Average Error: 31.3 → 17.3
Time: 4.7s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.070696817770049897362818226450973536409 \cdot 10^{119}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.290849821627844438172782342942280157051 \cdot 10^{117}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.070696817770049897362818226450973536409 \cdot 10^{119}:\\
\;\;\;\;\log \left(-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 r29259 = re;
        double r29260 = r29259 * r29259;
        double r29261 = im;
        double r29262 = r29261 * r29261;
        double r29263 = r29260 + r29262;
        double r29264 = sqrt(r29263);
        double r29265 = log(r29264);
        return r29265;
}
↓
double f(double re, double im) {
        double r29266 = re;
        double r29267 = -6.07069681777005e+119;
        bool r29268 = r29266 <= r29267;
        double r29269 = -r29266;
        double r29270 = log(r29269);
        double r29271 = 2.2908498216278444e+117;
        bool r29272 = r29266 <= r29271;
        double r29273 = r29266 * r29266;
        double r29274 = im;
        double r29275 = r29274 * r29274;
        double r29276 = r29273 + r29275;
        double r29277 = sqrt(r29276);
        double r29278 = log(r29277);
        double r29279 = log(r29266);
        double r29280 = r29272 ? r29278 : r29279;
        double r29281 = r29268 ? r29270 : r29280;
        return r29281;
}

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

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

    3. Simplified8.0

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

    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

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

Reproduce

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