Average Error: 31.3 → 17.3
Time: 4.6s
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 r23301 = re;
        double r23302 = r23301 * r23301;
        double r23303 = im;
        double r23304 = r23303 * r23303;
        double r23305 = r23302 + r23304;
        double r23306 = sqrt(r23305);
        double r23307 = log(r23306);
        return r23307;
}
↓
double f(double re, double im) {
        double r23308 = re;
        double r23309 = -6.07069681777005e+119;
        bool r23310 = r23308 <= r23309;
        double r23311 = -r23308;
        double r23312 = log(r23311);
        double r23313 = 2.2908498216278444e+117;
        bool r23314 = r23308 <= r23313;
        double r23315 = r23308 * r23308;
        double r23316 = im;
        double r23317 = r23316 * r23316;
        double r23318 = r23315 + r23317;
        double r23319 = sqrt(r23318);
        double r23320 = log(r23319);
        double r23321 = log(r23308);
        double r23322 = r23314 ? r23320 : r23321;
        double r23323 = r23310 ? r23312 : r23322;
        return r23323;
}

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)))))