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Derivation

1. Split input into 3 regimes

2. if re < -1.2094426055789201e+112

   1. Initial program 52.1

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

   2. Initial simplification52.1

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

   3. Taylor expanded around -inf 8.2

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

   4. Simplified8.2

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

   if -1.2094426055789201e+112 < re < 1.6002838359697833e+125

   1. Initial program 20.4

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

   2. Initial simplification20.4

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

   if 1.6002838359697833e+125 < re

   1. Initial program 53.8

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

   2. Initial simplification53.8

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

   3. Taylor expanded around inf 7.5

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

4. Final simplification16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2094426055789201 \cdot 10^{+112}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.6002838359697833 \cdot 10^{+125}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 4.4s)Debug logProfile

herbie shell --seed 2018263 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))