Average Error: 31.9 → 0.4
Time: 18.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\mathsf{hypot}\left(re, im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)
double f(double re, double im) {
        double r35963 = re;
        double r35964 = r35963 * r35963;
        double r35965 = im;
        double r35966 = r35965 * r35965;
        double r35967 = r35964 + r35966;
        double r35968 = sqrt(r35967);
        double r35969 = log(r35968);
        double r35970 = 10.0;
        double r35971 = log(r35970);
        double r35972 = r35969 / r35971;
        return r35972;
}
↓
double f(double re, double im) {
        double r35973 = 1.0;
        double r35974 = 10.0;
        double r35975 = log(r35974);
        double r35976 = sqrt(r35975);
        double r35977 = r35973 / r35976;
        double r35978 = re;
        double r35979 = im;
        double r35980 = hypot(r35978, r35979);
        double r35981 = log(r35980);
        double r35982 = r35981 * r35977;
        double r35983 = r35977 * r35982;
        return r35983;
}

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 31.9

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

  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-frac0.6

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

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))