Average Error: 32.1 → 18.0
Time: 17.4s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\log 10}{\log \left(-re\right)}\right)}^{3}}}\\ \mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{27} \cdot {\left(\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{\log 10}{-\log re}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\
\;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\log 10}{\log \left(-re\right)}\right)}^{3}}}\\

\mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\
\;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{27} \cdot {\left(\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}^{3}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{-\frac{\log 10}{-\log re}}\\

\end{array}
double f(double re, double im) {
        double r31247 = re;
        double r31248 = r31247 * r31247;
        double r31249 = im;
        double r31250 = r31249 * r31249;
        double r31251 = r31248 + r31250;
        double r31252 = sqrt(r31251);
        double r31253 = log(r31252);
        double r31254 = 10.0;
        double r31255 = log(r31254);
        double r31256 = r31253 / r31255;
        return r31256;
}
↓
double f(double re, double im) {
        double r31257 = re;
        double r31258 = -1.1564076018637175e+112;
        bool r31259 = r31257 <= r31258;
        double r31260 = 1.0;
        double r31261 = 10.0;
        double r31262 = log(r31261);
        double r31263 = -r31257;
        double r31264 = log(r31263);
        double r31265 = r31262 / r31264;
        double r31266 = 3.0;
        double r31267 = pow(r31265, r31266);
        double r31268 = cbrt(r31267);
        double r31269 = r31260 / r31268;
        double r31270 = 1.2449882138840628e+138;
        bool r31271 = r31257 <= r31270;
        double r31272 = 0.037037037037037035;
        double r31273 = r31257 * r31257;
        double r31274 = im;
        double r31275 = r31274 * r31274;
        double r31276 = r31273 + r31275;
        double r31277 = sqrt(r31276);
        double r31278 = cbrt(r31277);
        double r31279 = log(r31278);
        double r31280 = r31262 / r31279;
        double r31281 = pow(r31280, r31266);
        double r31282 = r31272 * r31281;
        double r31283 = cbrt(r31282);
        double r31284 = r31260 / r31283;
        double r31285 = log(r31257);
        double r31286 = -r31285;
        double r31287 = r31262 / r31286;
        double r31288 = -r31287;
        double r31289 = r31260 / r31288;
        double r31290 = r31271 ? r31284 : r31289;
        double r31291 = r31259 ? r31269 : r31290;
        return r31291;
}

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 < -1.1564076018637175e+112

    1. Initial program 52.8

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

    3. Applied clear-num52.8

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

    5. Applied add-cbrt-cube52.9

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

    6. Applied add-cbrt-cube53.0

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

    7. Applied cbrt-undiv52.9

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

    8. Simplified52.9

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

    9. Taylor expanded around -inf 8.7

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

    10. Simplified8.7

      \[\leadsto \frac{1}{\sqrt[3]{{\left(\frac{\log 10}{\log \color{blue}{\left(-re\right)}}\right)}^{3}}}\]

    if -1.1564076018637175e+112 < re < 1.2449882138840628e+138

    1. Initial program 22.1

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

    3. Applied clear-num22.1

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

    5. Applied add-cbrt-cube22.1

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

    6. Applied add-cbrt-cube22.6

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

    7. Applied cbrt-undiv22.1

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

    8. Simplified22.1

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}\right)}^{3}}}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt22.1

      \[\leadsto \frac{1}{\sqrt[3]{{\left(\frac{\log 10}{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}}\right)}^{3}}}\]
    11. Using strategy rm

    12. Applied pow122.1

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

    13. Applied pow122.1

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

    14. Applied pow122.1

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

    15. Applied pow-prod-up22.1

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

    16. Applied pow-prod-up22.1

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

    17. Applied log-pow22.2

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

    18. Applied pow122.2

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

    19. Applied log-pow22.2

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

    20. Applied times-frac22.2

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

    21. Applied unpow-prod-down22.2

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

    22. Simplified22.1

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\frac{1}{27}} \cdot {\left(\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}^{3}}}\]

    if 1.2449882138840628e+138 < re

    1. Initial program 58.8

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

    3. Applied clear-num58.8

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

    4. Taylor expanded around inf 8.1

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

    5. Simplified8.1

      \[\leadsto \frac{1}{\color{blue}{-\frac{\log 10}{-\log re}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\frac{\log 10}{\log \left(-re\right)}\right)}^{3}}}\\ \mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{1}{27} \cdot {\left(\frac{\log 10}{\log \left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{\log 10}{-\log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))