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Derivation

1. Split input into 4 regimes

2. if re < -6.53263513213185e+150

   1. Initial program 61.1

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

   2. Taylor expanded around -inf 5.8

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

   3. Simplified5.8

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

   if -6.53263513213185e+150 < re < -2.069088255243102e-259 or 1.7611866629175242e-146 < re < 1.4415508011465472e+127

   1. Initial program 18.2

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

   if -2.069088255243102e-259 < re < 1.7611866629175242e-146

   1. Initial program 29.6

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

   2. Taylor expanded around 0 33.9

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

   if 1.4415508011465472e+127 < re

   1. Initial program 55.1

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

   2. Taylor expanded around inf 6.9

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.53263513213185 \cdot 10^{+150}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.069088255243102 \cdot 10^{-259}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.7611866629175242 \cdot 10^{-146}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.4415508011465472 \cdot 10^{+127}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 4.2s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))