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Derivation

1. Split input into 4 regimes

2. if re < -7.759066996502054e+153

   1. Initial program 61.9

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

   2. Taylor expanded around -inf 6.7

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

   3. Simplified6.7

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

   if -7.759066996502054e+153 < re < -4.712388656371755e-181 or -4.286280745334311e-241 < re < 1.310065135699569e+96

   1. Initial program 19.9

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

   if -4.712388656371755e-181 < re < -4.286280745334311e-241

   1. Initial program 30.3

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

   2. Taylor expanded around 0 38.9

      \[\leadsto \log \color{blue}{im}\]

   if 1.310065135699569e+96 < re

   1. Initial program 49.6

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

   2. Taylor expanded around inf 8.7

      \[\leadsto \log \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.759066996502054 \cdot 10^{+153}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.712388656371755 \cdot 10^{-181}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -4.286280745334311 \cdot 10^{-241}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.310065135699569 \cdot 10^{+96}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.8s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))