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Derivation

1. Split input into 3 regimes

2. if re < -1.6842925293104357e+56

   1. Initial program 44.4

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

   2. Taylor expanded around -inf 9.9

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

   3. Simplified9.9

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

   if -1.6842925293104357e+56 < re < 2.23777221769595e-209 or 4.820775704044266e-164 < re < 3.8021600368862424e+111

   1. Initial program 20.9

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

   if 2.23777221769595e-209 < re < 4.820775704044266e-164 or 3.8021600368862424e+111 < re

   1. Initial program 46.6

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

   2. Taylor expanded around inf 14.8

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

4. Final simplification17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6842925293104357 \cdot 10^{+56}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.23777221769595 \cdot 10^{-209} \lor \neg \left(re \le 4.820775704044266 \cdot 10^{-164}\right) \land re \le 3.8021600368862424 \cdot 10^{+111}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.8s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))