Average Error: 31.7 → 0.3
Time: 20.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)
double f(double re, double im) {
        double r33175 = re;
        double r33176 = r33175 * r33175;
        double r33177 = im;
        double r33178 = r33177 * r33177;
        double r33179 = r33176 + r33178;
        double r33180 = sqrt(r33179);
        double r33181 = log(r33180);
        double r33182 = 10.0;
        double r33183 = log(r33182);
        double r33184 = r33181 / r33183;
        return r33184;
}
↓
double f(double re, double im) {
        double r33185 = 1.0;
        double r33186 = 10.0;
        double r33187 = log(r33186);
        double r33188 = sqrt(r33187);
        double r33189 = r33185 / r33188;
        double r33190 = re;
        double r33191 = im;
        double r33192 = hypot(r33190, r33191);
        double r33193 = cbrt(r33192);
        double r33194 = r33193 * r33193;
        double r33195 = r33194 * r33193;
        double r33196 = pow(r33195, r33189);
        double r33197 = log(r33196);
        double r33198 = r33189 * r33197;
        return r33198;
}

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 31.7

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

  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. Using strategy rm

  11. Applied add-log-exp0.4

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

  12. Simplified0.3

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

  14. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

  15. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))