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Derivation

1. Split input into 4 regimes

2. if re < -8.739157930024531e+82

   1. Initial program 48.2

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

   2. Initial simplification48.2

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

   3. Taylor expanded around -inf 9.5

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

   4. Simplified9.5

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

   if -8.739157930024531e+82 < re < -2.6401746271055864e-185 or 1.6510598051339403e-137 < re < 8.578878492887041e+100

   1. Initial program 15.8

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

   2. Initial simplification15.8

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

   if -2.6401746271055864e-185 < re < 1.6510598051339403e-137

   1. Initial program 29.8

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

   2. Initial simplification29.8

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

   3. Taylor expanded around 0 33.5

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

   if 8.578878492887041e+100 < re

   1. Initial program 49.8

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

   2. Initial simplification49.8

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

   3. Taylor expanded around inf 8.1

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.739157930024531 \cdot 10^{+82}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.6401746271055864 \cdot 10^{-185}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.6510598051339403 \cdot 10^{-137}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.578878492887041 \cdot 10^{+100}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.9s)Debug logProfile

herbie shell --seed 2018353 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))