Average Error: 30.8 → 17.1
Time: 3.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.795235363351034 \cdot 10^{+137}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.0866413834330731 \cdot 10^{-272}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 5.841766761279994 \cdot 10^{-190}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.740380038111227 \cdot 10^{+124}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.795235363351034 \cdot 10^{+137}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le 1.0866413834330731 \cdot 10^{-272}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \le 5.841766761279994 \cdot 10^{-190}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 9.740380038111227 \cdot 10^{+124}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r828259 = re;
        double r828260 = r828259 * r828259;
        double r828261 = im;
        double r828262 = r828261 * r828261;
        double r828263 = r828260 + r828262;
        double r828264 = sqrt(r828263);
        double r828265 = log(r828264);
        return r828265;
}
↓
double f(double re, double im) {
        double r828266 = re;
        double r828267 = -6.795235363351034e+137;
        bool r828268 = r828266 <= r828267;
        double r828269 = -r828266;
        double r828270 = log(r828269);
        double r828271 = 1.0866413834330731e-272;
        bool r828272 = r828266 <= r828271;
        double r828273 = im;
        double r828274 = r828273 * r828273;
        double r828275 = r828266 * r828266;
        double r828276 = r828274 + r828275;
        double r828277 = sqrt(r828276);
        double r828278 = log(r828277);
        double r828279 = 5.841766761279994e-190;
        bool r828280 = r828266 <= r828279;
        double r828281 = log(r828273);
        double r828282 = 9.740380038111227e+124;
        bool r828283 = r828266 <= r828282;
        double r828284 = log(r828266);
        double r828285 = r828283 ? r828278 : r828284;
        double r828286 = r828280 ? r828281 : r828285;
        double r828287 = r828272 ? r828278 : r828286;
        double r828288 = r828268 ? r828270 : r828287;
        return r828288;
}

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 < -6.795235363351034e+137

    1. Initial program 57.4

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

    2. Taylor expanded around -inf 7.2

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

    3. Simplified7.2

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

    if -6.795235363351034e+137 < re < 1.0866413834330731e-272 or 5.841766761279994e-190 < re < 9.740380038111227e+124

    1. Initial program 19.6

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

    if 1.0866413834330731e-272 < re < 5.841766761279994e-190

    1. Initial program 30.6

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

    2. Taylor expanded around 0 33.9

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

    if 9.740380038111227e+124 < re

    1. Initial program 55.1

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.795235363351034 \cdot 10^{+137}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.0866413834330731 \cdot 10^{-272}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 5.841766761279994 \cdot 10^{-190}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.740380038111227 \cdot 10^{+124}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))