Average Error: 31.8 → 0.3
Time: 5.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)
double f(double re, double im) {
        double r90392 = re;
        double r90393 = r90392 * r90392;
        double r90394 = im;
        double r90395 = r90394 * r90394;
        double r90396 = r90393 + r90395;
        double r90397 = sqrt(r90396);
        double r90398 = log(r90397);
        double r90399 = 10.0;
        double r90400 = log(r90399);
        double r90401 = r90398 / r90400;
        return r90401;
}
↓
double f(double re, double im) {
        double r90402 = 1.0;
        double r90403 = 10.0;
        double r90404 = log(r90403);
        double r90405 = sqrt(r90404);
        double r90406 = r90402 / r90405;
        double r90407 = re;
        double r90408 = im;
        double r90409 = hypot(r90407, r90408);
        double r90410 = pow(r90409, r90406);
        double r90411 = log(r90410);
        double r90412 = r90406 * r90411;
        return r90412;
}

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 31.8

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

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

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

  4. Applied sqrt-prod31.8

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

  5. Simplified31.8

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

  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-exp0.6

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))