Average Error: 0.8 → 0.7
Time: 15.2s
Precision: 64
Internal Precision: 320
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\log_* (1 + (e^{\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)} - 1)^*)\]

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. Initial program 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]

  2. Initial simplification0.8

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

  4. Applied log1p-expm1-u0.7

    \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}} - 1)^*)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.7

    \[\leadsto \log_* (1 + (e^{\frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}} - 1)^*)\]

  7. Applied *-un-lft-identity0.7

    \[\leadsto \log_* (1 + (e^{\frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}} - 1)^*)\]

  8. Applied times-frac0.7

    \[\leadsto \log_* (1 + (e^{\color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}} - 1)^*)\]
  9. Using strategy rm

  10. Applied div-inv0.7

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

  11. Final simplification0.7

    \[\leadsto \log_* (1 + (e^{\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)} - 1)^*)\]

Runtime

Time bar (total: 15.2s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))