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Derivation

1. Split input into 4 regimes

2. if re < -4.010025314837191e+152

   1. Initial program 59.2

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

   2. Initial simplification59.2

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

   3. Taylor expanded around -inf 8.5

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

   4. Simplified8.5

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

   if -4.010025314837191e+152 < re < -7.195014277572845e-271 or 4.170146857658731e-237 < re < 4.769476277215655e+147

   1. Initial program 18.3

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

   2. Initial simplification18.3

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

   if -7.195014277572845e-271 < re < 4.170146857658731e-237

   1. Initial program 29.6

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

   2. Initial simplification29.6

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

   3. Taylor expanded around 0 31.2

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

   if 4.769476277215655e+147 < re

   1. Initial program 57.4

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

   2. Initial simplification57.4

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

   3. Taylor expanded around inf 8.4

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

4. Final simplification16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.010025314837191 \cdot 10^{+152}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -7.195014277572845 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.170146857658731 \cdot 10^{-237}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.769476277215655 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 2.6s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))