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Derivation

1. Split input into 4 regimes

2. if re < -3.7719256724264393e+145

   1. Initial program 59.7

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

   2. Taylor expanded around -inf 7.4

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

   3. Simplified7.4

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

   if -3.7719256724264393e+145 < re < -9.94316899238669e-208 or 2.097899614946829e-237 < re < 1.2868907325585267e+79

   1. Initial program 18.4

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

   if -9.94316899238669e-208 < re < 2.097899614946829e-237

   1. Initial program 30.6

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

   2. Taylor expanded around 0 32.1

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

   if 1.2868907325585267e+79 < re

   1. Initial program 45.9

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

   2. Taylor expanded around inf 9.9

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

4. Final simplification17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.7719256724264393 \cdot 10^{+145}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -9.94316899238669 \cdot 10^{-208}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.097899614946829 \cdot 10^{-237}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.2868907325585267 \cdot 10^{+79}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.5s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))