Average Error: 32.0 → 17.9
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1982429734725978600:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1982429734725978600:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r88647 = re;
        double r88648 = r88647 * r88647;
        double r88649 = im;
        double r88650 = r88649 * r88649;
        double r88651 = r88648 + r88650;
        double r88652 = sqrt(r88651);
        double r88653 = log(r88652);
        return r88653;
}
↓
double f(double re, double im) {
        double r88654 = re;
        double r88655 = -2.6163751838198674e+143;
        bool r88656 = r88654 <= r88655;
        double r88657 = -1.0;
        double r88658 = r88657 * r88654;
        double r88659 = log(r88658);
        double r88660 = -2.0278572522938575e-184;
        bool r88661 = r88654 <= r88660;
        double r88662 = r88654 * r88654;
        double r88663 = im;
        double r88664 = r88663 * r88663;
        double r88665 = r88662 + r88664;
        double r88666 = sqrt(r88665);
        double r88667 = log(r88666);
        double r88668 = -8.804121592020497e-274;
        bool r88669 = r88654 <= r88668;
        double r88670 = log(r88663);
        double r88671 = 1.9824297347259786e+18;
        bool r88672 = r88654 <= r88671;
        double r88673 = log(r88654);
        double r88674 = r88672 ? r88667 : r88673;
        double r88675 = r88669 ? r88670 : r88674;
        double r88676 = r88661 ? r88667 : r88675;
        double r88677 = r88656 ? r88659 : r88676;
        return r88677;
}

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 < -2.6163751838198674e+143

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 6.8

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

    if -2.6163751838198674e+143 < re < -2.0278572522938575e-184 or -8.804121592020497e-274 < re < 1.9824297347259786e+18

    1. Initial program 20.3

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

    if -2.0278572522938575e-184 < re < -8.804121592020497e-274

    1. Initial program 31.1

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

    2. Taylor expanded around 0 36.8

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

    if 1.9824297347259786e+18 < re

    1. Initial program 42.8

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

    2. Taylor expanded around inf 13.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1982429734725978600:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))