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Derivation

1. Split input into 4 regimes

2. if re < -1.8480056928576853e+75

   1. Initial program 45.5

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

   2. Taylor expanded around -inf 9.2

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

   3. Simplified9.2

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

   if -1.8480056928576853e+75 < re < -1.3605484803417976e-159 or 3.367065929769179e-95 < re < 3.4810428221829335e+114

   1. Initial program 15.9

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

   if -1.3605484803417976e-159 < re < 3.367065929769179e-95

   1. Initial program 28.3

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

   2. Taylor expanded around 0 35.1

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

   if 3.4810428221829335e+114 < re

   1. Initial program 53.2

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

   2. Taylor expanded around inf 8.4

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

4. Final simplification19.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.8480056928576853 \cdot 10^{+75}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.3605484803417976 \cdot 10^{-159}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.367065929769179 \cdot 10^{-95}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 3.4810428221829335 \cdot 10^{+114}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.2s)Debug logProfile

herbie shell --seed 2018339 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))