Average Error: 31.8 → 17.5
Time: 7.9s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r40192 = re;
        double r40193 = r40192 * r40192;
        double r40194 = im;
        double r40195 = r40194 * r40194;
        double r40196 = r40193 + r40195;
        double r40197 = sqrt(r40196);
        double r40198 = log(r40197);
        return r40198;
}
↓
double f(double re, double im) {
        double r40199 = re;
        double r40200 = -1.1564076018637175e+112;
        bool r40201 = r40199 <= r40200;
        double r40202 = -r40199;
        double r40203 = log(r40202);
        double r40204 = 1.2449882138840628e+138;
        bool r40205 = r40199 <= r40204;
        double r40206 = r40199 * r40199;
        double r40207 = im;
        double r40208 = r40207 * r40207;
        double r40209 = r40206 + r40208;
        double r40210 = sqrt(r40209);
        double r40211 = log(r40210);
        double r40212 = log(r40199);
        double r40213 = r40205 ? r40211 : r40212;
        double r40214 = r40201 ? r40203 : r40213;
        return r40214;
}

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 < -1.1564076018637175e+112

    1. Initial program 52.8

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

    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 -1.1564076018637175e+112 < re < 1.2449882138840628e+138

    1. Initial program 21.7

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

    if 1.2449882138840628e+138 < re

    1. Initial program 58.8

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

    2. Taylor expanded around inf 7.6

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.244988213884062755522549209945596691708 \cdot 10^{138}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))