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Derivation

1. Split input into 3 regimes

2. if re < -7.95242969771968e+96

   1. Initial program 47.6

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

   2. Initial simplification47.6

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

   3. Taylor expanded around -inf 9.7

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

   4. Simplified9.7

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

   if -7.95242969771968e+96 < re < 2.558708822248632e+52

   1. Initial program 20.6

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

   2. Initial simplification20.6

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

   if 2.558708822248632e+52 < re

   1. Initial program 42.0

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

   2. Initial simplification42.0

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

   3. Taylor expanded around inf 12.5

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.95242969771968 \cdot 10^{+96}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.558708822248632 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 2.1s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))