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Derivation

1. Split input into 4 regimes

2. if re < -3.0987209229786233e+128

   1. Initial program 53.5

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around -inf 8.7

      \[\leadsto \color{blue}{-1 \cdot re}\]

   3. Simplified8.7

      \[\leadsto \color{blue}{-re}\]

   if -3.0987209229786233e+128 < re < -2.8891795525048955e-185 or 7.580755379127733e-178 < re < 8.930021655630846e+100

   1. Initial program 15.8

      \[\sqrt{re \cdot re + im \cdot im}\]

   if -2.8891795525048955e-185 < re < 7.580755379127733e-178

   1. Initial program 29.7

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around 0 33.2

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

   if 8.930021655630846e+100 < re

   1. Initial program 47.7

      \[\sqrt{re \cdot re + im \cdot im}\]

   2. Taylor expanded around inf 9.5

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.0987209229786233 \cdot 10^{+128}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.8891795525048955 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 7.580755379127733 \cdot 10^{-178}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.930021655630846 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.3s)Debug logProfile

herbie shell --seed 2018353 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))