Average Error: 31.6 → 17.1
Time: 9.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.0142287793804751 \cdot 10^{118}:\\ \;\;\;\;-\log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le 6.23122095692666968 \cdot 10^{141}:\\ \;\;\;\;\log \left(\sqrt{e^{\log \left(re \cdot re + im \cdot im\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.0142287793804751 \cdot 10^{118}:\\
\;\;\;\;-\log \left(\frac{-1}{re}\right)\\

\mathbf{elif}\;re \le 6.23122095692666968 \cdot 10^{141}:\\
\;\;\;\;\log \left(\sqrt{e^{\log \left(re \cdot re + im \cdot im\right)}}\right)\\

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

\end{array}
double f(double re, double im) {
        double r63153 = re;
        double r63154 = r63153 * r63153;
        double r63155 = im;
        double r63156 = r63155 * r63155;
        double r63157 = r63154 + r63156;
        double r63158 = sqrt(r63157);
        double r63159 = log(r63158);
        return r63159;
}
↓
double f(double re, double im) {
        double r63160 = re;
        double r63161 = -2.0142287793804751e+118;
        bool r63162 = r63160 <= r63161;
        double r63163 = -1.0;
        double r63164 = r63163 / r63160;
        double r63165 = log(r63164);
        double r63166 = -r63165;
        double r63167 = 6.23122095692667e+141;
        bool r63168 = r63160 <= r63167;
        double r63169 = r63160 * r63160;
        double r63170 = im;
        double r63171 = r63170 * r63170;
        double r63172 = r63169 + r63171;
        double r63173 = log(r63172);
        double r63174 = exp(r63173);
        double r63175 = sqrt(r63174);
        double r63176 = log(r63175);
        double r63177 = log(r63160);
        double r63178 = r63168 ? r63176 : r63177;
        double r63179 = r63162 ? r63166 : r63178;
        return r63179;
}

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 < -2.0142287793804751e+118

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 8.4

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

    if -2.0142287793804751e+118 < re < 6.23122095692667e+141

    1. Initial program 20.9

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
    2. Using strategy rm

    3. Applied add-exp-log20.9

      \[\leadsto \log \left(\sqrt{\color{blue}{e^{\log \left(re \cdot re + im \cdot im\right)}}}\right)\]

    if 6.23122095692667e+141 < re

    1. Initial program 60.0

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.0142287793804751 \cdot 10^{118}:\\ \;\;\;\;-\log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le 6.23122095692666968 \cdot 10^{141}:\\ \;\;\;\;\log \left(\sqrt{e^{\log \left(re \cdot re + im \cdot im\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))