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Derivation

1. Split input into 4 regimes

2. if re < -1.3589375087686996e+114

   1. Initial program 52.5

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

   2. Taylor expanded around -inf 8.5

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

   3. Simplified8.5

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

   if -1.3589375087686996e+114 < re < -5.324054775563344e-120 or -7.565496371114485e-306 < re < 1.9099951462284312e+87

   1. Initial program 18.4

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

   if -5.324054775563344e-120 < re < -7.565496371114485e-306

   1. Initial program 27.3

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

   2. Taylor expanded around 0 35.1

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

   if 1.9099951462284312e+87 < re

   1. Initial program 48.6

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

   2. Taylor expanded around inf 8.9

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

4. Final simplification17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3589375087686996 \cdot 10^{+114}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -5.324054775563344 \cdot 10^{-120}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -7.565496371114485 \cdot 10^{-306}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.9099951462284312 \cdot 10^{+87}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.4s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))