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Derivation

1. Split input into 3 regimes

2. if re < -5.117083376838441e+150

   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 7.6

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

   4. Simplified7.6

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

   if -5.117083376838441e+150 < re < 3.0822347793204883e+119

   1. Initial program 20.0

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

   2. Initial simplification20.0

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

   if 3.0822347793204883e+119 < re

   1. Initial program 51.5

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

   2. Initial simplification51.5

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

   3. Taylor expanded around inf 9.9

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.117083376838441 \cdot 10^{+150}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 3.0822347793204883 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 5.8s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))