Average Error: 32.4 → 17.4
Time: 9.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.143203660913673329850871473806326387498 \cdot 10^{92}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 2.153930701766585709222248503120339127843 \cdot 10^{145}:\\ \;\;\;\;\frac{\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.143203660913673329850871473806326387498 \cdot 10^{92}:\\
\;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\

\mathbf{elif}\;re \le 2.153930701766585709222248503120339127843 \cdot 10^{145}:\\
\;\;\;\;\frac{\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) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\

\end{array}
double f(double re, double im, double base) {
        double r49684 = re;
        double r49685 = r49684 * r49684;
        double r49686 = im;
        double r49687 = r49686 * r49686;
        double r49688 = r49685 + r49687;
        double r49689 = sqrt(r49688);
        double r49690 = log(r49689);
        double r49691 = base;
        double r49692 = log(r49691);
        double r49693 = r49690 * r49692;
        double r49694 = atan2(r49686, r49684);
        double r49695 = 0.0;
        double r49696 = r49694 * r49695;
        double r49697 = r49693 + r49696;
        double r49698 = r49692 * r49692;
        double r49699 = r49695 * r49695;
        double r49700 = r49698 + r49699;
        double r49701 = r49697 / r49700;
        return r49701;
}
↓
double f(double re, double im, double base) {
        double r49702 = re;
        double r49703 = -8.143203660913673e+92;
        bool r49704 = r49702 <= r49703;
        double r49705 = -1.0;
        double r49706 = r49705 * r49702;
        double r49707 = log(r49706);
        double r49708 = base;
        double r49709 = log(r49708);
        double r49710 = r49707 * r49709;
        double r49711 = im;
        double r49712 = atan2(r49711, r49702);
        double r49713 = 0.0;
        double r49714 = r49712 * r49713;
        double r49715 = r49710 + r49714;
        double r49716 = r49709 * r49709;
        double r49717 = r49713 * r49713;
        double r49718 = r49716 + r49717;
        double r49719 = r49715 / r49718;
        double r49720 = 2.1539307017665857e+145;
        bool r49721 = r49702 <= r49720;
        double r49722 = r49702 * r49702;
        double r49723 = r49711 * r49711;
        double r49724 = r49722 + r49723;
        double r49725 = sqrt(r49724);
        double r49726 = cbrt(r49725);
        double r49727 = r49726 * r49726;
        double r49728 = r49727 * r49726;
        double r49729 = log(r49728);
        double r49730 = r49729 * r49709;
        double r49731 = r49730 + r49714;
        double r49732 = sqrt(r49718);
        double r49733 = r49731 / r49732;
        double r49734 = r49733 / r49732;
        double r49735 = 1.0;
        double r49736 = r49735 / r49702;
        double r49737 = log(r49736);
        double r49738 = r49735 / r49708;
        double r49739 = log(r49738);
        double r49740 = r49737 / r49739;
        double r49741 = r49721 ? r49734 : r49740;
        double r49742 = r49704 ? r49719 : r49741;
        return r49742;
}

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 3 regimes

  2. if re < -8.143203660913673e+92

    1. Initial program 51.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around -inf 9.4

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    if -8.143203660913673e+92 < re < 2.1539307017665857e+145

    1. Initial program 21.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt21.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

    4. Applied associate-/r*21.6

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt21.7

      \[\leadsto \frac{\frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    if 2.1539307017665857e+145 < re

    1. Initial program 61.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

    2. Taylor expanded around inf 7.1

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.143203660913673329850871473806326387498 \cdot 10^{92}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 2.153930701766585709222248503120339127843 \cdot 10^{145}:\\ \;\;\;\;\frac{\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))