Average Error: 30.6 → 17.0
Time: 4.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.272776742038984 \cdot 10^{+36}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.5387418282241494 \cdot 10^{-275}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.259709475737078 \cdot 10^{-301}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.145199547234516 \cdot 10^{-246}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 4.071831069248947 \cdot 10^{-214}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.734502278787053 \cdot 10^{+66}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.272776742038984 \cdot 10^{+36}:\\
\;\;\;\;\log \left(-re\right)\\

\mathbf{elif}\;re \le -2.5387418282241494 \cdot 10^{-275}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \le 3.259709475737078 \cdot 10^{-301}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 8.145199547234516 \cdot 10^{-246}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

\mathbf{elif}\;re \le 4.071831069248947 \cdot 10^{-214}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 6.734502278787053 \cdot 10^{+66}:\\
\;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r891189 = re;
        double r891190 = r891189 * r891189;
        double r891191 = im;
        double r891192 = r891191 * r891191;
        double r891193 = r891190 + r891192;
        double r891194 = sqrt(r891193);
        double r891195 = log(r891194);
        return r891195;
}
↓
double f(double re, double im) {
        double r891196 = re;
        double r891197 = -1.272776742038984e+36;
        bool r891198 = r891196 <= r891197;
        double r891199 = -r891196;
        double r891200 = log(r891199);
        double r891201 = -2.5387418282241494e-275;
        bool r891202 = r891196 <= r891201;
        double r891203 = im;
        double r891204 = r891203 * r891203;
        double r891205 = r891196 * r891196;
        double r891206 = r891204 + r891205;
        double r891207 = sqrt(r891206);
        double r891208 = log(r891207);
        double r891209 = 3.259709475737078e-301;
        bool r891210 = r891196 <= r891209;
        double r891211 = log(r891203);
        double r891212 = 8.145199547234516e-246;
        bool r891213 = r891196 <= r891212;
        double r891214 = 4.071831069248947e-214;
        bool r891215 = r891196 <= r891214;
        double r891216 = 6.734502278787053e+66;
        bool r891217 = r891196 <= r891216;
        double r891218 = log(r891196);
        double r891219 = r891217 ? r891208 : r891218;
        double r891220 = r891215 ? r891211 : r891219;
        double r891221 = r891213 ? r891208 : r891220;
        double r891222 = r891210 ? r891211 : r891221;
        double r891223 = r891202 ? r891208 : r891222;
        double r891224 = r891198 ? r891200 : r891223;
        return r891224;
}

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 < -1.272776742038984e+36

    1. Initial program 41.3

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

    2. Taylor expanded around -inf 11.8

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

    3. Simplified11.8

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

    if -1.272776742038984e+36 < re < -2.5387418282241494e-275 or 3.259709475737078e-301 < re < 8.145199547234516e-246 or 4.071831069248947e-214 < re < 6.734502278787053e+66

    1. Initial program 20.8

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

    if -2.5387418282241494e-275 < re < 3.259709475737078e-301 or 8.145199547234516e-246 < re < 4.071831069248947e-214

    1. Initial program 33.3

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

    2. Taylor expanded around 0 32.3

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

    if 6.734502278787053e+66 < re

    1. Initial program 43.5

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.272776742038984 \cdot 10^{+36}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.5387418282241494 \cdot 10^{-275}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.259709475737078 \cdot 10^{-301}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.145199547234516 \cdot 10^{-246}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 4.071831069248947 \cdot 10^{-214}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.734502278787053 \cdot 10^{+66}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))