Average Error: 30.8 → 16.9
Time: 3.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.266849055505758 \cdot 10^{+89}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.762022248986236 \cdot 10^{+136}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.266849055505758 \cdot 10^{+89}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 7.762022248986236 \cdot 10^{+136}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1033066 = re;
        double r1033067 = r1033066 * r1033066;
        double r1033068 = im;
        double r1033069 = r1033068 * r1033068;
        double r1033070 = r1033067 + r1033069;
        double r1033071 = sqrt(r1033070);
        double r1033072 = log(r1033071);
        return r1033072;
}
↓
double f(double re, double im) {
        double r1033073 = re;
        double r1033074 = -7.266849055505758e+89;
        bool r1033075 = r1033073 <= r1033074;
        double r1033076 = -r1033073;
        double r1033077 = log(r1033076);
        double r1033078 = 7.762022248986236e+136;
        bool r1033079 = r1033073 <= r1033078;
        double r1033080 = im;
        double r1033081 = r1033080 * r1033080;
        double r1033082 = r1033073 * r1033073;
        double r1033083 = r1033081 + r1033082;
        double r1033084 = sqrt(r1033083);
        double r1033085 = log(r1033084);
        double r1033086 = log(r1033073);
        double r1033087 = r1033079 ? r1033085 : r1033086;
        double r1033088 = r1033075 ? r1033077 : r1033087;
        return r1033088;
}

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 < -7.266849055505758e+89

    1. Initial program 47.7

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

    2. Taylor expanded around -inf 9.0

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

    3. Simplified9.0

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

    if -7.266849055505758e+89 < re < 7.762022248986236e+136

    1. Initial program 21.0

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

    if 7.762022248986236e+136 < re

    1. Initial program 57.4

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

    2. Taylor expanded around inf 7.1

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.266849055505758 \cdot 10^{+89}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.762022248986236 \cdot 10^{+136}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))