Average Error: 31.9 → 17.8
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.989101613628458 \cdot 10^{44}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.744576864806113 \cdot 10^{-224}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1417370863594925 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.989101613628458 \cdot 10^{44}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 2.744576864806113 \cdot 10^{-224}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.1417370863594925 \cdot 10^{101}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r84712 = re;
        double r84713 = r84712 * r84712;
        double r84714 = im;
        double r84715 = r84714 * r84714;
        double r84716 = r84713 + r84715;
        double r84717 = sqrt(r84716);
        double r84718 = log(r84717);
        return r84718;
}
↓
double f(double re, double im) {
        double r84719 = re;
        double r84720 = -1.989101613628458e+44;
        bool r84721 = r84719 <= r84720;
        double r84722 = -1.0;
        double r84723 = r84722 * r84719;
        double r84724 = log(r84723);
        double r84725 = 7.942397244706197e-271;
        bool r84726 = r84719 <= r84725;
        double r84727 = r84719 * r84719;
        double r84728 = im;
        double r84729 = r84728 * r84728;
        double r84730 = r84727 + r84729;
        double r84731 = sqrt(r84730);
        double r84732 = log(r84731);
        double r84733 = 2.744576864806113e-224;
        bool r84734 = r84719 <= r84733;
        double r84735 = log(r84728);
        double r84736 = 1.1417370863594925e+101;
        bool r84737 = r84719 <= r84736;
        double r84738 = log(r84719);
        double r84739 = r84737 ? r84732 : r84738;
        double r84740 = r84734 ? r84735 : r84739;
        double r84741 = r84726 ? r84732 : r84740;
        double r84742 = r84721 ? r84724 : r84741;
        return r84742;
}

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 < -1.989101613628458e+44

    1. Initial program 43.3

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

    2. Taylor expanded around -inf 12.1

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

    if -1.989101613628458e+44 < re < 7.942397244706197e-271 or 2.744576864806113e-224 < re < 1.1417370863594925e+101

    1. Initial program 21.6

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

    if 7.942397244706197e-271 < re < 2.744576864806113e-224

    1. Initial program 29.7

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

    2. Taylor expanded around 0 31.9

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

    if 1.1417370863594925e+101 < re

    1. Initial program 52.5

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.989101613628458 \cdot 10^{44}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.744576864806113 \cdot 10^{-224}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1417370863594925 \cdot 10^{101}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))