Average Error: 31.7 → 17.6
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.2090901422421893 \cdot 10^{152}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 5087537.9202582333:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.2090901422421893 \cdot 10^{152}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 5087537.9202582333:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r28222 = re;
        double r28223 = r28222 * r28222;
        double r28224 = im;
        double r28225 = r28224 * r28224;
        double r28226 = r28223 + r28225;
        double r28227 = sqrt(r28226);
        double r28228 = log(r28227);
        return r28228;
}
↓
double f(double re, double im) {
        double r28229 = re;
        double r28230 = -4.209090142242189e+152;
        bool r28231 = r28229 <= r28230;
        double r28232 = -1.0;
        double r28233 = r28232 * r28229;
        double r28234 = log(r28233);
        double r28235 = 5087537.920258233;
        bool r28236 = r28229 <= r28235;
        double r28237 = r28229 * r28229;
        double r28238 = im;
        double r28239 = r28238 * r28238;
        double r28240 = r28237 + r28239;
        double r28241 = sqrt(r28240);
        double r28242 = log(r28241);
        double r28243 = log(r28229);
        double r28244 = r28236 ? r28242 : r28243;
        double r28245 = r28231 ? r28234 : r28244;
        return r28245;
}

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 < -4.209090142242189e+152

    1. Initial program 63.8

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

    2. Taylor expanded around -inf 6.3

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

    if -4.209090142242189e+152 < re < 5087537.920258233

    1. Initial program 21.5

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

    if 5087537.920258233 < re

    1. Initial program 40.8

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

    2. Taylor expanded around inf 13.7

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.2090901422421893 \cdot 10^{152}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 5087537.9202582333:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))