Error

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Derivation

1. Split input into 4 regimes

2. if re < -8.801722566171015e+86

   1. Initial program 47.7

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

   2. Taylor expanded around -inf 9.9

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

   3. Simplified9.9

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

   if -8.801722566171015e+86 < re < -2.2531648583739764e-146 or -2.4843513365715627e-188 < re < -2.6965385622040613e-280 or 2.4967545228125466e-154 < re < 8.434467310119467e+114

   1. Initial program 17.7

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

   if -2.2531648583739764e-146 < re < -2.4843513365715627e-188 or -2.6965385622040613e-280 < re < 2.4967545228125466e-154

   1. Initial program 29.4

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

   2. Taylor expanded around 0 35.5

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

   if 8.434467310119467e+114 < re

   1. Initial program 53.0

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

   2. Taylor expanded around inf 8.2

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

4. Final simplification18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.801722566171015 \cdot 10^{+86}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -2.2531648583739764 \cdot 10^{-146}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le -2.4843513365715627 \cdot 10^{-188}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le -2.6965385622040613 \cdot 10^{-280}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.4967545228125466 \cdot 10^{-154}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 8.434467310119467 \cdot 10^{+114}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))

Details

Time bar (total: 2.4s)Debug log