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Derivation

1. Split input into 3 regimes

2. if re < -7.1459028545956985e+149

   1. Initial program 58.6

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

   2. Initial simplification58.6

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

   3. Taylor expanded around -inf 7.2

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

   4. Simplified7.2

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

   if -7.1459028545956985e+149 < re < 2.937250069907319e-180 or 4.820775704044266e-164 < re < 9.389022784286144e+118

   1. Initial program 19.6

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

   2. Initial simplification19.6

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

   if 2.937250069907319e-180 < re < 4.820775704044266e-164 or 9.389022784286144e+118 < re

   1. Initial program 48.5

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

   2. Initial simplification48.5

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

   3. Taylor expanded around inf 12.2

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

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.1459028545956985 \cdot 10^{+149}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.937250069907319 \cdot 10^{-180} \lor \neg \left(re \le 4.820775704044266 \cdot 10^{-164}\right) \land re \le 9.389022784286144 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.6s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))