Average Error: 31.2 → 17.0
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r89969 = re;
        double r89970 = r89969 * r89969;
        double r89971 = im;
        double r89972 = r89971 * r89971;
        double r89973 = r89970 + r89972;
        double r89974 = sqrt(r89973);
        double r89975 = log(r89974);
        return r89975;
}
↓
double f(double re, double im) {
        double r89976 = re;
        double r89977 = -1.0674397664294253e+136;
        bool r89978 = r89976 <= r89977;
        double r89979 = -1.0;
        double r89980 = r89979 * r89976;
        double r89981 = log(r89980);
        double r89982 = -4.4039792789215395e-257;
        bool r89983 = r89976 <= r89982;
        double r89984 = r89976 * r89976;
        double r89985 = im;
        double r89986 = r89985 * r89985;
        double r89987 = r89984 + r89986;
        double r89988 = sqrt(r89987);
        double r89989 = log(r89988);
        double r89990 = 3.8197786805557845e-227;
        bool r89991 = r89976 <= r89990;
        double r89992 = log(r89985);
        double r89993 = 8.439330033545885e+67;
        bool r89994 = r89976 <= r89993;
        double r89995 = log(r89976);
        double r89996 = r89994 ? r89989 : r89995;
        double r89997 = r89991 ? r89992 : r89996;
        double r89998 = r89983 ? r89989 : r89997;
        double r89999 = r89978 ? r89981 : r89998;
        return r89999;
}

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 < -1.0674397664294253e+136

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 7.7

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

    if -1.0674397664294253e+136 < re < -4.4039792789215395e-257 or 3.8197786805557845e-227 < re < 8.439330033545885e+67

    1. Initial program 18.8

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

    if -4.4039792789215395e-257 < re < 3.8197786805557845e-227

    1. Initial program 31.0

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

    2. Taylor expanded around 0 32.6

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

    if 8.439330033545885e+67 < re

    1. Initial program 46.7

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.067439766429425256822678606355967347012 \cdot 10^{136}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.403979278921539526489078141768847052434 \cdot 10^{-257}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))