Average Error: 32.0 → 17.7
Time: 2.3s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.0370240534066732 \cdot 10^{-273}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.9546023522807356 \cdot 10^{-186}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 7.34377542514503093 \cdot 10^{133}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.0370240534066732 \cdot 10^{-273}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.9546023522807356 \cdot 10^{-186}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 7.34377542514503093 \cdot 10^{133}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r178 = re;
        double r179 = r178 * r178;
        double r180 = im;
        double r181 = r180 * r180;
        double r182 = r179 + r181;
        double r183 = sqrt(r182);
        double r184 = log(r183);
        return r184;
}
↓
double f(double re, double im) {
        double r185 = re;
        double r186 = -9.16501881147336e+142;
        bool r187 = r185 <= r186;
        double r188 = -1.0;
        double r189 = r188 * r185;
        double r190 = log(r189);
        double r191 = -2.037024053406673e-273;
        bool r192 = r185 <= r191;
        double r193 = r185 * r185;
        double r194 = im;
        double r195 = r194 * r194;
        double r196 = r193 + r195;
        double r197 = sqrt(r196);
        double r198 = log(r197);
        double r199 = 3.954602352280736e-186;
        bool r200 = r185 <= r199;
        double r201 = log(r194);
        double r202 = 7.343775425145031e+133;
        bool r203 = r185 <= r202;
        double r204 = log(r185);
        double r205 = r203 ? r198 : r204;
        double r206 = r200 ? r201 : r205;
        double r207 = r192 ? r198 : r206;
        double r208 = r187 ? r190 : r207;
        return r208;
}

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 < -9.16501881147336e+142

    1. Initial program 61.3

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

    2. Taylor expanded around -inf 7.6

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

    if -9.16501881147336e+142 < re < -2.037024053406673e-273 or 3.954602352280736e-186 < re < 7.343775425145031e+133

    1. Initial program 19.0

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

    if -2.037024053406673e-273 < re < 3.954602352280736e-186

    1. Initial program 31.3

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

    2. Taylor expanded around 0 34.6

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

    if 7.343775425145031e+133 < re

    1. Initial program 58.6

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

    2. Taylor expanded around inf 7.7

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.0370240534066732 \cdot 10^{-273}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.9546023522807356 \cdot 10^{-186}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 7.34377542514503093 \cdot 10^{133}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))