Average Error: 31.9 → 0.3
Time: 8.6s
Precision: binary64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]
↓
\[-\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{-\log base}\right)\right) \]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
-\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{-\log base}\right)\right)

(FPCore (re im base)
 :precision binary64
 (/
  (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))
↓
(FPCore (re im base)
 :precision binary64
 (- (log1p (expm1 (/ (atan2 im re) (- (log base)))))))
double code(double re, double im, double base) {
	return ((atan2(im, re) * log(base)) - (log(sqrt((re * re) + (im * im))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0));
}
↓
double code(double re, double im, double base) {
	return -log1p(expm1(atan2(im, re) / -log(base)));
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

    \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0} \]

  2. Simplified0.3

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

  3. Taylor expanded in base around inf 0.3

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

  4. Applied add-cube-cbrt_binary640.3

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

  5. Applied add-cube-cbrt_binary640.3

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

  6. Applied times-frac_binary640.3

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

  7. Applied log-prod_binary640.4

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

  8. Simplified0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\log \left(\sqrt[3]{base}\right) \cdot -2} + \log \left(\frac{\sqrt[3]{1}}{\sqrt[3]{base}}\right)} \]

  9. Simplified0.4

    \[\leadsto -1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\sqrt[3]{base}\right) \cdot -2 + \color{blue}{\left(-\log \left(\sqrt[3]{base}\right)\right)}} \]

  10. Applied log1p-expm1-u_binary640.4

    \[\leadsto -1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\sqrt[3]{base}\right) \cdot -2 + \left(-\log \left(\sqrt[3]{base}\right)\right)}\right)\right)} \]

  11. Simplified0.3

    \[\leadsto -1 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{-\log base}\right)}\right) \]

  12. Final simplification0.3

    \[\leadsto -\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{-\log base}\right)\right) \]

Reproduce

herbie shell --seed 2022067 
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))