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Derivation

1. Split input into 4 regimes

2. if re < -1.5189369300320487e+101

   1. Initial program 50.5

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

   2. Taylor expanded around -inf 8.5

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

   3. Simplified8.5

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

   if -1.5189369300320487e+101 < re < -2.1623795656049374e-236 or 5.321773148611292e-160 < re < 2.4827261809125983e+86

   1. Initial program 17.8

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

   if -2.1623795656049374e-236 < re < 5.321773148611292e-160

   1. Initial program 30.5

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

   2. Taylor expanded around 0 33.5

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

   if 2.4827261809125983e+86 < re

   1. Initial program 46.2

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

   2. Taylor expanded around inf 9.1

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

4. Final simplification17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.5189369300320487 \cdot 10^{+101}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.1623795656049374 \cdot 10^{-236}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 5.321773148611292 \cdot 10^{-160}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 2.4827261809125983 \cdot 10^{+86}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Runtime

Time bar (total: 4.0s)Debug logProfile

herbie shell --seed 2018219 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))