Average Error: 31.8 → 17.5
Time: 4.1s
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 r19473 = re;
        double r19474 = r19473 * r19473;
        double r19475 = im;
        double r19476 = r19475 * r19475;
        double r19477 = r19474 + r19476;
        double r19478 = sqrt(r19477);
        double r19479 = log(r19478);
        return r19479;
}
↓
double f(double re, double im) {
        double r19480 = re;
        double r19481 = -1.1564076018637175e+112;
        bool r19482 = r19480 <= r19481;
        double r19483 = -r19480;
        double r19484 = log(r19483);
        double r19485 = 1.2449882138840628e+138;
        bool r19486 = r19480 <= r19485;
        double r19487 = r19480 * r19480;
        double r19488 = im;
        double r19489 = r19488 * r19488;
        double r19490 = r19487 + r19489;
        double r19491 = sqrt(r19490);
        double r19492 = log(r19491);
        double r19493 = log(r19480);
        double r19494 = r19486 ? r19492 : r19493;
        double r19495 = r19482 ? r19484 : r19494;
        return r19495;
}

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)))))