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Derivation

1. Split input into 4 regimes

2. if re < -4.502193544579294e+153

   1. Initial program 59.3

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

   2. Taylor expanded around -inf 7.8

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

   3. Simplified7.8

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

   if -4.502193544579294e+153 < re < -9.387058305577293e-172 or 4.890309052171191e-287 < re < 2.7623865262755493e+132

   1. Initial program 17.3

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

   if -9.387058305577293e-172 < re < 4.890309052171191e-287

   1. Initial program 29.4

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

   3. Applied add-sqr-sqrt29.4

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

   4. Applied sqrt-prod29.6

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

   5. Taylor expanded around 0 33.7

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

   if 2.7623865262755493e+132 < re

   1. Initial program 54.0

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

   2. Taylor expanded around inf 8.4

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.502193544579294 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -9.387058305577293 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.890309052171191 \cdot 10^{-287}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.7623865262755493 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 4.6s)Debug logProfile

herbie shell --seed 2018340 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))