Average Error: 31.8 → 17.7
Time: 1.2s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -5.15311307602505082 \cdot 10^{-257}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.71398425882254737 \cdot 10^{53}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\
\;\;\;\;\log \left(-1 \cdot re\right)\\

\mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

\mathbf{elif}\;re \le -5.15311307602505082 \cdot 10^{-257}:\\
\;\;\;\;\log im\\

\mathbf{elif}\;re \le 9.71398425882254737 \cdot 10^{53}:\\
\;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\

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

\end{array}
double f(double re, double im) {
        double r23086 = re;
        double r23087 = r23086 * r23086;
        double r23088 = im;
        double r23089 = r23088 * r23088;
        double r23090 = r23087 + r23089;
        double r23091 = sqrt(r23090);
        double r23092 = log(r23091);
        return r23092;
}
↓
double f(double re, double im) {
        double r23093 = re;
        double r23094 = -8.76230374871451e+143;
        bool r23095 = r23093 <= r23094;
        double r23096 = -1.0;
        double r23097 = r23096 * r23093;
        double r23098 = log(r23097);
        double r23099 = -1.180833083521909e-161;
        bool r23100 = r23093 <= r23099;
        double r23101 = r23093 * r23093;
        double r23102 = im;
        double r23103 = r23102 * r23102;
        double r23104 = r23101 + r23103;
        double r23105 = sqrt(r23104);
        double r23106 = log(r23105);
        double r23107 = -5.153113076025051e-257;
        bool r23108 = r23093 <= r23107;
        double r23109 = log(r23102);
        double r23110 = 9.713984258822547e+53;
        bool r23111 = r23093 <= r23110;
        double r23112 = log(r23093);
        double r23113 = r23111 ? r23106 : r23112;
        double r23114 = r23108 ? r23109 : r23113;
        double r23115 = r23100 ? r23106 : r23114;
        double r23116 = r23095 ? r23098 : r23115;
        return r23116;
}

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 < -8.76230374871451e+143

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 6.7

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

    if -8.76230374871451e+143 < re < -1.180833083521909e-161 or -5.153113076025051e-257 < re < 9.713984258822547e+53

    1. Initial program 20.6

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

    if -1.180833083521909e-161 < re < -5.153113076025051e-257

    1. Initial program 32.3

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

    2. Taylor expanded around 0 35.5

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

    if 9.713984258822547e+53 < re

    1. Initial program 43.8

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -5.15311307602505082 \cdot 10^{-257}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 9.71398425882254737 \cdot 10^{53}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))