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Derivation

1. Split input into 4 regimes

2. if re < -6.115666643649848e+106

   1. Initial program 50.4

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

   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 -6.115666643649848e+106 < re < -7.2009850172815975e-180 or 1.315376429877619e-295 < re < 1.1322433632218456e+124

   1. Initial program 19.1

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

   if -7.2009850172815975e-180 < re < 1.315376429877619e-295

   1. Initial program 28.8

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

   2. Taylor expanded around 0 32.7

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

   if 1.1322433632218456e+124 < re

   1. Initial program 53.8

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

   2. Taylor expanded around inf 8.3

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.115666643649848 \cdot 10^{+106}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -7.2009850172815975 \cdot 10^{-180}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.315376429877619 \cdot 10^{-295}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1322433632218456 \cdot 10^{+124}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 5.5s)Debug logProfile

herbie shell --seed 2018242 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))