Average Error: 31.6 → 17.5
Time: 7.1s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.552702335522331775782681989067170419459 \cdot 10^{114}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 1.241086545067041095458499268505642473269 \cdot 10^{67}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.552702335522331775782681989067170419459 \cdot 10^{114}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\

\mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\

\mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\
\;\;\;\;\frac{\log im}{\log 10}\\

\mathbf{elif}\;re \le 1.241086545067041095458499268505642473269 \cdot 10^{67}:\\
\;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\

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

\end{array}
double f(double re, double im) {
        double r99347 = re;
        double r99348 = r99347 * r99347;
        double r99349 = im;
        double r99350 = r99349 * r99349;
        double r99351 = r99348 + r99350;
        double r99352 = sqrt(r99351);
        double r99353 = log(r99352);
        double r99354 = 10.0;
        double r99355 = log(r99354);
        double r99356 = r99353 / r99355;
        return r99356;
}
↓
double f(double re, double im) {
        double r99357 = re;
        double r99358 = -2.552702335522332e+114;
        bool r99359 = r99357 <= r99358;
        double r99360 = 1.0;
        double r99361 = 10.0;
        double r99362 = log(r99361);
        double r99363 = sqrt(r99362);
        double r99364 = r99360 / r99363;
        double r99365 = -1.0;
        double r99366 = r99365 * r99357;
        double r99367 = pow(r99366, r99364);
        double r99368 = log(r99367);
        double r99369 = r99360 * r99368;
        double r99370 = r99364 * r99369;
        double r99371 = -8.357854099413859e-296;
        bool r99372 = r99357 <= r99371;
        double r99373 = r99357 * r99357;
        double r99374 = im;
        double r99375 = r99374 * r99374;
        double r99376 = r99373 + r99375;
        double r99377 = sqrt(r99376);
        double r99378 = pow(r99377, r99364);
        double r99379 = log(r99378);
        double r99380 = r99360 * r99379;
        double r99381 = r99364 * r99380;
        double r99382 = 7.758435713562232e-281;
        bool r99383 = r99357 <= r99382;
        double r99384 = log(r99374);
        double r99385 = r99384 / r99362;
        double r99386 = 1.2410865450670411e+67;
        bool r99387 = r99357 <= r99386;
        double r99388 = pow(r99357, r99364);
        double r99389 = log(r99388);
        double r99390 = r99360 * r99389;
        double r99391 = r99364 * r99390;
        double r99392 = r99387 ? r99381 : r99391;
        double r99393 = r99383 ? r99385 : r99392;
        double r99394 = r99372 ? r99381 : r99393;
        double r99395 = r99359 ? r99370 : r99394;
        return r99395;
}

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 4 regimes

  2. if re < -2.552702335522332e+114

    1. Initial program 53.6

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

    3. Applied add-sqr-sqrt53.6

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

    4. Applied pow153.6

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

    5. Applied log-pow53.6

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

    6. Applied times-frac53.6

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

    8. Applied add-log-exp53.6

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

    9. Simplified53.5

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

    11. Applied *-un-lft-identity53.5

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

    12. Taylor expanded around -inf 8.4

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

    if -2.552702335522332e+114 < re < -8.357854099413859e-296 or 7.758435713562232e-281 < re < 1.2410865450670411e+67

    1. Initial program 21.6

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

    3. Applied add-sqr-sqrt21.6

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

    4. Applied pow121.6

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

    5. Applied log-pow21.6

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

    6. Applied times-frac21.6

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

    8. Applied add-log-exp21.6

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

    9. Simplified21.4

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

    11. Applied *-un-lft-identity21.4

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

    if -8.357854099413859e-296 < re < 7.758435713562232e-281

    1. Initial program 29.8

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

    2. Taylor expanded around 0 34.6

      \[\leadsto \frac{\log \color{blue}{im}}{\log 10}\]

    if 1.2410865450670411e+67 < re

    1. Initial program 45.2

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

    3. Applied add-sqr-sqrt45.2

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

    4. Applied pow145.2

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

    5. Applied log-pow45.2

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

    6. Applied times-frac45.2

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

    8. Applied add-log-exp45.2

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

    9. Simplified45.1

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

    11. Applied *-un-lft-identity45.1

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

    12. Taylor expanded around inf 9.9

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.552702335522331775782681989067170419459 \cdot 10^{114}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 1.241086545067041095458499268505642473269 \cdot 10^{67}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(1 \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))