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Derivation

1. Split input into 4 regimes

2. if re < -4.7451005407218806e+151

   1. Initial program 58.6

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

   2. Initial simplification58.6

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

   3. Taylor expanded around -inf 7.7

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

   4. Simplified7.7

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

   if -4.7451005407218806e+151 < re < 9.88691132157457e-297 or 1.5776960121797607e-157 < re < 8.38874173978045e+123

   1. Initial program 17.4

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

   2. Initial simplification17.4

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

   if 9.88691132157457e-297 < re < 1.5776960121797607e-157

   1. Initial program 30.9

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

   2. Initial simplification30.9

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

   3. Taylor expanded around 0 35.0

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

   if 8.38874173978045e+123 < re

   1. Initial program 52.0

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

   2. Initial simplification52.0

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

   3. Taylor expanded around inf 10.3

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

4. Final simplification17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.7451005407218806 \cdot 10^{+151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 9.88691132157457 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.5776960121797607 \cdot 10^{-157}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.38874173978045 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 12.1s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))