Average Error: 31.6 → 17.6
Time: 3.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6312992236024163726688387878937257276604000:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.949891623707017456131986292760721140859 \cdot 10^{80}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6312992236024163726688387878937257276604000:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.949891623707017456131986292760721140859 \cdot 10^{80}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r43341 = re;
        double r43342 = r43341 * r43341;
        double r43343 = im;
        double r43344 = r43343 * r43343;
        double r43345 = r43342 + r43344;
        double r43346 = sqrt(r43345);
        double r43347 = log(r43346);
        return r43347;
}
↓
double f(double re, double im) {
        double r43348 = re;
        double r43349 = -6.312992236024164e+42;
        bool r43350 = r43348 <= r43349;
        double r43351 = -r43348;
        double r43352 = log(r43351);
        double r43353 = 1.9498916237070175e+80;
        bool r43354 = r43348 <= r43353;
        double r43355 = r43348 * r43348;
        double r43356 = im;
        double r43357 = r43356 * r43356;
        double r43358 = r43355 + r43357;
        double r43359 = sqrt(r43358);
        double r43360 = log(r43359);
        double r43361 = log(r43348);
        double r43362 = r43354 ? r43360 : r43361;
        double r43363 = r43350 ? r43352 : r43362;
        return r43363;
}

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 < -6.312992236024164e+42

    1. Initial program 43.4

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

    2. Taylor expanded around -inf 10.9

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

    3. Simplified10.9

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

    if -6.312992236024164e+42 < re < 1.9498916237070175e+80

    1. Initial program 22.5

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

    if 1.9498916237070175e+80 < re

    1. Initial program 48.6

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

    2. Taylor expanded around inf 8.5

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6312992236024163726688387878937257276604000:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.949891623707017456131986292760721140859 \cdot 10^{80}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))