Average Error: 32.0 → 17.6
Time: 4.5s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.981250171867664465389722520730842139436 \cdot 10^{79}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.270821738026904207339185332013542781266 \cdot 10^{102}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.981250171867664465389722520730842139436 \cdot 10^{79}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 2.270821738026904207339185332013542781266 \cdot 10^{102}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1591605 = re;
        double r1591606 = r1591605 * r1591605;
        double r1591607 = im;
        double r1591608 = r1591607 * r1591607;
        double r1591609 = r1591606 + r1591608;
        double r1591610 = sqrt(r1591609);
        double r1591611 = log(r1591610);
        return r1591611;
}
↓
double f(double re, double im) {
        double r1591612 = re;
        double r1591613 = -3.9812501718676645e+79;
        bool r1591614 = r1591612 <= r1591613;
        double r1591615 = -r1591612;
        double r1591616 = log(r1591615);
        double r1591617 = 2.270821738026904e+102;
        bool r1591618 = r1591612 <= r1591617;
        double r1591619 = im;
        double r1591620 = r1591619 * r1591619;
        double r1591621 = r1591612 * r1591612;
        double r1591622 = r1591620 + r1591621;
        double r1591623 = sqrt(r1591622);
        double r1591624 = log(r1591623);
        double r1591625 = log(r1591612);
        double r1591626 = r1591618 ? r1591624 : r1591625;
        double r1591627 = r1591614 ? r1591616 : r1591626;
        return r1591627;
}

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 < -3.9812501718676645e+79

    1. Initial program 49.2

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

    2. Taylor expanded around -inf 10.1

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

    3. Simplified10.1

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

    if -3.9812501718676645e+79 < re < 2.270821738026904e+102

    1. Initial program 22.0

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

    if 2.270821738026904e+102 < re

    1. Initial program 52.1

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.981250171867664465389722520730842139436 \cdot 10^{79}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.270821738026904207339185332013542781266 \cdot 10^{102}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))