Average Error: 0.8 → 0.1
Time: 5.4s
Precision: binary64
Cost: 45504
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)
(FPCore (re im) :precision binary64 (/ (atan2 im re) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (*
  (sqrt (/ 1.0 (sqrt (log 10.0))))
  (* (atan2 im re) (pow (/ 1.0 (sqrt (log 10.0))) 1.5))))
double code(double re, double im) {
	return atan2(im, re) / log(10.0);
}
↓
double code(double re, double im) {
	return sqrt(1.0 / sqrt(log(10.0))) * (atan2(im, re) * pow((1.0 / sqrt(log(10.0))), 1.5));
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.0
Cost97024
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \left(\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}\right)\]
Alternative 2
Error30.6
Cost96832
\[\frac{\frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}\]
Alternative 3
Error1.1
Cost90752
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}} \cdot \left(\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\right)\right)\]
Alternative 4
Error30.4
Cost90688
\[\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\right)\]
Alternative 5
Error1.3
Cost90624
\[\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}} \cdot \sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}}{\sqrt{\log 10}}\]
Alternative 6
Error2.2
Cost77696
\[\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\log 10}}\]
Alternative 7
Error30.8
Cost64896
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\tan^{-1}_* \frac{im}{re}} \cdot \left(\sqrt{\tan^{-1}_* \frac{im}{re}} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)\right)\]
Alternative 8
Error2.0
Cost64768
\[\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}}\]
Alternative 9
Error1.5
Cost58496
\[\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}} \cdot \left(\sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}} \cdot \sqrt[3]{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}}\right)\]
Alternative 10
Error6.3
Cost58368
\[\sqrt[3]{{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)\right)}^{3}}\]
Alternative 11
Error0.9
Cost58368
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)\]
Alternative 12
Error30.8
Cost51904
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}}\right)\]
Alternative 13
Error1.9
Cost45440
\[\left(\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}\right) \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\log 10}\]
Alternative 14
Error1.9
Cost45440
\[\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\frac{\log 10}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}}\]
Alternative 15
Error0.8
Cost45376
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}}{\sqrt{\sqrt{\log 10}}}\]
Alternative 16
Error30.4
Cost38976
\[\sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}} \cdot \sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}}\]
Alternative 17
Error0.8
Cost32640
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)\]
Alternative 18
Error1.0
Cost32640
\[\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\]
Alternative 19
Error0.8
Cost32512
\[\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\]
Alternative 20
Error30.8
Cost32448
\[\frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\frac{\log 10}{\sqrt{\tan^{-1}_* \frac{im}{re}}}}\]
Alternative 21
Error30.8
Cost32448
\[\sqrt{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\log 10}\]
Alternative 22
Error6.3
Cost25920
\[\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\]
Alternative 23
Error11.3
Cost25856
\[\log \left(e^{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}}\right)\]
Alternative 24
Error1.0
Cost13184
\[\frac{1}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\]
Alternative 25
Error0.8
Cost13056
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
Alternative 26
Error58.2
Cost64
\[1\]
Alternative 27
Error58.0
Cost64
\[0\]
Alternative 28
Error58.1
Cost64
\[-1\]

Error

Derivation

  1. Initial program 0.8

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

  3. Applied add-sqr-sqrt_binary64_930.8

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

  4. Applied *-un-lft-identity_binary64_710.8

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

  5. Applied times-frac_binary64_770.8

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

  7. Applied div-inv_binary64_680.8

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

  9. Applied add-sqr-sqrt_binary64_930.8

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

  10. Applied associate-*l*_binary64_120.8

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

  11. Simplified0.1

    \[\leadsto \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)}\]

  12. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)}\]

  13. Final simplification0.1

    \[\leadsto \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)\]

Reproduce

herbie shell --seed 2021042 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))