Average Error: 31.6 → 17.5
Time: 2.6s
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 r43838 = re;
        double r43839 = r43838 * r43838;
        double r43840 = im;
        double r43841 = r43840 * r43840;
        double r43842 = r43839 + r43841;
        double r43843 = sqrt(r43842);
        double r43844 = log(r43843);
        return r43844;
}
↓
double f(double re, double im) {
        double r43845 = re;
        double r43846 = -1.9401195934627838e+70;
        bool r43847 = r43845 <= r43846;
        double r43848 = -r43845;
        double r43849 = log(r43848);
        double r43850 = 7.747777771049568e+94;
        bool r43851 = r43845 <= r43850;
        double r43852 = im;
        double r43853 = r43852 * r43852;
        double r43854 = r43845 * r43845;
        double r43855 = r43853 + r43854;
        double r43856 = sqrt(r43855);
        double r43857 = log(r43856);
        double r43858 = log(r43845);
        double r43859 = r43851 ? r43857 : r43858;
        double r43860 = r43847 ? r43849 : r43859;
        return r43860;
}

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)))))