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Derivation

1. Split input into 4 regimes

2. if re < -3.512834698096542e+153

   1. Initial program 59.0

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

   2. Initial simplification59.0

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

   3. Taylor expanded around -inf 7.4

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

   4. Simplified7.4

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

   if -3.512834698096542e+153 < re < 3.122530532110095e-218 or 2.20388124317475e-168 < re < 2.4414305271140054e+144

   1. Initial program 19.2

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

   2. Initial simplification19.2

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

   if 3.122530532110095e-218 < re < 2.20388124317475e-168

   1. Initial program 29.3

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

   2. Initial simplification29.3

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

   3. Taylor expanded around 0 34.2

      \[\leadsto \color{blue}{im}\]

   if 2.4414305271140054e+144 < re

   1. Initial program 56.9

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

   2. Initial simplification56.9

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

   3. Taylor expanded around inf 7.3

      \[\leadsto \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.512834698096542 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 3.122530532110095 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.20388124317475 \cdot 10^{-168}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.4414305271140054 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 7.5s)Debug logProfile

herbie shell --seed 2018297 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))