Average Error: 30.3 → 16.6
Time: 7.6s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.388360913720003 \cdot 10^{+94}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.421534942183472 \cdot 10^{-187}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -5.407728963531234 \cdot 10^{-222}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.5503547602037399 \cdot 10^{+103}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.388360913720003 \cdot 10^{+94}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -4.421534942183472 \cdot 10^{-187}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \le -5.407728963531234 \cdot 10^{-222}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.5503547602037399 \cdot 10^{+103}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r898287 = re;
        double r898288 = r898287 * r898287;
        double r898289 = im;
        double r898290 = r898289 * r898289;
        double r898291 = r898288 + r898290;
        double r898292 = sqrt(r898291);
        double r898293 = log(r898292);
        return r898293;
}
↓
double f(double re, double im) {
        double r898294 = re;
        double r898295 = -1.388360913720003e+94;
        bool r898296 = r898294 <= r898295;
        double r898297 = -r898294;
        double r898298 = log(r898297);
        double r898299 = -4.421534942183472e-187;
        bool r898300 = r898294 <= r898299;
        double r898301 = im;
        double r898302 = r898301 * r898301;
        double r898303 = r898294 * r898294;
        double r898304 = r898302 + r898303;
        double r898305 = sqrt(r898304);
        double r898306 = log(r898305);
        double r898307 = -5.407728963531234e-222;
        bool r898308 = r898294 <= r898307;
        double r898309 = log(r898301);
        double r898310 = 1.5503547602037399e+103;
        bool r898311 = r898294 <= r898310;
        double r898312 = log(r898294);
        double r898313 = r898311 ? r898306 : r898312;
        double r898314 = r898308 ? r898309 : r898313;
        double r898315 = r898300 ? r898306 : r898314;
        double r898316 = r898296 ? r898298 : r898315;
        return r898316;
}

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 < -1.388360913720003e+94

    1. Initial program 48.5

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

    2. Taylor expanded around -inf 8.0

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

    3. Simplified8.0

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

    if -1.388360913720003e+94 < re < -4.421534942183472e-187 or -5.407728963531234e-222 < re < 1.5503547602037399e+103

    1. Initial program 20.1

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

    if -4.421534942183472e-187 < re < -5.407728963531234e-222

    1. Initial program 31.6

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

    2. Taylor expanded around 0 34.5

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

    if 1.5503547602037399e+103 < re

    1. Initial program 49.8

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

    2. Taylor expanded around inf 8.6

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

    3. Simplified8.6

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.388360913720003 \cdot 10^{+94}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.421534942183472 \cdot 10^{-187}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -5.407728963531234 \cdot 10^{-222}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.5503547602037399 \cdot 10^{+103}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))