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Derivation

1. Split input into 4 regimes

2. if re < -1.2266429946375697e+141

   1. Initial program 58.1

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

   2. Taylor expanded around -inf 6.5

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

   3. Simplified6.5

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

   if -1.2266429946375697e+141 < re < -3.2900832584178346e-245 or 3.080764931580134e-244 < re < 2.330994921200347e+104

   1. Initial program 18.4

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

   if -3.2900832584178346e-245 < re < 3.080764931580134e-244

   1. Initial program 31.1

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

   2. Taylor expanded around 0 32.0

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

   if 2.330994921200347e+104 < re

   1. Initial program 51.3

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

   2. Taylor expanded around inf 8.5

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

4. Final simplification16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2266429946375697 \cdot 10^{+141}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -3.2900832584178346 \cdot 10^{-245}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.080764931580134 \cdot 10^{-244}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.330994921200347 \cdot 10^{+104}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 4.1s)Debug logProfile

herbie shell --seed 2018355 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))