 × 2-51]
pow1⁻₁(x, y, z)⊆	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]
pow1⁻₂(x, ]-∞, ∞[, z)⊆	]-∞, ∞[	]-∞, ∞[	]-∞, ∞[
pow1⁻₂(x, y, z)⊆	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]

pow2

pow2(x, y)⊆	[1.0, 4.0000000000000018]	[1.0, 4.0000000000000017763568394002504646778106689453125]	[1, 2251799813685249 × 2-49]
pow2⁻₁(]-∞, ∞[, y, z)⊆	[-1.4142135623730954, 2.0000000000000005]	[-1.414213562373095367519226783770136535167694091796875, 2.000000000000000444089209850062616169452667236328125]	[-3184525836262887 × 2-51, 4503599627370497 × 2-51]
pow2⁻₁(x, y, z)⊆	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]
pow2⁻₂(x, ]-∞, ∞[, z)⊆	]-∞, ∞[	]-∞, ∞[	]-∞, ∞[
pow2⁻₂(x, y, z)⊆	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]

pow3

pow3(x, y)⊆	[1.0, 4.0000000000000018]	[1.0, 4.0000000000000017763568394002504646778106689453125]	[1, 2251799813685249 × 2-49]
pow3⁻₁(]-∞, ∞[, y, z)⊆	[-2.0000000000000005, 2.0000000000000005]	[-2.000000000000000444089209850062616169452667236328125, 2.000000000000000444089209850062616169452667236328125]	[-4503599627370497 × 2-51, 4503599627370497 × 2-51]
pow3⁻₁(x, y, z)⊆	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]
pow3⁻₂(x, ]-∞, ∞[, z)⊆	]-∞, ∞[	]-∞, ∞[	]-∞, ∞[
pow3⁻₂(x, y, z)⊆	[1.0, 2.0]	[1.0, 2.0]	[1, 1 × 21]

A conversion to double precision of the interval boundaries may result in initial errors. The entered boundaries are rounded to double precision accordingly, i.e., upper/lower boundaries are rounded up/down. Output of interval boundaries is always exact and can therefore be very verbose: all relevant decimal digits are displayed.

The functions pow1, pow2, pow3, and their reverse functions have been implemented in PHP and their results are quite inaccurate. This is due to the lack of pow, log and exp functions with directed rounding in PHP.

Function Definitions

Point Functions

$\begin{array}{rl}\mathrm{pow1}:& \mathrm{ℝ⁺}\times ℝ\to ℝ\\ & (x,y)\mapsto \exp (y\cdot \log x)\\ \mathrm{pow2}:& (\mathrm{ℝ⁺}\times ℝ)\cup \left(\right\{0\}\times \mathrm{ℝ⁺})\cup (\mathrm{ℝ⁻}\times \mathbb{Z})\to ℝ\\ & (x,y)\mapsto \{\begin{array}{ll}\exp (y\cdot \log x)& \text{if }x\text{ positive,}\\ 0& \text{if }x\text{ zero,}\\ \exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y\text{ even,}\\ -\exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y\text{ odd.}\end{array}\\ \mathrm{pow3}:& (\mathrm{ℝ⁺}\times ℝ)\cup \left(\right\{0\}\times \mathrm{ℝ⁺})\cup (\mathrm{ℝ⁻}\times \{\frac{m}{n}|m,n\in \mathbb{Z}\text{, }n\text{ odd}\})\to ℝ\\ & (x,y)\mapsto \{\begin{array}{ll}\exp (y\cdot \log x)& \text{if }x\text{ positive,}\\ 0& \text{if }x\text{ zero,}\\ \exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y=\frac{m}{n}\text{ with }m\text{ even and }n\text{ odd,}\\ -\exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y=\frac{m}{n}\text{ with }m\text{ odd and }n\text{ odd.}\end{array}\end{array}$

Interval Functions

For $f\in \{\mathrm{pow1},\mathrm{pow2},\mathrm{pow3}\}$ it is
$$\begin{array}{rrl}\mathit{f}:& (\mathit{x},\mathit{y})& \mapsto \mathrm{hull}\left(\right\{\mathit{f}(x,y)\in ℝ|x\in \mathit{x}\text{ and }y\in \mathit{y}\})\text{,}\\ {\mathit{f}}_1^{-}:& (\mathit{x},\mathit{y},\mathit{z})& \mapsto \mathrm{hull}\left(\right\{x\in \mathit{x}|\text{there exists }y\in \mathit{y}\text{ >with }f(x,y)\in \mathit{z}\})\text{,}\\ {\mathit{f}}_2^{-}:& (\mathit{x},\mathit{y},\mathit{z})& \mapsto \mathrm{hull}\left(\right\{y\in \mathit{y}|\text{there exists }x\in \mathit{x}\text{ with }f(x,y)\in \mathit{z}\})\text{,}\end{array}$$
where
$$\begin{array}{rl}\mathrm{hull}:& \wp \left(ℝ\right)\to \wp \left(ℝ\right)\\ & \mathit{s}\mapsto \cap \{\mathit{x}\in \wp (ℝ\left)\right|\mathit{s}\subseteq \mathit{x}\text{ and }\mathit{x}\text{ is a closed real interval}\}\text{.}\end{array}$$