Average Error: 31.2 → 17.0
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \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 r31559 = re;
        double r31560 = r31559 * r31559;
        double r31561 = im;
        double r31562 = r31561 * r31561;
        double r31563 = r31560 + r31562;
        double r31564 = sqrt(r31563);
        double r31565 = log(r31564);
        return r31565;
}
↓
double f(double re, double im) {
        double r31566 = re;
        double r31567 = -1.0674397664294253e+136;
        bool r31568 = r31566 <= r31567;
        double r31569 = -1.0;
        double r31570 = r31569 * r31566;
        double r31571 = log(r31570);
        double r31572 = -4.4039792789215395e-257;
        bool r31573 = r31566 <= r31572;
        double r31574 = r31566 * r31566;
        double r31575 = im;
        double r31576 = r31575 * r31575;
        double r31577 = r31574 + r31576;
        double r31578 = sqrt(r31577);
        double r31579 = log(r31578);
        double r31580 = 3.8197786805557845e-227;
        bool r31581 = r31566 <= r31580;
        double r31582 = log(r31575);
        double r31583 = 8.439330033545885e+67;
        bool r31584 = r31566 <= r31583;
        double r31585 = log(r31566);
        double r31586 = r31584 ? r31579 : r31585;
        double r31587 = r31581 ? r31582 : r31586;
        double r31588 = r31573 ? r31579 : r31587;
        double r31589 = r31568 ? r31571 : r31588;
        return r31589;
}

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 < -1.0674397664294253e+136

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 7.7

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

    if -1.0674397664294253e+136 < re < -4.4039792789215395e-257 or 3.8197786805557845e-227 < re < 8.439330033545885e+67

    1. Initial program 18.8

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

    if -4.4039792789215395e-257 < re < 3.8197786805557845e-227

    1. Initial program 31.0

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

    2. Taylor expanded around 0 32.6

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

    if 8.439330033545885e+67 < re

    1. Initial program 46.7

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))