Error

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Derivation

1. Split input into 4 regimes

2. if im < -3.7077807277074944e-202 or 1.5629530404064845e-115 < im < 2.3473699638254033e+139

   1. Initial program 25.8

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

   if -3.7077807277074944e-202 < im < -8.313973292319987e-286 or 7.871394234672321e-253 < im < 1.5629530404064845e-115

   1. Initial program 26.6

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

   2. Taylor expanded around inf 35.9

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

   if -8.313973292319987e-286 < im < 7.871394234672321e-253

   1. Initial program 30.9

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

   2. Taylor expanded around -inf 32.6

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

   3. Simplified32.6

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

   if 2.3473699638254033e+139 < im

   1. Initial program 57.0

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

   2. Taylor expanded around 0 5.8

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

4. Final simplification25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -3.7077807277074944 \cdot 10^{-202}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;im \le -8.313973292319987 \cdot 10^{-286}:\\ \;\;\;\;\log re\\ \mathbf{elif}\;im \le 7.871394234672321 \cdot 10^{-253}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;im \le 1.5629530404064845 \cdot 10^{-115}:\\ \;\;\;\;\log re\\ \mathbf{elif}\;im \le 2.3473699638254033 \cdot 10^{+139}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log im\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))

Details

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