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Derivation

1. Split input into 3 regimes

2. if re < -3.6713244494953644e+65

   1. Initial program 45.5

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

   2. Taylor expanded around -inf 9.6

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

   3. Simplified9.6

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

   if -3.6713244494953644e+65 < re < 2.011001543074701e+123

   1. Initial program 20.7

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

   if 2.011001543074701e+123 < re

   1. Initial program 53.2

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

   2. Taylor expanded around inf 8.3

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

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.6713244494953644 \cdot 10^{+65}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 2.011001543074701 \cdot 10^{+123}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.7s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))