Average Error: 31.1 → 17.4
Time: 19.2s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2774730308138672 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 9.535766378899208 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{2}} \cdot \left(2 \cdot {\left(\frac{1}{\log 10 \cdot \left(\log 10 \cdot \log 10\right)}\right)}^{\frac{1}{4}}\right)\right) \cdot \log re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2774730308138672 \cdot 10^{+123}:\\
\;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\

\mathbf{elif}\;re \le 9.535766378899208 \cdot 10^{+128}:\\
\;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\sqrt{\frac{1}{2}} \cdot \left(2 \cdot {\left(\frac{1}{\log 10 \cdot \left(\log 10 \cdot \log 10\right)}\right)}^{\frac{1}{4}}\right)\right) \cdot \log re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\

\end{array}
double f(double re, double im) {
        double r702296 = re;
        double r702297 = r702296 * r702296;
        double r702298 = im;
        double r702299 = r702298 * r702298;
        double r702300 = r702297 + r702299;
        double r702301 = sqrt(r702300);
        double r702302 = log(r702301);
        double r702303 = 10.0;
        double r702304 = log(r702303);
        double r702305 = r702302 / r702304;
        return r702305;
}
↓
double f(double re, double im) {
        double r702306 = re;
        double r702307 = -1.2774730308138672e+123;
        bool r702308 = r702306 <= r702307;
        double r702309 = 0.5;
        double r702310 = 10.0;
        double r702311 = log(r702310);
        double r702312 = sqrt(r702311);
        double r702313 = r702309 / r702312;
        double r702314 = 1.0;
        double r702315 = r702314 / r702311;
        double r702316 = sqrt(r702315);
        double r702317 = -1.0;
        double r702318 = r702317 / r702306;
        double r702319 = log(r702318);
        double r702320 = r702316 * r702319;
        double r702321 = -2.0;
        double r702322 = r702320 * r702321;
        double r702323 = r702313 * r702322;
        double r702324 = 9.535766378899208e+128;
        bool r702325 = r702306 <= r702324;
        double r702326 = sqrt(r702313);
        double r702327 = r702306 * r702306;
        double r702328 = im;
        double r702329 = r702328 * r702328;
        double r702330 = r702327 + r702329;
        double r702331 = log(r702330);
        double r702332 = r702331 / r702312;
        double r702333 = r702332 * r702326;
        double r702334 = r702326 * r702333;
        double r702335 = sqrt(r702309);
        double r702336 = 2.0;
        double r702337 = r702311 * r702311;
        double r702338 = r702311 * r702337;
        double r702339 = r702314 / r702338;
        double r702340 = 0.25;
        double r702341 = pow(r702339, r702340);
        double r702342 = r702336 * r702341;
        double r702343 = r702335 * r702342;
        double r702344 = log(r702306);
        double r702345 = r702343 * r702344;
        double r702346 = r702345 * r702326;
        double r702347 = r702325 ? r702334 : r702346;
        double r702348 = r702308 ? r702323 : r702347;
        return r702348;
}

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 < -1.2774730308138672e+123

    1. Initial program 54.8

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

    3. Applied add-sqr-sqrt54.8

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

    4. Applied pow1/254.8

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow54.8

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac54.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]

    7. Taylor expanded around -inf 9.0

      \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

    if -1.2774730308138672e+123 < re < 9.535766378899208e+128

    1. Initial program 21.1

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

    3. Applied add-sqr-sqrt21.1

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

    4. Applied pow1/221.1

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow21.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac21.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt21.1

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

    9. Applied associate-*l*21.0

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]

    if 9.535766378899208e+128 < re

    1. Initial program 55.4

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

    3. Applied add-sqr-sqrt55.4

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

    4. Applied pow1/255.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    5. Applied log-pow55.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

    6. Applied times-frac55.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt55.4

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\]

    9. Applied associate-*l*55.4

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)}\]
    10. Using strategy rm

    11. Applied div-inv55.4

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\right)\]

    12. Applied associate-*r*55.4

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}\]

    13. Taylor expanded around inf 8.4

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(-2 \cdot \left({\left(\frac{1}{{\left(\log 10\right)}^{3}}\right)}^{\frac{1}{4}} \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)}\]

    14. Simplified8.4

      \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\left(\left(-2 \cdot {\left(\frac{1}{\log 10 \cdot \left(\log 10 \cdot \log 10\right)}\right)}^{\frac{1}{4}}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \left(-\log re\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2774730308138672 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \log \left(\frac{-1}{re}\right)\right) \cdot -2\right)\\ \mathbf{elif}\;re \le 9.535766378899208 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}} \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{2}} \cdot \left(2 \cdot {\left(\frac{1}{\log 10 \cdot \left(\log 10 \cdot \log 10\right)}\right)}^{\frac{1}{4}}\right)\right) \cdot \log re\right) \cdot \sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))