Average Error: 31.1 → 17.1
Time: 4.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.68163596973405975259895298385316105053 \cdot 10^{102}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.545380571942664302984715356869784321431 \cdot 10^{140}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.68163596973405975259895298385316105053 \cdot 10^{102}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 3.545380571942664302984715356869784321431 \cdot 10^{140}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r2461307 = re;
        double r2461308 = r2461307 * r2461307;
        double r2461309 = im;
        double r2461310 = r2461309 * r2461309;
        double r2461311 = r2461308 + r2461310;
        double r2461312 = sqrt(r2461311);
        double r2461313 = log(r2461312);
        return r2461313;
}
↓
double f(double re, double im) {
        double r2461314 = re;
        double r2461315 = -9.68163596973406e+102;
        bool r2461316 = r2461314 <= r2461315;
        double r2461317 = -r2461314;
        double r2461318 = log(r2461317);
        double r2461319 = 3.5453805719426643e+140;
        bool r2461320 = r2461314 <= r2461319;
        double r2461321 = im;
        double r2461322 = r2461321 * r2461321;
        double r2461323 = r2461314 * r2461314;
        double r2461324 = r2461322 + r2461323;
        double r2461325 = sqrt(r2461324);
        double r2461326 = log(r2461325);
        double r2461327 = log(r2461314);
        double r2461328 = r2461320 ? r2461326 : r2461327;
        double r2461329 = r2461316 ? r2461318 : r2461328;
        return r2461329;
}

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 < -9.68163596973406e+102

    1. Initial program 51.9

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified8.5

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

    if -9.68163596973406e+102 < re < 3.5453805719426643e+140

    1. Initial program 21.0

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

    if 3.5453805719426643e+140 < re

    1. Initial program 59.5

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.68163596973405975259895298385316105053 \cdot 10^{102}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 3.545380571942664302984715356869784321431 \cdot 10^{140}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))