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Derivation

1. Split input into 3 regimes

2. if re < -1.2872897085244062e+154 or -1.6983590970193986e-164 < re < -3.151634490341881e-222

   1. Initial program 51.5

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

   3. Applied add-exp-log52.2

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

   4. Taylor expanded around -inf 23.3

      \[\leadsto \color{blue}{e^{-\log \left(\frac{-1}{re}\right)}}\]

   5. Simplified18.8

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

   if -1.2872897085244062e+154 < re < -1.6983590970193986e-164 or -3.151634490341881e-222 < re < 6.937989018162917e+152

   1. Initial program 19.0

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

   if 6.937989018162917e+152 < re

   1. Initial program 59.0

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

   3. Applied add-exp-log59.1

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

   4. Taylor expanded around inf 13.4

      \[\leadsto \color{blue}{e^{-\log \left(\frac{1}{re}\right)}}\]

   5. Simplified7.5

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2872897085244062 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.6983590970193986 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.151634490341881 \cdot 10^{-222}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 6.937989018162917 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 8.6s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))