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Derivation

1. Split input into 4 regimes

2. if re < -1.3429377186161065e+154

   1. Initial program 59.4

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

   2. Taylor expanded around -inf 7.1

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

   3. Simplified7.1

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

   if -1.3429377186161065e+154 < re < -2.8680269864911846e-245 or 2.1797926685456316e-244 < re < 1.8925171695161476e+105

   1. Initial program 17.7

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

   if -2.8680269864911846e-245 < re < 2.1797926685456316e-244

   1. Initial program 30.1

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

   2. Taylor expanded around 0 32.4

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

   if 1.8925171695161476e+105 < re

   1. Initial program 49.3

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

   2. Taylor expanded around inf 10.0

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

4. Final simplification16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3429377186161065 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.8680269864911846 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.1797926685456316 \cdot 10^{-244}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.8925171695161476 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.6s)Debug logProfile

herbie shell --seed 2018355 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))