Error

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Derivation

1. Split input into 4 regimes

2. if re < -3.1984365912010734e+91

   1. Initial program 48.9

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

   2. Taylor expanded around -inf 9.2

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

   3. Simplified9.2

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

   if -3.1984365912010734e+91 < re < -4.064052350582338e-291 or 3.0369500767927204e-242 < re < 1.1367817676099328e+115

   1. Initial program 20.2

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

   if -4.064052350582338e-291 < re < 3.0369500767927204e-242

   1. Initial program 32.5

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

   2. Taylor expanded around 0 33.3

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

   if 1.1367817676099328e+115 < re

   1. Initial program 52.0

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

   2. Taylor expanded around inf 7.6

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.1984365912010734 \cdot 10^{+91}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -4.064052350582338 \cdot 10^{-291}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 3.0369500767927204 \cdot 10^{-242}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.1367817676099328 \cdot 10^{+115}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))

Details

Time bar (total: 2.1s)Debug log