Average Error: 32.7 → 0.3
Time: 4.4s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \log \left(\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)
double f(double re, double im) {
        double r33724 = re;
        double r33725 = r33724 * r33724;
        double r33726 = im;
        double r33727 = r33726 * r33726;
        double r33728 = r33725 + r33727;
        double r33729 = sqrt(r33728);
        double r33730 = log(r33729);
        double r33731 = 10.0;
        double r33732 = log(r33731);
        double r33733 = r33730 / r33732;
        return r33733;
}
↓
double f(double re, double im) {
        double r33734 = 1.0;
        double r33735 = 10.0;
        double r33736 = log(r33735);
        double r33737 = sqrt(r33736);
        double r33738 = r33734 / r33737;
        double r33739 = re;
        double r33740 = im;
        double r33741 = hypot(r33739, r33740);
        double r33742 = pow(r33741, r33738);
        double r33743 = cbrt(r33742);
        double r33744 = r33743 * r33743;
        double r33745 = r33744 * r33743;
        double r33746 = log(r33745);
        double r33747 = r33738 * r33746;
        return r33747;
}

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 32.7

    \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity32.7

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

  4. Applied sqrt-prod32.7

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

  5. Simplified32.7

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

  6. Simplified0.6

    \[\leadsto \frac{\log \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)}{\log 10}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.6

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

  9. Applied pow10.6

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

  10. Applied pow10.6

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

  11. Applied pow-prod-down0.6

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

  12. Applied log-pow0.6

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

  13. Applied times-frac0.5

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

  15. Applied add-log-exp0.5

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

  16. Simplified0.3

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

  18. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)}\]

  19. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left(\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))