Average Error: 30.6 → 17.0
Time: 5.9s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.327249381423175 \cdot 10^{+109}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.515107223737886 \cdot 10^{+123}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.327249381423175 \cdot 10^{+109}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 2.515107223737886 \cdot 10^{+123}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r819602 = re;
        double r819603 = r819602 * r819602;
        double r819604 = im;
        double r819605 = r819604 * r819604;
        double r819606 = r819603 + r819605;
        double r819607 = sqrt(r819606);
        double r819608 = log(r819607);
        return r819608;
}
↓
double f(double re, double im) {
        double r819609 = re;
        double r819610 = -2.327249381423175e+109;
        bool r819611 = r819609 <= r819610;
        double r819612 = -r819609;
        double r819613 = log(r819612);
        double r819614 = 2.515107223737886e+123;
        bool r819615 = r819609 <= r819614;
        double r819616 = im;
        double r819617 = r819616 * r819616;
        double r819618 = r819609 * r819609;
        double r819619 = r819617 + r819618;
        double r819620 = sqrt(r819619);
        double r819621 = log(r819620);
        double r819622 = log(r819609);
        double r819623 = r819615 ? r819621 : r819622;
        double r819624 = r819611 ? r819613 : r819623;
        return r819624;
}

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 < -2.327249381423175e+109

    1. Initial program 51.7

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

    2. Taylor expanded around -inf 8.8

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

    3. Simplified8.8

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

    if -2.327249381423175e+109 < re < 2.515107223737886e+123

    1. Initial program 20.8

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

    if 2.515107223737886e+123 < re

    1. Initial program 53.3

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

    2. Taylor expanded around inf 8.5

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.327249381423175 \cdot 10^{+109}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.515107223737886 \cdot 10^{+123}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))