Average Error: 31.7 → 0.2
Time: 6.8s
Precision: binary64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
↓
\[\begin{array}{l} t_0 := \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right)\\ t_0 + t_0 \end{array} \]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
t_0 := \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right)\\
t_0 + t_0
\end{array}

(FPCore (re im)
 :precision binary64
 (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (log
          (pow
           (pow (hypot re im) (sqrt (/ 1.0 (log 10.0))))
           (/ (/ 1.0 (sqrt (log 10.0))) 2.0)))))
   (+ t_0 t_0)))
double code(double re, double im) {
	return log(sqrt((re * re) + (im * im))) / log(10.0);
}
↓
double code(double re, double im) {
	double t_0 = log(pow(pow(hypot(re, im), sqrt(1.0 / log(10.0))), ((1.0 / sqrt(log(10.0))) / 2.0)));
	return t_0 + t_0;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

  3. Applied add-sqr-sqrt_binary640.6

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}} \]

  4. Applied pow1_binary640.6

    \[\leadsto \frac{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \]

  5. Applied log-pow_binary640.6

    \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}} \]

  6. Applied times-frac_binary640.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]

  7. Applied add-log-exp_binary640.6

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\right)} \]

  8. Simplified0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)} \]

  9. Applied add-log-exp_binary640.3

    \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}\right)} \]

  10. Simplified0.2

    \[\leadsto \log \color{blue}{\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)} \]

  11. Applied sqr-pow_binary640.2

    \[\leadsto \log \color{blue}{\left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)} \cdot {\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right)} \]

  12. Applied log-prod_binary640.2

    \[\leadsto \color{blue}{\log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right) + \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right)} \]

  13. Final simplification0.2

    \[\leadsto \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right) + \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right) \]

Reproduce

herbie shell --seed 2022082 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))