Average Error: 0.8 → 0.8
Time: 3.5s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\sqrt{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\sqrt[3]{1} \cdot \frac{\sqrt[3]{1}}{\sqrt{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}\right)
(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (*
  (* (cbrt 1.0) (/ (cbrt 1.0) (sqrt (log 10.0))))
  (* (/ (cbrt 1.0) (sqrt (log 10.0))) (atan2 im re))))
double code(double re, double im) {
	return (((double) atan2(im, re)) / ((double) log(10.0)));
}
↓
double code(double re, double im) {
	return ((double) (((double) (((double) cbrt(1.0)) * (((double) cbrt(1.0)) / ((double) sqrt(((double) log(10.0))))))) * ((double) ((((double) cbrt(1.0)) / ((double) sqrt(((double) log(10.0))))) * ((double) atan2(im, re))))));
}

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 Error: 0.8 bits

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

  3. Applied clear-numError: 1.0 bits

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

  5. Applied *-un-lft-identityError: 1.0 bits

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

  6. Applied add-sqr-sqrtError: 1.0 bits

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

  7. Applied times-fracError: 1.1 bits

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

  8. Applied add-cube-cbrtError: 1.1 bits

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

  9. Applied times-fracError: 1.0 bits

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

  10. SimplifiedError: 1.0 bits

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

  11. SimplifiedError: 0.8 bits

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

  12. Final simplificationError: 0.8 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))