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Derivation

1. Split input into 3 regimes

2. if re < -6.1157351543722565e+62

   1. Initial program 44.3

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

   2. Initial simplification44.3

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

   3. Taylor expanded around -inf 10.0

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

   4. Simplified10.0

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

   if -6.1157351543722565e+62 < re < 2.194524025777904e+126

   1. Initial program 21.5

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

   2. Initial simplification21.5

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

   if 2.194524025777904e+126 < re

   1. Initial program 55.6

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

   2. Initial simplification55.6

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

   3. Taylor expanded around inf 8.1

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

4. Final simplification17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.1157351543722565 \cdot 10^{+62}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.194524025777904 \cdot 10^{+126}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.6s)Debug logProfile

herbie shell --seed 2018351 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))