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Derivation

1. Split input into 4 regimes

2. if re < -3.6871908419745517e+118

   1. Initial program 50.5

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

   2. Initial simplification50.5

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

   3. Taylor expanded around -inf 10.2

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

   4. Simplified10.2

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

   if -3.6871908419745517e+118 < re < -1.766606193124001e-186 or 3.026564794600837e-272 < re < 5.359130068716829e+143

   1. Initial program 17.8

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

   2. Initial simplification17.8

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

   if -1.766606193124001e-186 < re < 3.026564794600837e-272

   1. Initial program 29.8

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

   2. Initial simplification29.8

      \[\leadsto \sqrt{re \cdot re + im \cdot im}\]
   3. Using strategy rm

   4. Applied add-exp-log32.2

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   5. Taylor expanded around 0 32.0

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

   if 5.359130068716829e+143 < re

   1. Initial program 56.7

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

   2. Initial simplification56.7

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

   3. Taylor expanded around inf 8.8

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

4. Final simplification17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.6871908419745517 \cdot 10^{+118}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.766606193124001 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 3.026564794600837 \cdot 10^{-272}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.359130068716829 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 11.1s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))