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Derivation

1. Split input into 3 regimes

2. if re < -5.1510501608275733e+120

   1. Initial program 53.7

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

   2. Initial simplification53.7

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

   3. Taylor expanded around -inf 7.2

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

   4. Simplified7.2

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

   if -5.1510501608275733e+120 < re < 6.706260153190329e+103

   1. Initial program 21.5

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

   2. Initial simplification21.5

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

   if 6.706260153190329e+103 < re

   1. Initial program 50.9

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

   2. Initial simplification50.9

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

   3. Taylor expanded around inf 9.0

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.1510501608275733 \cdot 10^{+120}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.706260153190329 \cdot 10^{+103}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.8s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))