# 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]=∅.

### Your Input Converted to Double Precision

x= |
[1, 2] | [1, 2] | [1, 1 × 2^{1}] |

y= |
[1, 2] | [1, 2] | [1, 1 × 2^{1}] |

z= |
[1, 2] | [1, 2] | [1, 1 × 2^{1}] |

### pow (IEEE Std 1788-2015)

pow(x, y)⊆ |
[1, 4] |
[1, 4] |
[1, 1 × 2^{2}] |

pow⁻₁(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

pow⁻₂(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

### pow1

pow1(x, y)⊆ |
[1, 4.0000000000000018] |
[1, 4.0000000000000017763568394002504646778106689453125] |
[1, 2251799813685249 × 2^{-49}] |

pow1⁻₁(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

pow1⁻₂(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

### pow2

pow2(x, y)⊆ |
[1, 4.0000000000000018] |
[1, 4.0000000000000017763568394002504646778106689453125] |
[1, 2251799813685249 × 2^{-49}] |

in GNU Octave |
[1, 4] |
[1, 4] |
[1, 1 × 2^{2}] |

pow2⁻₁(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

pow2⁻₂(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

### pow3

pow3(x, y)⊆ |
[1, 4.0000000000000018] |
[1, 4.0000000000000017763568394002504646778106689453125] |
[1, 2251799813685249 × 2^{-49}] |

pow3⁻₁(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

pow3⁻₂(x, y, z)⊆ |
[1, 2] |
[1, 2] |
[1, 1 × 2^{1}] |

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.

The functions pow and pow2 provide best possible accuracy and the reverse operations of pow are also quite accurate. These functions have been implemented by the GNU Octave interval package as free software.