Average Error: 31.6 → 17.5
Time: 3.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.542162861981963339451786530680392909626 \cdot 10^{56}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.681783727083966440338519619684767953495 \cdot 10^{54}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.542162861981963339451786530680392909626 \cdot 10^{56}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 4.681783727083966440338519619684767953495 \cdot 10^{54}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r40064 = re;
        double r40065 = r40064 * r40064;
        double r40066 = im;
        double r40067 = r40066 * r40066;
        double r40068 = r40065 + r40067;
        double r40069 = sqrt(r40068);
        double r40070 = log(r40069);
        return r40070;
}
↓
double f(double re, double im) {
        double r40071 = re;
        double r40072 = -4.542162861981963e+56;
        bool r40073 = r40071 <= r40072;
        double r40074 = -r40071;
        double r40075 = log(r40074);
        double r40076 = 4.6817837270839664e+54;
        bool r40077 = r40071 <= r40076;
        double r40078 = im;
        double r40079 = r40078 * r40078;
        double r40080 = r40071 * r40071;
        double r40081 = r40079 + r40080;
        double r40082 = sqrt(r40081);
        double r40083 = log(r40082);
        double r40084 = log(r40071);
        double r40085 = r40077 ? r40083 : r40084;
        double r40086 = r40073 ? r40075 : r40085;
        return r40086;
}

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 < -4.542162861981963e+56

    1. Initial program 45.9

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

    2. Taylor expanded around -inf 10.7

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

    3. Simplified10.7

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

    if -4.542162861981963e+56 < re < 4.6817837270839664e+54

    1. Initial program 21.9

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

    if 4.6817837270839664e+54 < re

    1. Initial program 45.3

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

    2. Taylor expanded around inf 11.5

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.542162861981963339451786530680392909626 \cdot 10^{56}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.681783727083966440338519619684767953495 \cdot 10^{54}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))