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Derivation

1. Split input into 4 regimes

2. if re < -9.534618369590614e+92

   1. Initial program 48.6

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

   2. Taylor expanded around -inf 8.8

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

   3. Simplified8.8

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

   if -9.534618369590614e+92 < re < 7.0324657533777544e-301 or 1.5776960121797607e-157 < re < 1.0588804460692496e+101

   1. Initial program 18.3

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

   if 7.0324657533777544e-301 < re < 1.5776960121797607e-157

   1. Initial program 31.8

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

   2. Taylor expanded around 0 34.0

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

   if 1.0588804460692496e+101 < re

   1. Initial program 50.1

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

   2. Taylor expanded around inf 9.1

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

4. Final simplification17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.534618369590614 \cdot 10^{+92}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 7.0324657533777544 \cdot 10^{-301}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.5776960121797607 \cdot 10^{-157}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.0588804460692496 \cdot 10^{+101}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 5.6s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))