Average Error: 32.0 → 17.0
Time: 2.6s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.682619343593359335222412458502304029603 \cdot 10^{146}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.948735873616171181106179890855443778222 \cdot 10^{127}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.682619343593359335222412458502304029603 \cdot 10^{146}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 1.948735873616171181106179890855443778222 \cdot 10^{127}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r23076 = re;
        double r23077 = r23076 * r23076;
        double r23078 = im;
        double r23079 = r23078 * r23078;
        double r23080 = r23077 + r23079;
        double r23081 = sqrt(r23080);
        double r23082 = log(r23081);
        return r23082;
}
↓
double f(double re, double im) {
        double r23083 = re;
        double r23084 = -5.682619343593359e+146;
        bool r23085 = r23083 <= r23084;
        double r23086 = -1.0;
        double r23087 = r23086 * r23083;
        double r23088 = log(r23087);
        double r23089 = 1.9487358736161712e+127;
        bool r23090 = r23083 <= r23089;
        double r23091 = r23083 * r23083;
        double r23092 = im;
        double r23093 = r23092 * r23092;
        double r23094 = r23091 + r23093;
        double r23095 = sqrt(r23094);
        double r23096 = log(r23095);
        double r23097 = log(r23083);
        double r23098 = r23090 ? r23096 : r23097;
        double r23099 = r23085 ? r23088 : r23098;
        return r23099;
}

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 < -5.682619343593359e+146

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 7.5

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

    if -5.682619343593359e+146 < re < 1.9487358736161712e+127

    1. Initial program 21.0

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

    if 1.9487358736161712e+127 < re

    1. Initial program 56.3

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.682619343593359335222412458502304029603 \cdot 10^{146}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.948735873616171181106179890855443778222 \cdot 10^{127}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))