Average Error: 30.8 → 16.9
Time: 2.8s
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 r1033015 = re;
        double r1033016 = r1033015 * r1033015;
        double r1033017 = im;
        double r1033018 = r1033017 * r1033017;
        double r1033019 = r1033016 + r1033018;
        double r1033020 = sqrt(r1033019);
        double r1033021 = log(r1033020);
        return r1033021;
}
↓
double f(double re, double im) {
        double r1033022 = re;
        double r1033023 = -7.266849055505758e+89;
        bool r1033024 = r1033022 <= r1033023;
        double r1033025 = -r1033022;
        double r1033026 = log(r1033025);
        double r1033027 = 7.762022248986236e+136;
        bool r1033028 = r1033022 <= r1033027;
        double r1033029 = im;
        double r1033030 = r1033029 * r1033029;
        double r1033031 = r1033022 * r1033022;
        double r1033032 = r1033030 + r1033031;
        double r1033033 = sqrt(r1033032);
        double r1033034 = log(r1033033);
        double r1033035 = log(r1033022);
        double r1033036 = r1033028 ? r1033034 : r1033035;
        double r1033037 = r1033024 ? r1033026 : r1033036;
        return r1033037;
}

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)))))