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Derivation

1. Split input into 3 regimes

2. if re < -6.247790985273033e+93

   1. Initial program 50.0

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

   2. Initial simplification50.0

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

   3. Taylor expanded around -inf 9.1

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

   4. Simplified9.1

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

   if -6.247790985273033e+93 < re < 1.3202313428826321e+122

   1. Initial program 21.3

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

   2. Initial simplification21.3

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

   if 1.3202313428826321e+122 < re

   1. Initial program 54.7

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

   2. Initial simplification54.7

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

   3. Taylor expanded around inf 8.3

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.247790985273033 \cdot 10^{+93}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.3202313428826321 \cdot 10^{+122}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.6s)Debug logProfile

herbie shell --seed 2018285 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))