Average Error: 32.1 → 17.7
Time: 1.4s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.00847790071649149 \cdot 10^{147}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 4.7344679219365152 \cdot 10^{65}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.00847790071649149 \cdot 10^{147}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 4.7344679219365152 \cdot 10^{65}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r78113 = re;
        double r78114 = r78113 * r78113;
        double r78115 = im;
        double r78116 = r78115 * r78115;
        double r78117 = r78114 + r78116;
        double r78118 = sqrt(r78117);
        double r78119 = log(r78118);
        return r78119;
}
↓
double f(double re, double im) {
        double r78120 = re;
        double r78121 = -4.0084779007164915e+147;
        bool r78122 = r78120 <= r78121;
        double r78123 = -1.0;
        double r78124 = r78123 * r78120;
        double r78125 = log(r78124);
        double r78126 = 4.734467921936515e+65;
        bool r78127 = r78120 <= r78126;
        double r78128 = r78120 * r78120;
        double r78129 = im;
        double r78130 = r78129 * r78129;
        double r78131 = r78128 + r78130;
        double r78132 = sqrt(r78131);
        double r78133 = log(r78132);
        double r78134 = log(r78120);
        double r78135 = r78127 ? r78133 : r78134;
        double r78136 = r78122 ? r78125 : r78135;
        return r78136;
}

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 < -4.0084779007164915e+147

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 7.1

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

    if -4.0084779007164915e+147 < re < 4.734467921936515e+65

    1. Initial program 21.9

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

    if 4.734467921936515e+65 < re

    1. Initial program 47.4

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

    2. Taylor expanded around inf 10.1

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.00847790071649149 \cdot 10^{147}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 4.7344679219365152 \cdot 10^{65}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))