Average Error: 31.5 → 18.5
Time: 5.5s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.482912996481695209350075344359753892544 \cdot 10^{-250}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.50977017724907722738153182022955067076 \cdot 10^{55}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.482912996481695209350075344359753892544 \cdot 10^{-250}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 6.50977017724907722738153182022955067076 \cdot 10^{55}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r22527 = re;
        double r22528 = r22527 * r22527;
        double r22529 = im;
        double r22530 = r22529 * r22529;
        double r22531 = r22528 + r22530;
        double r22532 = sqrt(r22531);
        double r22533 = log(r22532);
        return r22533;
}
↓
double f(double re, double im) {
        double r22534 = re;
        double r22535 = -5.330091552844717e+114;
        bool r22536 = r22534 <= r22535;
        double r22537 = -r22534;
        double r22538 = log(r22537);
        double r22539 = -4.2156616274993736e-144;
        bool r22540 = r22534 <= r22539;
        double r22541 = r22534 * r22534;
        double r22542 = im;
        double r22543 = r22542 * r22542;
        double r22544 = r22541 + r22543;
        double r22545 = sqrt(r22544);
        double r22546 = log(r22545);
        double r22547 = 3.482912996481695e-250;
        bool r22548 = r22534 <= r22547;
        double r22549 = log(r22542);
        double r22550 = 6.509770177249077e+55;
        bool r22551 = r22534 <= r22550;
        double r22552 = log(r22534);
        double r22553 = r22551 ? r22546 : r22552;
        double r22554 = r22548 ? r22549 : r22553;
        double r22555 = r22540 ? r22546 : r22554;
        double r22556 = r22536 ? r22538 : r22555;
        return r22556;
}

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 < -5.330091552844717e+114

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 7.4

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

    3. Simplified7.4

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

    if -5.330091552844717e+114 < re < -4.2156616274993736e-144 or 3.482912996481695e-250 < re < 6.509770177249077e+55

    1. Initial program 18.7

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

    if -4.2156616274993736e-144 < re < 3.482912996481695e-250

    1. Initial program 31.0

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

    2. Taylor expanded around 0 35.8

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

    if 6.509770177249077e+55 < re

    1. Initial program 44.0

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

    2. Taylor expanded around inf 11.1

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.482912996481695209350075344359753892544 \cdot 10^{-250}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.50977017724907722738153182022955067076 \cdot 10^{55}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))