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Derivation

1. Split input into 3 regimes

2. if re < -1.2479508588260587e+154

   1. Initial program 59.4

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

   2. Initial simplification59.4

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

   3. Taylor expanded around -inf 7.1

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

   4. Simplified7.1

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

   if -1.2479508588260587e+154 < re < 1.9728387301565315e+146

   1. Initial program 19.4

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

   2. Initial simplification19.4

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

   if 1.9728387301565315e+146 < re

   1. Initial program 57.8

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

   2. Initial simplification57.8

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

   3. Taylor expanded around inf 8.0

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

4. Final simplification16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2479508588260587 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.9728387301565315 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 4.3s)Debug logProfile

herbie shell --seed 2018286 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))