Average Error: 31.9 → 17.8
Time: 1.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 7.01675339023471452 \cdot 10^{131}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 7.01675339023471452 \cdot 10^{131}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r26622 = re;
        double r26623 = r26622 * r26622;
        double r26624 = im;
        double r26625 = r26624 * r26624;
        double r26626 = r26623 + r26625;
        double r26627 = sqrt(r26626);
        double r26628 = log(r26627);
        return r26628;
}
↓
double f(double re, double im) {
        double r26629 = re;
        double r26630 = -2.2710920961423764e+59;
        bool r26631 = r26629 <= r26630;
        double r26632 = -1.0;
        double r26633 = r26632 * r26629;
        double r26634 = log(r26633);
        double r26635 = 7.0167533902347145e+131;
        bool r26636 = r26629 <= r26635;
        double r26637 = r26629 * r26629;
        double r26638 = im;
        double r26639 = r26638 * r26638;
        double r26640 = r26637 + r26639;
        double r26641 = sqrt(r26640);
        double r26642 = log(r26641);
        double r26643 = log(r26629);
        double r26644 = r26636 ? r26642 : r26643;
        double r26645 = r26631 ? r26634 : r26644;
        return r26645;
}

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 < -2.2710920961423764e+59

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 11.0

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

    if -2.2710920961423764e+59 < re < 7.0167533902347145e+131

    1. Initial program 22.2

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

    if 7.0167533902347145e+131 < re

    1. Initial program 57.7

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

    2. Taylor expanded around inf 7.1

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 7.01675339023471452 \cdot 10^{131}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))