Average Error: 31.5 → 17.4
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.12229097909121671 \cdot 10^{44}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.8012497291896643 \cdot 10^{-291}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.64838694760230754 \cdot 10^{-196}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.58395198674246248 \cdot 10^{49}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.12229097909121671 \cdot 10^{44}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le 1.8012497291896643 \cdot 10^{-291}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.64838694760230754 \cdot 10^{-196}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1.58395198674246248 \cdot 10^{49}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r81289 = re;
        double r81290 = r81289 * r81289;
        double r81291 = im;
        double r81292 = r81291 * r81291;
        double r81293 = r81290 + r81292;
        double r81294 = sqrt(r81293);
        double r81295 = log(r81294);
        return r81295;
}
↓
double f(double re, double im) {
        double r81296 = re;
        double r81297 = -6.122290979091217e+44;
        bool r81298 = r81296 <= r81297;
        double r81299 = -1.0;
        double r81300 = r81299 * r81296;
        double r81301 = log(r81300);
        double r81302 = 1.8012497291896643e-291;
        bool r81303 = r81296 <= r81302;
        double r81304 = r81296 * r81296;
        double r81305 = im;
        double r81306 = r81305 * r81305;
        double r81307 = r81304 + r81306;
        double r81308 = sqrt(r81307);
        double r81309 = log(r81308);
        double r81310 = 3.6483869476023075e-196;
        bool r81311 = r81296 <= r81310;
        double r81312 = log(r81305);
        double r81313 = 1.5839519867424625e+49;
        bool r81314 = r81296 <= r81313;
        double r81315 = log(r81296);
        double r81316 = r81314 ? r81309 : r81315;
        double r81317 = r81311 ? r81312 : r81316;
        double r81318 = r81303 ? r81309 : r81317;
        double r81319 = r81298 ? r81301 : r81318;
        return r81319;
}

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 < -6.122290979091217e+44

    1. Initial program 45.2

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

    2. Taylor expanded around -inf 10.9

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

    if -6.122290979091217e+44 < re < 1.8012497291896643e-291 or 3.6483869476023075e-196 < re < 1.5839519867424625e+49

    1. Initial program 20.0

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

    if 1.8012497291896643e-291 < re < 3.6483869476023075e-196

    1. Initial program 32.9

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

    2. Taylor expanded around 0 34.3

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

    if 1.5839519867424625e+49 < re

    1. Initial program 44.4

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

    2. Taylor expanded around inf 11.0

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.12229097909121671 \cdot 10^{44}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le 1.8012497291896643 \cdot 10^{-291}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.64838694760230754 \cdot 10^{-196}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.58395198674246248 \cdot 10^{49}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))