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Derivation

1. Split input into 3 regimes

2. if re < -6.807171614999129e+153

   1. Initial program 59.3

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

   2. Initial simplification59.3

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

   3. Taylor expanded around -inf 6.9

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

   4. Simplified6.9

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

   if -6.807171614999129e+153 < re < 1.313127260661707e+106

   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 1.313127260661707e+106 < re

   1. Initial program 47.7

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

   2. Initial simplification47.7

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

   3. Taylor expanded around inf 10.9

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.807171614999129 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.313127260661707 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 2.1s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))