Average Error: 32.1 → 17.5
Time: 2.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.240082689521645815217683079061051755514 \cdot 10^{136}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.213355571208327936170048634465133347109 \cdot 10^{-281}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.169888596740019630894497558694608225431 \cdot 10^{-187}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.727696643549419805924866097503983245008 \cdot 10^{127}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.240082689521645815217683079061051755514 \cdot 10^{136}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 2.213355571208327936170048634465133347109 \cdot 10^{-281}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.169888596740019630894497558694608225431 \cdot 10^{-187}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 5.727696643549419805924866097503983245008 \cdot 10^{127}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r88616 = re;
        double r88617 = r88616 * r88616;
        double r88618 = im;
        double r88619 = r88618 * r88618;
        double r88620 = r88617 + r88619;
        double r88621 = sqrt(r88620);
        double r88622 = log(r88621);
        return r88622;
}
↓
double f(double re, double im) {
        double r88623 = re;
        double r88624 = -1.2400826895216458e+136;
        bool r88625 = r88623 <= r88624;
        double r88626 = -r88623;
        double r88627 = log(r88626);
        double r88628 = 2.213355571208328e-281;
        bool r88629 = r88623 <= r88628;
        double r88630 = r88623 * r88623;
        double r88631 = im;
        double r88632 = r88631 * r88631;
        double r88633 = r88630 + r88632;
        double r88634 = sqrt(r88633);
        double r88635 = log(r88634);
        double r88636 = 1.1698885967400196e-187;
        bool r88637 = r88623 <= r88636;
        double r88638 = log(r88631);
        double r88639 = 5.72769664354942e+127;
        bool r88640 = r88623 <= r88639;
        double r88641 = log(r88623);
        double r88642 = r88640 ? r88635 : r88641;
        double r88643 = r88637 ? r88638 : r88642;
        double r88644 = r88629 ? r88635 : r88643;
        double r88645 = r88625 ? r88627 : r88644;
        return r88645;
}

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 4 regimes

  2. if re < -1.2400826895216458e+136

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 7.6

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

    3. Simplified7.6

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

    if -1.2400826895216458e+136 < re < 2.213355571208328e-281 or 1.1698885967400196e-187 < re < 5.72769664354942e+127

    1. Initial program 20.1

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

    if 2.213355571208328e-281 < re < 1.1698885967400196e-187

    1. Initial program 31.6

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

    2. Taylor expanded around 0 35.0

      \[\leadsto \log \color{blue}{im}\]

    if 5.72769664354942e+127 < re

    1. Initial program 57.1

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

    2. Taylor expanded around inf 7.2

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.240082689521645815217683079061051755514 \cdot 10^{136}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.213355571208327936170048634465133347109 \cdot 10^{-281}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.169888596740019630894497558694608225431 \cdot 10^{-187}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.727696643549419805924866097503983245008 \cdot 10^{127}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))