Error

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Derivation

1. Split input into 4 regimes

2. if re < -1.3769342465113995e+148

   1. Initial program 59.9

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

   2. Taylor expanded around -inf 5.8

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

   3. Simplified5.8

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

   if -1.3769342465113995e+148 < re < -9.888754662830056e-214 or -7.432754596401146e-269 < re < 4.670873727021844e-211 or 1.942710743904727e-125 < re < 4.1181592821076165e+123

   1. Initial program 18.9

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

   if -9.888754662830056e-214 < re < -7.432754596401146e-269 or 4.670873727021844e-211 < re < 1.942710743904727e-125

   1. Initial program 26.5

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

   2. Taylor expanded around 0 36.3

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

   if 4.1181592821076165e+123 < re

   1. Initial program 54.6

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

   2. Taylor expanded around inf 7.8

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3769342465113995 \cdot 10^{+148}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -9.888754662830056 \cdot 10^{-214}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -7.432754596401146 \cdot 10^{-269}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.670873727021844 \cdot 10^{-211}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.942710743904727 \cdot 10^{-125}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.1181592821076165 \cdot 10^{+123}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))

Details

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