Average Error: 31.8 → 0.5
Time: 8.3s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot \mathsf{hypot}\left(\log base, 0.0\right)}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot \mathsf{hypot}\left(\log base, 0.0\right)}
double code(double re, double im, double base) {
	return (((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0)));
}
↓
double code(double re, double im, double base) {
	return (fma(log(base), log(hypot(re, im)), (atan2(im, re) * 0.0)) / (hypot(log(base), 0.0) * hypot(log(base), 0.0)));
}

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 31.8

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity31.8

    \[\leadsto \frac{\log \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  4. Applied sqrt-prod31.8

    \[\leadsto \frac{\log \color{blue}{\left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  5. Simplified31.8

    \[\leadsto \frac{\log \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  6. Simplified0.5

    \[\leadsto \frac{\log \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

  9. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

  10. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

  11. Simplified0.5

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}}} \cdot \frac{\log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

  12. Simplified0.5

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1}} \cdot \color{blue}{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}\]
  13. Using strategy rm

  14. Applied frac-times0.5

    \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1} \cdot \left(\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1\right)}}\]

  15. Simplified0.5

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}}{\frac{\mathsf{hypot}\left(\log base, 0.0\right)}{1} \cdot \left(\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1\right)}\]

  16. Simplified0.5

    \[\leadsto \frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\color{blue}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot \mathsf{hypot}\left(\log base, 0.0\right)}}\]

  17. Final simplification0.5

    \[\leadsto \frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot \mathsf{hypot}\left(\log base, 0.0\right)}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))