Average Error: 31.1 → 18.8
Time: 1.0m
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -3.723507598105854 \cdot 10^{+150}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{elif}\;im \le -2.973946629754585 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}{\log base}\\ \mathbf{elif}\;im \le 3.3440052807984994 \cdot 10^{-56}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
\begin{array}{l}
\mathbf{if}\;im \le -3.723507598105854 \cdot 10^{+150}:\\
\;\;\;\;\frac{\log \left(-re\right) \cdot \log base}{\log base \cdot \log base}\\

\mathbf{elif}\;im \le -2.973946629754585 \cdot 10^{-88}:\\
\;\;\;\;\frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}{\log base}\\

\mathbf{elif}\;im \le 3.3440052807984994 \cdot 10^{-56}:\\
\;\;\;\;\frac{\log \left(-re\right)}{\log base}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log im}{\log base}\\

\end{array}
double f(double re, double im, double base) {
        double r1467501 = re;
        double r1467502 = r1467501 * r1467501;
        double r1467503 = im;
        double r1467504 = r1467503 * r1467503;
        double r1467505 = r1467502 + r1467504;
        double r1467506 = sqrt(r1467505);
        double r1467507 = log(r1467506);
        double r1467508 = base;
        double r1467509 = log(r1467508);
        double r1467510 = r1467507 * r1467509;
        double r1467511 = atan2(r1467503, r1467501);
        double r1467512 = 0.0;
        double r1467513 = r1467511 * r1467512;
        double r1467514 = r1467510 + r1467513;
        double r1467515 = r1467509 * r1467509;
        double r1467516 = r1467512 * r1467512;
        double r1467517 = r1467515 + r1467516;
        double r1467518 = r1467514 / r1467517;
        return r1467518;
}
↓
double f(double re, double im, double base) {
        double r1467519 = im;
        double r1467520 = -3.723507598105854e+150;
        bool r1467521 = r1467519 <= r1467520;
        double r1467522 = re;
        double r1467523 = -r1467522;
        double r1467524 = log(r1467523);
        double r1467525 = base;
        double r1467526 = log(r1467525);
        double r1467527 = r1467524 * r1467526;
        double r1467528 = r1467526 * r1467526;
        double r1467529 = r1467527 / r1467528;
        double r1467530 = -2.973946629754585e-88;
        bool r1467531 = r1467519 <= r1467530;
        double r1467532 = r1467522 * r1467522;
        double r1467533 = r1467519 * r1467519;
        double r1467534 = r1467532 + r1467533;
        double r1467535 = sqrt(r1467534);
        double r1467536 = sqrt(r1467535);
        double r1467537 = r1467536 * r1467536;
        double r1467538 = log(r1467537);
        double r1467539 = r1467538 / r1467526;
        double r1467540 = 3.3440052807984994e-56;
        bool r1467541 = r1467519 <= r1467540;
        double r1467542 = r1467524 / r1467526;
        double r1467543 = log(r1467519);
        double r1467544 = r1467543 / r1467526;
        double r1467545 = r1467541 ? r1467542 : r1467544;
        double r1467546 = r1467531 ? r1467539 : r1467545;
        double r1467547 = r1467521 ? r1467529 : r1467546;
        return r1467547;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if im < -3.723507598105854e+150

    1. Initial program 60.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified60.9

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

    3. Taylor expanded around -inf 51.6

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

    4. Simplified51.6

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

    if -3.723507598105854e+150 < im < -2.973946629754585e-88

    1. Initial program 16.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified16.5

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

    4. Applied associate-/r*16.4

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

    5. Simplified16.4

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

    7. Applied add-sqr-sqrt16.4

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

    8. Applied sqrt-prod16.4

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

    if -2.973946629754585e-88 < im < 3.3440052807984994e-56

    1. Initial program 25.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified25.7

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

    4. Applied associate-/r*25.6

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

    5. Simplified25.6

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

    6. Taylor expanded around -inf 11.2

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

    7. Simplified11.2

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

    if 3.3440052807984994e-56 < im

    1. Initial program 34.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified34.9

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

    4. Applied associate-/r*34.9

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

    5. Simplified34.9

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

    6. Taylor expanded around 0 16.3

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -3.723507598105854 \cdot 10^{+150}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base}{\log base \cdot \log base}\\ \mathbf{elif}\;im \le -2.973946629754585 \cdot 10^{-88}:\\ \;\;\;\;\frac{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}{\log base}\\ \mathbf{elif}\;im \le 3.3440052807984994 \cdot 10^{-56}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))