Average Error: 31.4 → 17.1
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 3.914068510197849137483898167156006374927 \cdot 10^{109}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 3.914068510197849137483898167156006374927 \cdot 10^{109}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r34080 = re;
        double r34081 = r34080 * r34080;
        double r34082 = im;
        double r34083 = r34082 * r34082;
        double r34084 = r34081 + r34083;
        double r34085 = sqrt(r34084);
        double r34086 = log(r34085);
        return r34086;
}
↓
double f(double re, double im) {
        double r34087 = re;
        double r34088 = -1.9802675205288276e+108;
        bool r34089 = r34087 <= r34088;
        double r34090 = -1.0;
        double r34091 = r34090 * r34087;
        double r34092 = log(r34091);
        double r34093 = 3.914068510197849e+109;
        bool r34094 = r34087 <= r34093;
        double r34095 = r34087 * r34087;
        double r34096 = im;
        double r34097 = r34096 * r34096;
        double r34098 = r34095 + r34097;
        double r34099 = sqrt(r34098);
        double r34100 = log(r34099);
        double r34101 = log(r34087);
        double r34102 = r34094 ? r34100 : r34101;
        double r34103 = r34089 ? r34092 : r34102;
        return r34103;
}

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 < -1.9802675205288276e+108

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 8.4

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

    if -1.9802675205288276e+108 < re < 3.914068510197849e+109

    1. Initial program 21.3

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

    if 3.914068510197849e+109 < re

    1. Initial program 53.6

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

    2. Taylor expanded around inf 8.3

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 3.914068510197849137483898167156006374927 \cdot 10^{109}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))