Average Error: 32.0 → 17.4
Time: 1.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -210362893173384.781:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.4595461172925864 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -210362893173384.781:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 2.4595461172925864 \cdot 10^{94}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r27039 = re;
        double r27040 = r27039 * r27039;
        double r27041 = im;
        double r27042 = r27041 * r27041;
        double r27043 = r27040 + r27042;
        double r27044 = sqrt(r27043);
        double r27045 = log(r27044);
        return r27045;
}
↓
double f(double re, double im) {
        double r27046 = re;
        double r27047 = -210362893173384.78;
        bool r27048 = r27046 <= r27047;
        double r27049 = -1.0;
        double r27050 = r27049 * r27046;
        double r27051 = log(r27050);
        double r27052 = 2.4595461172925864e+94;
        bool r27053 = r27046 <= r27052;
        double r27054 = r27046 * r27046;
        double r27055 = im;
        double r27056 = r27055 * r27055;
        double r27057 = r27054 + r27056;
        double r27058 = sqrt(r27057);
        double r27059 = log(r27058);
        double r27060 = log(r27046);
        double r27061 = r27053 ? r27059 : r27060;
        double r27062 = r27048 ? r27051 : r27061;
        return r27062;
}

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 < -210362893173384.78

    1. Initial program 41.8

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

    2. Taylor expanded around -inf 11.7

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

    if -210362893173384.78 < re < 2.4595461172925864e+94

    1. Initial program 22.5

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

    if 2.4595461172925864e+94 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 8.1

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -210362893173384.781:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.4595461172925864 \cdot 10^{94}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))