Average Error: 30.0 → 17.6
Time: 8.5s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.3951279250903065 \cdot 10^{+116}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.2552454715818077 \cdot 10^{-176}:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;re \le 8.33451305802747 \cdot 10^{-256}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8235632211164311.0:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.3951279250903065 \cdot 10^{+116}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -1.2552454715818077 \cdot 10^{-176}:\\
\;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\

\mathbf{elif}\;re \le 8.33451305802747 \cdot 10^{-256}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8235632211164311.0:\\
\;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\

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

\end{array}
double f(double re, double im) {
        double r2922190 = re;
        double r2922191 = r2922190 * r2922190;
        double r2922192 = im;
        double r2922193 = r2922192 * r2922192;
        double r2922194 = r2922191 + r2922193;
        double r2922195 = sqrt(r2922194);
        double r2922196 = log(r2922195);
        return r2922196;
}
↓
double f(double re, double im) {
        double r2922197 = re;
        double r2922198 = -1.3951279250903065e+116;
        bool r2922199 = r2922197 <= r2922198;
        double r2922200 = -r2922197;
        double r2922201 = log(r2922200);
        double r2922202 = -1.2552454715818077e-176;
        bool r2922203 = r2922197 <= r2922202;
        double r2922204 = im;
        double r2922205 = r2922204 * r2922204;
        double r2922206 = r2922197 * r2922197;
        double r2922207 = r2922205 + r2922206;
        double r2922208 = log(r2922207);
        double r2922209 = 0.5;
        double r2922210 = r2922208 * r2922209;
        double r2922211 = 8.33451305802747e-256;
        bool r2922212 = r2922197 <= r2922211;
        double r2922213 = log(r2922204);
        double r2922214 = 8235632211164311.0;
        bool r2922215 = r2922197 <= r2922214;
        double r2922216 = log(r2922197);
        double r2922217 = r2922215 ? r2922210 : r2922216;
        double r2922218 = r2922212 ? r2922213 : r2922217;
        double r2922219 = r2922203 ? r2922210 : r2922218;
        double r2922220 = r2922199 ? r2922201 : r2922219;
        return r2922220;
}

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 4 regimes

  2. if re < -1.3951279250903065e+116

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 7.5

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

    3. Simplified7.5

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

    if -1.3951279250903065e+116 < re < -1.2552454715818077e-176 or 8.33451305802747e-256 < re < 8235632211164311.0

    1. Initial program 18.0

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
    2. Using strategy rm

    3. Applied pow1/218.0

      \[\leadsto \log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}\]

    4. Applied log-pow18.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}\]

    if -1.2552454715818077e-176 < re < 8.33451305802747e-256

    1. Initial program 30.2

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

    2. Taylor expanded around 0 32.5

      \[\leadsto \log \color{blue}{im}\]

    if 8235632211164311.0 < re

    1. Initial program 39.5

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

    2. Taylor expanded around inf 12.8

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3951279250903065 \cdot 10^{+116}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.2552454715818077 \cdot 10^{-176}:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{elif}\;re \le 8.33451305802747 \cdot 10^{-256}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8235632211164311.0:\\ \;\;\;\;\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))