Average Error: 30.6 → 17.1
Time: 8.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.1373638352484895 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 441347509.722609:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.1373638352484895 \cdot 10^{+151}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 441347509.722609:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r429075 = re;
        double r429076 = r429075 * r429075;
        double r429077 = im;
        double r429078 = r429077 * r429077;
        double r429079 = r429076 + r429078;
        double r429080 = sqrt(r429079);
        double r429081 = log(r429080);
        return r429081;
}
↓
double f(double re, double im) {
        double r429082 = re;
        double r429083 = -2.1373638352484895e+151;
        bool r429084 = r429082 <= r429083;
        double r429085 = -r429082;
        double r429086 = log(r429085);
        double r429087 = 441347509.722609;
        bool r429088 = r429082 <= r429087;
        double r429089 = im;
        double r429090 = r429089 * r429089;
        double r429091 = r429082 * r429082;
        double r429092 = r429090 + r429091;
        double r429093 = sqrt(r429092);
        double r429094 = log(r429093);
        double r429095 = log(r429082);
        double r429096 = r429088 ? r429094 : r429095;
        double r429097 = r429084 ? r429086 : r429096;
        return r429097;
}

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 < -2.1373638352484895e+151

    1. Initial program 61.2

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

    2. Taylor expanded around -inf 5.7

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

    3. Simplified5.7

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

    if -2.1373638352484895e+151 < re < 441347509.722609

    1. Initial program 21.1

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

    if 441347509.722609 < re

    1. Initial program 39.2

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

    2. Taylor expanded around inf 13.0

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.1373638352484895 \cdot 10^{+151}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 441347509.722609:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))