Average Error: 31.4 → 17.2
Time: 1.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2199670469480383 \cdot 10^{95}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 3.8849079706542796 \cdot 10^{45}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2199670469480383 \cdot 10^{95}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 3.8849079706542796 \cdot 10^{45}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r86286 = re;
        double r86287 = r86286 * r86286;
        double r86288 = im;
        double r86289 = r86288 * r86288;
        double r86290 = r86287 + r86289;
        double r86291 = sqrt(r86290);
        double r86292 = log(r86291);
        return r86292;
}
↓
double f(double re, double im) {
        double r86293 = re;
        double r86294 = -1.2199670469480383e+95;
        bool r86295 = r86293 <= r86294;
        double r86296 = -1.0;
        double r86297 = r86296 * r86293;
        double r86298 = log(r86297);
        double r86299 = 3.8849079706542796e+45;
        bool r86300 = r86293 <= r86299;
        double r86301 = r86293 * r86293;
        double r86302 = im;
        double r86303 = r86302 * r86302;
        double r86304 = r86301 + r86303;
        double r86305 = sqrt(r86304);
        double r86306 = log(r86305);
        double r86307 = log(r86293);
        double r86308 = r86300 ? r86306 : r86307;
        double r86309 = r86295 ? r86298 : r86308;
        return r86309;
}

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 < -1.2199670469480383e+95

    1. Initial program 50.3

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

    2. Taylor expanded around -inf 8.8

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

    if -1.2199670469480383e+95 < re < 3.8849079706542796e+45

    1. Initial program 21.5

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

    if 3.8849079706542796e+45 < re

    1. Initial program 44.7

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

    2. Taylor expanded around inf 11.5

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2199670469480383 \cdot 10^{95}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 3.8849079706542796 \cdot 10^{45}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))