Average Error: 32.7 → 0.3
Time: 4.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)
double f(double re, double im) {
        double r89484 = re;
        double r89485 = r89484 * r89484;
        double r89486 = im;
        double r89487 = r89486 * r89486;
        double r89488 = r89485 + r89487;
        double r89489 = sqrt(r89488);
        double r89490 = log(r89489);
        double r89491 = 10.0;
        double r89492 = log(r89491);
        double r89493 = r89490 / r89492;
        return r89493;
}
↓
double f(double re, double im) {
        double r89494 = 1.0;
        double r89495 = 10.0;
        double r89496 = log(r89495);
        double r89497 = sqrt(r89496);
        double r89498 = r89494 / r89497;
        double r89499 = re;
        double r89500 = im;
        double r89501 = hypot(r89499, r89500);
        double r89502 = log1p(r89501);
        double r89503 = expm1(r89502);
        double r89504 = pow(r89503, r89498);
        double r89505 = log(r89504);
        double r89506 = r89498 * r89505;
        return r89506;
}

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 hypot-def0.6

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

  5. Applied add-sqr-sqrt0.6

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

  6. Applied pow10.6

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

  7. Applied log-pow0.6

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

  8. Applied times-frac0.5

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

  10. Applied add-log-exp0.5

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

  11. Simplified0.3

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

  13. Applied expm1-log1p-u0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

  14. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(re, im\right)\right)\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))