Average Error: 31.1 → 16.9
Time: 4.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.1674722654441826 \cdot 10^{+113}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.4251927726542934 \cdot 10^{+129}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.1674722654441826 \cdot 10^{+113}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.4251927726542934 \cdot 10^{+129}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1101154 = re;
        double r1101155 = r1101154 * r1101154;
        double r1101156 = im;
        double r1101157 = r1101156 * r1101156;
        double r1101158 = r1101155 + r1101157;
        double r1101159 = sqrt(r1101158);
        double r1101160 = log(r1101159);
        return r1101160;
}
↓
double f(double re, double im) {
        double r1101161 = re;
        double r1101162 = -5.1674722654441826e+113;
        bool r1101163 = r1101161 <= r1101162;
        double r1101164 = -r1101161;
        double r1101165 = log(r1101164);
        double r1101166 = 1.4251927726542934e+129;
        bool r1101167 = r1101161 <= r1101166;
        double r1101168 = im;
        double r1101169 = r1101168 * r1101168;
        double r1101170 = r1101161 * r1101161;
        double r1101171 = r1101169 + r1101170;
        double r1101172 = sqrt(r1101171);
        double r1101173 = log(r1101172);
        double r1101174 = log(r1101161);
        double r1101175 = r1101167 ? r1101173 : r1101174;
        double r1101176 = r1101163 ? r1101165 : r1101175;
        return r1101176;
}

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 < -5.1674722654441826e+113

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 8.1

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

    3. Simplified8.1

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

    if -5.1674722654441826e+113 < re < 1.4251927726542934e+129

    1. Initial program 21.0

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

    if 1.4251927726542934e+129 < re

    1. Initial program 55.4

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

    2. Taylor expanded around inf 7.2

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.1674722654441826 \cdot 10^{+113}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.4251927726542934 \cdot 10^{+129}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))