Average Error: 32.0 → 17.6
Time: 18.6s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.567655793297945903315979089691981730874 \cdot 10^{135}:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 6.558844456481168673429335625584593485645 \cdot 10^{63}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.567655793297945903315979089691981730874 \cdot 10^{135}:\\
\;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\

\mathbf{elif}\;re \le 6.558844456481168673429335625584593485645 \cdot 10^{63}:\\
\;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\

\end{array}
double f(double re, double im) {
        double r31200 = re;
        double r31201 = r31200 * r31200;
        double r31202 = im;
        double r31203 = r31202 * r31202;
        double r31204 = r31201 + r31203;
        double r31205 = sqrt(r31204);
        double r31206 = log(r31205);
        double r31207 = 10.0;
        double r31208 = log(r31207);
        double r31209 = r31206 / r31208;
        return r31209;
}
↓
double f(double re, double im) {
        double r31210 = re;
        double r31211 = -5.567655793297946e+135;
        bool r31212 = r31210 <= r31211;
        double r31213 = -1.0;
        double r31214 = 10.0;
        double r31215 = log(r31214);
        double r31216 = sqrt(r31215);
        double r31217 = r31213 / r31216;
        double r31218 = r31213 / r31210;
        double r31219 = log(r31218);
        double r31220 = 1.0;
        double r31221 = r31220 / r31215;
        double r31222 = sqrt(r31221);
        double r31223 = r31219 * r31222;
        double r31224 = r31217 * r31223;
        double r31225 = 6.558844456481169e+63;
        bool r31226 = r31210 <= r31225;
        double r31227 = r31210 * r31210;
        double r31228 = im;
        double r31229 = r31228 * r31228;
        double r31230 = r31227 + r31229;
        double r31231 = sqrt(r31230);
        double r31232 = cbrt(r31231);
        double r31233 = r31232 * r31232;
        double r31234 = r31233 * r31232;
        double r31235 = log(r31234);
        double r31236 = r31235 / r31215;
        double r31237 = r31220 / r31216;
        double r31238 = log(r31210);
        double r31239 = r31222 * r31238;
        double r31240 = r31237 * r31239;
        double r31241 = r31226 ? r31236 : r31240;
        double r31242 = r31212 ? r31224 : r31241;
        return r31242;
}

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. Split input into 3 regimes

  2. if re < -5.567655793297946e+135

    1. Initial program 58.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt58.9

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

    4. Applied pow158.9

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

    5. Applied log-pow58.9

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

    6. Applied times-frac58.9

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

    7. Taylor expanded around -inf 7.5

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-1 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

    8. Simplified7.5

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

    if -5.567655793297946e+135 < re < 6.558844456481169e+63

    1. Initial program 22.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt22.0

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

    if 6.558844456481169e+63 < re

    1. Initial program 46.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt46.0

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

    4. Applied pow146.0

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

    5. Applied log-pow46.0

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

    6. Applied times-frac46.0

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

    7. Taylor expanded around inf 10.3

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-1 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

    8. Simplified10.3

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-\sqrt{\frac{1}{\log 10}} \cdot \left(-\log re\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.567655793297945903315979089691981730874 \cdot 10^{135}:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;re \le 6.558844456481168673429335625584593485645 \cdot 10^{63}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))