Average Error: 31.0 → 0.4
Time: 20.3s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)
double f(double re, double im) {
        double r1088711 = re;
        double r1088712 = r1088711 * r1088711;
        double r1088713 = im;
        double r1088714 = r1088713 * r1088713;
        double r1088715 = r1088712 + r1088714;
        double r1088716 = sqrt(r1088715);
        double r1088717 = log(r1088716);
        double r1088718 = 10.0;
        double r1088719 = log(r1088718);
        double r1088720 = r1088717 / r1088719;
        return r1088720;
}
↓
double f(double re, double im) {
        double r1088721 = 1.0;
        double r1088722 = 10.0;
        double r1088723 = log(r1088722);
        double r1088724 = sqrt(r1088723);
        double r1088725 = r1088721 / r1088724;
        double r1088726 = re;
        double r1088727 = im;
        double r1088728 = hypot(r1088726, r1088727);
        double r1088729 = log(r1088728);
        double r1088730 = r1088725 * r1088729;
        double r1088731 = r1088725 * r1088730;
        return r1088731;
}

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  2. Simplified0.6

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied pow10.6

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

  6. Applied log-pow0.6

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

  7. Applied times-frac0.5

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

  9. Applied div-inv0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))