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Derivation

1. Split input into 3 regimes

2. if re < -2.2287473210148045e+154

   1. Initial program 59.4

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

   2. Initial simplification59.4

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

   3. Taylor expanded around -inf 6.8

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

   4. Simplified6.8

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

   if -2.2287473210148045e+154 < re < 1.1303110575557633e+92

   1. Initial program 19.9

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

   2. Initial simplification19.9

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

   if 1.1303110575557633e+92 < re

   1. Initial program 45.4

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

   2. Initial simplification45.4

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

   3. Taylor expanded around inf 11.5

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.2287473210148045 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.1303110575557633 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.6s)Debug logProfile

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