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Derivation

1. Split input into 3 regimes

2. if re < -6.98919469234238e+142

   1. Initial program 55.9

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

   2. Initial simplification55.9

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

   3. Taylor expanded around -inf 8.1

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

   4. Simplified8.1

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

   if -6.98919469234238e+142 < re < 1.1329987674401443e+143

   1. Initial program 19.8

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

   2. Initial simplification19.8

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

   if 1.1329987674401443e+143 < re

   1. Initial program 55.6

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

   2. Initial simplification55.6

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

   3. Taylor expanded around inf 8.8

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

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.98919469234238 \cdot 10^{+142}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.1329987674401443 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.3s)Debug logProfile

herbie shell --seed 2018242 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))