Average Error: 0.9 → 0.5
Time: 23.1s
Precision: 64
Internal Precision: 576
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \le -0.6821881769208953:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \frac{\tan^{-1}_* \frac{im}{re}}{\log 10})} - 1)^*\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (atan2 im re) (log 10)) < -0.6821881769208953

    1. Initial program 1.0

      \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube1.6

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    4. Applied add-cbrt-cube1.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \tan^{-1}_* \frac{im}{re}}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

    5. Applied cbrt-undiv0.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \tan^{-1}_* \frac{im}{re}}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    6. Simplified0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}}\]

    if -0.6821881769208953 < (/ (atan2 im re) (log 10))

    1. Initial program 0.8

      \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
    2. Using strategy rm

    3. Applied expm1-log1p-u0.8

      \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{\tan^{-1}_* \frac{im}{re}}{\log 10})} - 1)^*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \le -0.6821881769208953:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \frac{\tan^{-1}_* \frac{im}{re}}{\log 10})} - 1)^*\\ \end{array}\]

Runtime

Time bar (total: 23.1s)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))