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Derivation

1. Split input into 3 regimes

2. if re < -3.6133336050225706e+141

   1. Initial program 55.5

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

   2. Initial simplification55.5

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

   3. Taylor expanded around -inf 8.2

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

   4. Simplified8.2

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

   if -3.6133336050225706e+141 < re < 4.207204443620986e+85

   1. Initial program 19.7

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

   2. Initial simplification19.7

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

   if 4.207204443620986e+85 < re

   1. Initial program 45.0

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

   2. Initial simplification45.0

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

   3. Taylor expanded around inf 10.7

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

4. Final simplification16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.6133336050225706 \cdot 10^{+141}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.207204443620986 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 11.9s)Debug logProfile

herbie shell --seed 2018258 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))