Average Error: 31.8 → 17.5
Time: 3.8s
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 r23331 = re;
        double r23332 = r23331 * r23331;
        double r23333 = im;
        double r23334 = r23333 * r23333;
        double r23335 = r23332 + r23334;
        double r23336 = sqrt(r23335);
        double r23337 = log(r23336);
        return r23337;
}
↓
double f(double re, double im) {
        double r23338 = re;
        double r23339 = -1.1564076018637175e+112;
        bool r23340 = r23338 <= r23339;
        double r23341 = -r23338;
        double r23342 = log(r23341);
        double r23343 = 1.2449882138840628e+138;
        bool r23344 = r23338 <= r23343;
        double r23345 = r23338 * r23338;
        double r23346 = im;
        double r23347 = r23346 * r23346;
        double r23348 = r23345 + r23347;
        double r23349 = sqrt(r23348);
        double r23350 = log(r23349);
        double r23351 = log(r23338);
        double r23352 = r23344 ? r23350 : r23351;
        double r23353 = r23340 ? r23342 : r23352;
        return r23353;
}

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)))))