Average Error: 32.1 → 17.2
Time: 3.8s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.844506481309793477015811086474695118347 \cdot 10^{101}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.3735182711102215089283798883990759293 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.844506481309793477015811086474695118347 \cdot 10^{101}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 6.3735182711102215089283798883990759293 \cdot 10^{95}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1402097 = re;
        double r1402098 = r1402097 * r1402097;
        double r1402099 = im;
        double r1402100 = r1402099 * r1402099;
        double r1402101 = r1402098 + r1402100;
        double r1402102 = sqrt(r1402101);
        double r1402103 = log(r1402102);
        return r1402103;
}
↓
double f(double re, double im) {
        double r1402104 = re;
        double r1402105 = -4.8445064813097935e+101;
        bool r1402106 = r1402104 <= r1402105;
        double r1402107 = -r1402104;
        double r1402108 = log(r1402107);
        double r1402109 = 6.373518271110222e+95;
        bool r1402110 = r1402104 <= r1402109;
        double r1402111 = im;
        double r1402112 = r1402111 * r1402111;
        double r1402113 = r1402104 * r1402104;
        double r1402114 = r1402112 + r1402113;
        double r1402115 = sqrt(r1402114);
        double r1402116 = log(r1402115);
        double r1402117 = log(r1402104);
        double r1402118 = r1402110 ? r1402116 : r1402117;
        double r1402119 = r1402106 ? r1402108 : r1402118;
        return r1402119;
}

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 < -4.8445064813097935e+101

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 7.7

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

    3. Simplified7.7

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

    if -4.8445064813097935e+101 < re < 6.373518271110222e+95

    1. Initial program 21.9

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

    if 6.373518271110222e+95 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 8.5

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.844506481309793477015811086474695118347 \cdot 10^{101}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.3735182711102215089283798883990759293 \cdot 10^{95}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))