Average Error: 31.8 → 18.1
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.80536176757501775 \cdot 10^{-229}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 6.80536176757501775 \cdot 10^{-229}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r98128 = re;
        double r98129 = r98128 * r98128;
        double r98130 = im;
        double r98131 = r98130 * r98130;
        double r98132 = r98129 + r98131;
        double r98133 = sqrt(r98132);
        double r98134 = log(r98133);
        return r98134;
}
↓
double f(double re, double im) {
        double r98135 = re;
        double r98136 = -1.103215696956926e+72;
        bool r98137 = r98135 <= r98136;
        double r98138 = -1.0;
        double r98139 = r98138 * r98135;
        double r98140 = log(r98139);
        double r98141 = -1.3504253849915568e-194;
        bool r98142 = r98135 <= r98141;
        double r98143 = r98135 * r98135;
        double r98144 = im;
        double r98145 = r98144 * r98144;
        double r98146 = r98143 + r98145;
        double r98147 = sqrt(r98146);
        double r98148 = log(r98147);
        double r98149 = -2.968956980813959e-266;
        bool r98150 = r98135 <= r98149;
        double r98151 = log(r98144);
        double r98152 = 6.805361767575018e-229;
        bool r98153 = r98135 <= r98152;
        double r98154 = 1.3057340609530177e-191;
        bool r98155 = r98135 <= r98154;
        double r98156 = 5.156219500915728e+39;
        bool r98157 = r98135 <= r98156;
        double r98158 = log(r98135);
        double r98159 = r98157 ? r98148 : r98158;
        double r98160 = r98155 ? r98151 : r98159;
        double r98161 = r98153 ? r98148 : r98160;
        double r98162 = r98150 ? r98151 : r98161;
        double r98163 = r98142 ? r98148 : r98162;
        double r98164 = r98137 ? r98140 : r98163;
        return r98164;
}

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 < -1.103215696956926e+72

    1. Initial program 46.8

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

    2. Taylor expanded around -inf 9.3

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

    if -1.103215696956926e+72 < re < -1.3504253849915568e-194 or -2.968956980813959e-266 < re < 6.805361767575018e-229 or 1.3057340609530177e-191 < re < 5.156219500915728e+39

    1. Initial program 21.1

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

    if -1.3504253849915568e-194 < re < -2.968956980813959e-266 or 6.805361767575018e-229 < re < 1.3057340609530177e-191

    1. Initial program 32.2

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

    2. Taylor expanded around 0 34.8

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

    if 5.156219500915728e+39 < re

    1. Initial program 43.9

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

    2. Taylor expanded around inf 11.6

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.80536176757501775 \cdot 10^{-229}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))