Average Error: 32.0 → 17.3
Time: 4.5s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.983262521343274363476799981820042586015 \cdot 10^{136}:\\
\;\;\;\;\log \left(-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 r26040 = re;
        double r26041 = r26040 * r26040;
        double r26042 = im;
        double r26043 = r26042 * r26042;
        double r26044 = r26041 + r26043;
        double r26045 = sqrt(r26044);
        double r26046 = log(r26045);
        return r26046;
}
↓
double f(double re, double im) {
        double r26047 = re;
        double r26048 = -9.983262521343274e+136;
        bool r26049 = r26047 <= r26048;
        double r26050 = -r26047;
        double r26051 = log(r26050);
        double r26052 = 2.7153468834491098e+73;
        bool r26053 = r26047 <= r26052;
        double r26054 = r26047 * r26047;
        double r26055 = im;
        double r26056 = r26055 * r26055;
        double r26057 = r26054 + r26056;
        double r26058 = sqrt(r26057);
        double r26059 = log(r26058);
        double r26060 = log(r26047);
        double r26061 = r26053 ? r26059 : r26060;
        double r26062 = r26049 ? r26051 : r26061;
        return r26062;
}

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

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

    3. Simplified7.8

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

    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

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

Reproduce

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