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Derivation

1. Split input into 3 regimes

2. if re < -1.7618980710262054e+71

   1. Initial program 46.3

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

   2. Taylor expanded around -inf 9.8

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

   3. Simplified9.8

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

   if -1.7618980710262054e+71 < re < 1.4560974996417245e+83

   1. Initial program 21.1

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

   if 1.4560974996417245e+83 < re

   1. Initial program 47.5

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

   2. Taylor expanded around inf 9.0

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

   3. Simplified9.0

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.7618980710262054 \cdot 10^{+71}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 1.4560974996417245 \cdot 10^{+83}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 4.8s)Debug logProfile

herbie shell --seed 2018277 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))