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Derivation

1. Split input into 4 regimes

2. if re < -2.0035406798386108e+118

   1. Initial program 52.8

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

   2. Taylor expanded around -inf 8.6

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

   3. Simplified8.6

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

   if -2.0035406798386108e+118 < re < -1.9812566793834287e-186 or 3.4378810267307503e-261 < re < 6.634416192647577e+134

   1. Initial program 18.3

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

   if -1.9812566793834287e-186 < re < 3.4378810267307503e-261

   1. Initial program 30.9

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

   2. Taylor expanded around 0 31.4

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

   if 6.634416192647577e+134 < re

   1. Initial program 57.1

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

   2. Taylor expanded around inf 7.5

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

4. Final simplification17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.0035406798386108 \cdot 10^{+118}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.9812566793834287 \cdot 10^{-186}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.4378810267307503 \cdot 10^{-261}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 6.634416192647577 \cdot 10^{+134}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.9s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))