Average Error: 31.0 → 16.9
Time: 3.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.194751856245766 \cdot 10^{+143}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.7317281282196852 \cdot 10^{+140}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.194751856245766 \cdot 10^{+143}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.7317281282196852 \cdot 10^{+140}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r1367107 = re;
        double r1367108 = r1367107 * r1367107;
        double r1367109 = im;
        double r1367110 = r1367109 * r1367109;
        double r1367111 = r1367108 + r1367110;
        double r1367112 = sqrt(r1367111);
        double r1367113 = log(r1367112);
        return r1367113;
}
↓
double f(double re, double im) {
        double r1367114 = re;
        double r1367115 = -3.194751856245766e+143;
        bool r1367116 = r1367114 <= r1367115;
        double r1367117 = -r1367114;
        double r1367118 = log(r1367117);
        double r1367119 = 1.7317281282196852e+140;
        bool r1367120 = r1367114 <= r1367119;
        double r1367121 = im;
        double r1367122 = r1367121 * r1367121;
        double r1367123 = r1367114 * r1367114;
        double r1367124 = r1367122 + r1367123;
        double r1367125 = sqrt(r1367124);
        double r1367126 = log(r1367125);
        double r1367127 = log(r1367114);
        double r1367128 = r1367120 ? r1367126 : r1367127;
        double r1367129 = r1367116 ? r1367118 : r1367128;
        return r1367129;
}

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 < -3.194751856245766e+143

    1. Initial program 59.5

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

    2. Taylor expanded around -inf 6.9

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

    3. Simplified6.9

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

    if -3.194751856245766e+143 < re < 1.7317281282196852e+140

    1. Initial program 20.5

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

    if 1.7317281282196852e+140 < re

    1. Initial program 57.9

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

    2. Taylor expanded around inf 7.6

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.194751856245766 \cdot 10^{+143}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.7317281282196852 \cdot 10^{+140}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))