Average Error: 0.9 → 0.1
Time: 8.8s
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
Error1.1
Cost109952
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}\right)\]
Alternative 2
Error1.3
Cost109824
\[\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\log 10}}\]
Alternative 3
Error0.2
Cost97152
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}\right)\right)\right)\]
Alternative 4
Error1.0
Cost97024
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\]
Alternative 5
Error30.1
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 6
Error1.4
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 7
Error2.4
Cost90496
\[\frac{\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt[3]{\log 10}}}\]
Alternative 8
Error2.3
Cost84032
\[\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\frac{{\left(\sqrt{\sqrt{\log 10}}\right)}^{3}}{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt{\log 10}}}\]
Alternative 9
Error0.9
Cost77760
\[\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\]
Alternative 10
Error1.1
Cost77568
\[\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{\sqrt[3]{\sqrt{\log 10}}}\]
Alternative 11
Error30.3
Cost64832
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\right)\]
Alternative 12
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 13
Error6.7
Cost58368
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt[3]{{\left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\right)}^{3}}\]
Alternative 14
Error0.9
Cost58368
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\]
Alternative 15
Error0.1
Cost45504
\[\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}\]
Alternative 16
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 17
Error0.8
Cost45376
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\sqrt{\log 10}}\right)}^{1.5}}{\sqrt{\sqrt{\log 10}}}\]
Alternative 18
Error7.4
Cost45376
\[\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)}^{3}}\]
Alternative 19
Error30.1
Cost38976
\[\sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}} \cdot \sqrt{\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}}\]
Alternative 20
Error0.8
Cost32640
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)\]
Alternative 21
Error1.0
Cost32640
\[\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\]
Alternative 22
Error0.8
Cost32512
\[\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\]
Alternative 23
Error30.5
Cost32448
\[\frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\frac{\log 10}{\sqrt{\tan^{-1}_* \frac{im}{re}}}}\]
Alternative 24
Error30.5
Cost32448
\[\sqrt{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt{\tan^{-1}_* \frac{im}{re}}}{\log 10}\]
Alternative 25
Error0.9
Cost32384
\[\frac{\frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\]
Alternative 26
Error6.7
Cost25920
\[\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\]
Alternative 27
Error30.8
Cost25856
\[e^{\log \left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}\]
Alternative 28
Error1.0
Cost13184
\[\frac{1}{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\]
Alternative 29
Error0.9
Cost13056
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
Alternative 30
Error58.1
Cost64
\[1\]
Alternative 31
Error58.2
Cost64
\[0\]
Alternative 32
Error58.2
Cost64
\[-1\]

Error

Derivation

  1. Initial program 0.9

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

  3. Applied add-sqr-sqrt_binary64_11230.9

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

  4. Applied *-un-lft-identity_binary64_11010.9

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

  5. Applied times-frac_binary64_11070.8

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

  7. Applied div-inv_binary64_10980.8

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

  9. Applied add-sqr-sqrt_binary64_11230.8

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

  10. Applied associate-*l*_binary64_10420.8

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

  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 2021022 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10.0)))