Average Error: 32.0 → 17.9
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1982429734725978600:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 1982429734725978600:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r27917 = re;
        double r27918 = r27917 * r27917;
        double r27919 = im;
        double r27920 = r27919 * r27919;
        double r27921 = r27918 + r27920;
        double r27922 = sqrt(r27921);
        double r27923 = log(r27922);
        return r27923;
}
↓
double f(double re, double im) {
        double r27924 = re;
        double r27925 = -2.6163751838198674e+143;
        bool r27926 = r27924 <= r27925;
        double r27927 = -1.0;
        double r27928 = r27927 * r27924;
        double r27929 = log(r27928);
        double r27930 = -2.0278572522938575e-184;
        bool r27931 = r27924 <= r27930;
        double r27932 = r27924 * r27924;
        double r27933 = im;
        double r27934 = r27933 * r27933;
        double r27935 = r27932 + r27934;
        double r27936 = sqrt(r27935);
        double r27937 = log(r27936);
        double r27938 = -8.804121592020497e-274;
        bool r27939 = r27924 <= r27938;
        double r27940 = log(r27933);
        double r27941 = 1.9824297347259786e+18;
        bool r27942 = r27924 <= r27941;
        double r27943 = log(r27924);
        double r27944 = r27942 ? r27937 : r27943;
        double r27945 = r27939 ? r27940 : r27944;
        double r27946 = r27931 ? r27937 : r27945;
        double r27947 = r27926 ? r27929 : r27946;
        return r27947;
}

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 < -2.6163751838198674e+143

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 6.8

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

    if -2.6163751838198674e+143 < re < -2.0278572522938575e-184 or -8.804121592020497e-274 < re < 1.9824297347259786e+18

    1. Initial program 20.3

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

    if -2.0278572522938575e-184 < re < -8.804121592020497e-274

    1. Initial program 31.1

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

    2. Taylor expanded around 0 36.8

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

    if 1.9824297347259786e+18 < re

    1. Initial program 42.8

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

    2. Taylor expanded around inf 13.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1982429734725978600:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))