Average Error: 31.6 → 17.5
Time: 3.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r2219117 = re;
        double r2219118 = r2219117 * r2219117;
        double r2219119 = im;
        double r2219120 = r2219119 * r2219119;
        double r2219121 = r2219118 + r2219120;
        double r2219122 = sqrt(r2219121);
        double r2219123 = log(r2219122);
        return r2219123;
}
↓
double f(double re, double im) {
        double r2219124 = re;
        double r2219125 = -1.9401195934627838e+70;
        bool r2219126 = r2219124 <= r2219125;
        double r2219127 = -r2219124;
        double r2219128 = log(r2219127);
        double r2219129 = 7.747777771049568e+94;
        bool r2219130 = r2219124 <= r2219129;
        double r2219131 = im;
        double r2219132 = r2219131 * r2219131;
        double r2219133 = r2219124 * r2219124;
        double r2219134 = r2219132 + r2219133;
        double r2219135 = sqrt(r2219134);
        double r2219136 = log(r2219135);
        double r2219137 = log(r2219124);
        double r2219138 = r2219130 ? r2219136 : r2219137;
        double r2219139 = r2219126 ? r2219128 : r2219138;
        return r2219139;
}

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 < -1.9401195934627838e+70

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 10.3

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

    3. Simplified10.3

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

    if -1.9401195934627838e+70 < re < 7.747777771049568e+94

    1. Initial program 22.1

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

    if 7.747777771049568e+94 < re

    1. Initial program 49.0

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))