Average Error: 31.2 → 17.0
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{else}:\\
\;\;\;\;\log re\\

\end{array}
double f(double re, double im) {
        double r31697 = re;
        double r31698 = r31697 * r31697;
        double r31699 = im;
        double r31700 = r31699 * r31699;
        double r31701 = r31698 + r31700;
        double r31702 = sqrt(r31701);
        double r31703 = log(r31702);
        return r31703;
}
↓
double f(double re, double im) {
        double r31704 = re;
        double r31705 = -1.0674397664294253e+136;
        bool r31706 = r31704 <= r31705;
        double r31707 = -1.0;
        double r31708 = r31707 * r31704;
        double r31709 = log(r31708);
        double r31710 = -4.4039792789215395e-257;
        bool r31711 = r31704 <= r31710;
        double r31712 = r31704 * r31704;
        double r31713 = im;
        double r31714 = r31713 * r31713;
        double r31715 = r31712 + r31714;
        double r31716 = sqrt(r31715);
        double r31717 = log(r31716);
        double r31718 = 3.8197786805557845e-227;
        bool r31719 = r31704 <= r31718;
        double r31720 = log(r31713);
        double r31721 = 8.439330033545885e+67;
        bool r31722 = r31704 <= r31721;
        double r31723 = log(r31704);
        double r31724 = r31722 ? r31717 : r31723;
        double r31725 = r31719 ? r31720 : r31724;
        double r31726 = r31711 ? r31717 : r31725;
        double r31727 = r31706 ? r31709 : r31726;
        return r31727;
}

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 < -1.0674397664294253e+136

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 7.7

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

    if -1.0674397664294253e+136 < re < -4.4039792789215395e-257 or 3.8197786805557845e-227 < re < 8.439330033545885e+67

    1. Initial program 18.8

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

    if -4.4039792789215395e-257 < re < 3.8197786805557845e-227

    1. Initial program 31.0

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

    2. Taylor expanded around 0 32.6

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

    if 8.439330033545885e+67 < re

    1. Initial program 46.7

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))