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Derivation

1. Split input into 3 regimes

2. if re < -6.40181675753278e+116

   1. Initial program 52.1

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

   2. Taylor expanded around -inf 7.5

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

   3. Simplified7.5

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

   if -6.40181675753278e+116 < re < 6.1478219919101515e+91

   1. Initial program 20.8

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

   if 6.1478219919101515e+91 < re

   1. Initial program 47.2

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

   2. Taylor expanded around inf 9.8

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.40181675753278 \cdot 10^{+116}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.1478219919101515 \cdot 10^{+91}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.3s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))