Average Error: 31.4 → 17.3
Time: 3.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r80560 = re;
        double r80561 = r80560 * r80560;
        double r80562 = im;
        double r80563 = r80562 * r80562;
        double r80564 = r80561 + r80563;
        double r80565 = sqrt(r80564);
        double r80566 = log(r80565);
        return r80566;
}
↓
double f(double re, double im) {
        double r80567 = re;
        double r80568 = -2.4740957117892876e+117;
        bool r80569 = r80567 <= r80568;
        double r80570 = -r80567;
        double r80571 = log(r80570);
        double r80572 = 5.435137585363575e+84;
        bool r80573 = r80567 <= r80572;
        double r80574 = r80567 * r80567;
        double r80575 = im;
        double r80576 = r80575 * r80575;
        double r80577 = r80574 + r80576;
        double r80578 = sqrt(r80577);
        double r80579 = log(r80578);
        double r80580 = log(r80567);
        double r80581 = r80573 ? r80579 : r80580;
        double r80582 = r80569 ? r80571 : r80581;
        return r80582;
}

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 < -2.4740957117892876e+117

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 8.0

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

    3. Simplified8.0

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

    if -2.4740957117892876e+117 < re < 5.435137585363575e+84

    1. Initial program 21.6

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

    if 5.435137585363575e+84 < re

    1. Initial program 47.8

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))