Average Error: 32.0 → 17.7
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.0370240534066732 \cdot 10^{-273}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.9546023522807356 \cdot 10^{-186}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 7.34377542514503093 \cdot 10^{133}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.0370240534066732 \cdot 10^{-273}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.9546023522807356 \cdot 10^{-186}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 7.34377542514503093 \cdot 10^{133}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r31582 = re;
        double r31583 = r31582 * r31582;
        double r31584 = im;
        double r31585 = r31584 * r31584;
        double r31586 = r31583 + r31585;
        double r31587 = sqrt(r31586);
        double r31588 = log(r31587);
        return r31588;
}
↓
double f(double re, double im) {
        double r31589 = re;
        double r31590 = -9.16501881147336e+142;
        bool r31591 = r31589 <= r31590;
        double r31592 = -1.0;
        double r31593 = r31592 * r31589;
        double r31594 = log(r31593);
        double r31595 = -2.037024053406673e-273;
        bool r31596 = r31589 <= r31595;
        double r31597 = r31589 * r31589;
        double r31598 = im;
        double r31599 = r31598 * r31598;
        double r31600 = r31597 + r31599;
        double r31601 = sqrt(r31600);
        double r31602 = log(r31601);
        double r31603 = 3.954602352280736e-186;
        bool r31604 = r31589 <= r31603;
        double r31605 = log(r31598);
        double r31606 = 7.343775425145031e+133;
        bool r31607 = r31589 <= r31606;
        double r31608 = log(r31589);
        double r31609 = r31607 ? r31602 : r31608;
        double r31610 = r31604 ? r31605 : r31609;
        double r31611 = r31596 ? r31602 : r31610;
        double r31612 = r31591 ? r31594 : r31611;
        return r31612;
}

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 < -9.16501881147336e+142

    1. Initial program 61.3

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

    2. Taylor expanded around -inf 7.6

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

    if -9.16501881147336e+142 < re < -2.037024053406673e-273 or 3.954602352280736e-186 < re < 7.343775425145031e+133

    1. Initial program 19.0

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

    if -2.037024053406673e-273 < re < 3.954602352280736e-186

    1. Initial program 31.3

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

    2. Taylor expanded around 0 34.6

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

    if 7.343775425145031e+133 < re

    1. Initial program 58.6

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

    2. Taylor expanded around inf 7.7

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.0370240534066732 \cdot 10^{-273}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.9546023522807356 \cdot 10^{-186}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 7.34377542514503093 \cdot 10^{133}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))