Average Error: 31.3 → 17.3
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.552702335522331775782681989067170419459 \cdot 10^{114}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.241086545067041095458499268505642473269 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.552702335522331775782681989067170419459 \cdot 10^{114}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.241086545067041095458499268505642473269 \cdot 10^{67}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r36519 = re;
        double r36520 = r36519 * r36519;
        double r36521 = im;
        double r36522 = r36521 * r36521;
        double r36523 = r36520 + r36522;
        double r36524 = sqrt(r36523);
        double r36525 = log(r36524);
        return r36525;
}
↓
double f(double re, double im) {
        double r36526 = re;
        double r36527 = -2.552702335522332e+114;
        bool r36528 = r36526 <= r36527;
        double r36529 = -1.0;
        double r36530 = r36529 * r36526;
        double r36531 = log(r36530);
        double r36532 = -8.357854099413859e-296;
        bool r36533 = r36526 <= r36532;
        double r36534 = r36526 * r36526;
        double r36535 = im;
        double r36536 = r36535 * r36535;
        double r36537 = r36534 + r36536;
        double r36538 = sqrt(r36537);
        double r36539 = log(r36538);
        double r36540 = 7.758435713562232e-281;
        bool r36541 = r36526 <= r36540;
        double r36542 = log(r36535);
        double r36543 = 1.2410865450670411e+67;
        bool r36544 = r36526 <= r36543;
        double r36545 = log(r36526);
        double r36546 = r36544 ? r36539 : r36545;
        double r36547 = r36541 ? r36542 : r36546;
        double r36548 = r36533 ? r36539 : r36547;
        double r36549 = r36528 ? r36531 : r36548;
        return r36549;
}

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 < -2.552702335522332e+114

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.2

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

    if -2.552702335522332e+114 < re < -8.357854099413859e-296 or 7.758435713562232e-281 < re < 1.2410865450670411e+67

    1. Initial program 21.3

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

    if -8.357854099413859e-296 < re < 7.758435713562232e-281

    1. Initial program 29.5

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

    2. Taylor expanded around 0 34.3

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

    if 1.2410865450670411e+67 < re

    1. Initial program 45.0

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

    2. Taylor expanded around inf 9.6

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.552702335522331775782681989067170419459 \cdot 10^{114}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.241086545067041095458499268505642473269 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))