Average Error: 32.0 → 17.3
Time: 2.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r55054 = re;
        double r55055 = r55054 * r55054;
        double r55056 = im;
        double r55057 = r55056 * r55056;
        double r55058 = r55055 + r55057;
        double r55059 = sqrt(r55058);
        double r55060 = log(r55059);
        return r55060;
}
↓
double f(double re, double im) {
        double r55061 = re;
        double r55062 = -9.983262521343274e+136;
        bool r55063 = r55061 <= r55062;
        double r55064 = -1.0;
        double r55065 = r55064 * r55061;
        double r55066 = log(r55065);
        double r55067 = 2.7153468834491098e+73;
        bool r55068 = r55061 <= r55067;
        double r55069 = r55061 * r55061;
        double r55070 = im;
        double r55071 = r55070 * r55070;
        double r55072 = r55069 + r55071;
        double r55073 = sqrt(r55072);
        double r55074 = log(r55073);
        double r55075 = log(r55061);
        double r55076 = r55068 ? r55074 : r55075;
        double r55077 = r55063 ? r55066 : r55076;
        return r55077;
}

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 < -9.983262521343274e+136

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 7.8

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

    if -9.983262521343274e+136 < re < 2.7153468834491098e+73

    1. Initial program 21.8

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

    if 2.7153468834491098e+73 < re

    1. Initial program 47.0

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

    2. Taylor expanded around inf 8.9

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))