Average Error: 31.0 → 0.4
Time: 30.0s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]
double f(double re, double im) {
        double r796478 = re;
        double r796479 = r796478 * r796478;
        double r796480 = im;
        double r796481 = r796480 * r796480;
        double r796482 = r796479 + r796481;
        double r796483 = sqrt(r796482);
        double r796484 = log(r796483);
        double r796485 = 10.0;
        double r796486 = log(r796485);
        double r796487 = r796484 / r796486;
        return r796487;
}
↓
double f(double re, double im) {
        double r796488 = 1.0;
        double r796489 = 10.0;
        double r796490 = log(r796489);
        double r796491 = sqrt(r796490);
        double r796492 = r796488 / r796491;
        double r796493 = re;
        double r796494 = im;
        double r796495 = hypot(r796493, r796494);
        double r796496 = cbrt(r796495);
        double r796497 = log(r796496);
        double r796498 = r796492 * r796497;
        double r796499 = r796496 * r796496;
        double r796500 = log(r796499);
        double r796501 = r796492 * r796500;
        double r796502 = r796498 + r796501;
        double r796503 = r796502 * r796492;
        return r796503;
}


\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 31.0

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.6

    \[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  5. Applied pow10.6

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

  6. Applied log-pow0.6

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

  7. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\sqrt{\log 10}}}\]
  8. Using strategy rm

  9. Applied div-inv0.4

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

  10. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right) \cdot \frac{1}{\sqrt{\log 10}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt0.4

    \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

  13. Applied log-prod0.4

    \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

  14. Applied distribute-lft-in0.4

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

  15. Final simplification0.4

    \[\leadsto \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))