Average Error: 31.8 → 17.5
Time: 4.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r89510 = re;
        double r89511 = r89510 * r89510;
        double r89512 = im;
        double r89513 = r89512 * r89512;
        double r89514 = r89511 + r89513;
        double r89515 = sqrt(r89514);
        double r89516 = log(r89515);
        return r89516;
}
↓
double f(double re, double im) {
        double r89517 = re;
        double r89518 = -1.1564076018637175e+112;
        bool r89519 = r89517 <= r89518;
        double r89520 = -r89517;
        double r89521 = log(r89520);
        double r89522 = 1.2449882138840628e+138;
        bool r89523 = r89517 <= r89522;
        double r89524 = r89517 * r89517;
        double r89525 = im;
        double r89526 = r89525 * r89525;
        double r89527 = r89524 + r89526;
        double r89528 = sqrt(r89527);
        double r89529 = log(r89528);
        double r89530 = log(r89517);
        double r89531 = r89523 ? r89529 : r89530;
        double r89532 = r89519 ? r89521 : r89531;
        return r89532;
}

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 < -1.1564076018637175e+112

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

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

    if -1.1564076018637175e+112 < re < 1.2449882138840628e+138

    1. Initial program 21.7

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

    if 1.2449882138840628e+138 < re

    1. Initial program 58.8

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

    2. Taylor expanded around inf 7.6

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))