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Derivation

1. Split input into 3 regimes

2. if re < -9.487910074454056e+128

   1. Initial program 56.5

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

   2. Initial simplification56.5

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

   3. Taylor expanded around -inf 7.4

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

   4. Simplified7.4

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

   if -9.487910074454056e+128 < re < 4.87295810596755e+120

   1. Initial program 20.1

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

   2. Initial simplification20.1

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

   if 4.87295810596755e+120 < re

   1. Initial program 54.7

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

   2. Initial simplification54.7

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

   3. Taylor expanded around inf 8.2

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

4. Final simplification16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.487910074454056 \cdot 10^{+128}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 4.87295810596755 \cdot 10^{+120}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 3.8s)Debug logProfile

herbie shell --seed 2018286 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))