Error

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Derivation

1. Split input into 4 regimes

2. if re < -7.61821412696286e+150

   1. Initial program 61.3

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

   2. Taylor expanded around -inf 6.5

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

   3. Simplified6.5

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

   if -7.61821412696286e+150 < re < -5.536443801166738e-261 or 2.727916884694162e-303 < re < 5.667469564790048e+145

   1. Initial program 19.6

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

   if -5.536443801166738e-261 < re < 2.727916884694162e-303

   1. Initial program 32.3

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

   2. Taylor expanded around 0 31.4

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

   if 5.667469564790048e+145 < re

   1. Initial program 59.9

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

   2. Taylor expanded around inf 6.4

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

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.61821412696286 \cdot 10^{+150}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -5.536443801166738 \cdot 10^{-261}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 2.727916884694162 \cdot 10^{-303}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 5.667469564790048 \cdot 10^{+145}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))

Details

Time bar (total: 2.3s)Debug log