Average Error: 31.3 → 17.3
Time: 3.1s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.070696817770049897362818226450973536409 \cdot 10^{119}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.290849821627844438172782342942280157051 \cdot 10^{117}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.070696817770049897362818226450973536409 \cdot 10^{119}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 2.290849821627844438172782342942280157051 \cdot 10^{117}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r18421 = re;
        double r18422 = r18421 * r18421;
        double r18423 = im;
        double r18424 = r18423 * r18423;
        double r18425 = r18422 + r18424;
        double r18426 = sqrt(r18425);
        double r18427 = log(r18426);
        return r18427;
}
↓
double f(double re, double im) {
        double r18428 = re;
        double r18429 = -6.07069681777005e+119;
        bool r18430 = r18428 <= r18429;
        double r18431 = -1.0;
        double r18432 = r18431 * r18428;
        double r18433 = log(r18432);
        double r18434 = 2.2908498216278444e+117;
        bool r18435 = r18428 <= r18434;
        double r18436 = r18428 * r18428;
        double r18437 = im;
        double r18438 = r18437 * r18437;
        double r18439 = r18436 + r18438;
        double r18440 = sqrt(r18439);
        double r18441 = log(r18440);
        double r18442 = log(r18428);
        double r18443 = r18435 ? r18441 : r18442;
        double r18444 = r18430 ? r18433 : r18443;
        return r18444;
}

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 < -6.07069681777005e+119

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 8.0

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

    if -6.07069681777005e+119 < re < 2.2908498216278444e+117

    1. Initial program 21.3

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

    if 2.2908498216278444e+117 < re

    1. Initial program 53.6

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

    2. Taylor expanded around inf 7.8

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.070696817770049897362818226450973536409 \cdot 10^{119}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 2.290849821627844438172782342942280157051 \cdot 10^{117}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))