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Derivation

1. Split input into 3 regimes

2. if re < -3.826179680151437

   1. Initial program 38.0

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

   2. Initial simplification38.0

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

   3. Taylor expanded around -inf 12.4

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

   4. Simplified12.4

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

   if -3.826179680151437 < re < 6.24742936036835e+17

   1. Initial program 22.8

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

   2. Initial simplification22.8

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

   if 6.24742936036835e+17 < re

   1. Initial program 40.3

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

   2. Initial simplification40.3

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

   3. Taylor expanded around inf 12.1

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

4. Final simplification17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.826179680151437:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.24742936036835 \cdot 10^{+17}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 2.5s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))