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Derivation

1. Split input into 3 regimes

2. if re < -1.2572156992507207e+154

   1. Initial program 59.4

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

   3. Applied add-exp-log59.4

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

   5. Applied pow159.4

      \[\leadsto e^{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}\]

   6. Applied log-pow59.4

      \[\leadsto e^{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   7. Applied exp-prod59.4

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\]

   8. Simplified59.4

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

   9. Taylor expanded around -inf 8.0

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

   10. Simplified8.0

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

   if -1.2572156992507207e+154 < re < 2.585496216458394e+134

   1. Initial program 19.8

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

   if 2.585496216458394e+134 < re

   1. Initial program 54.1

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

   3. Applied add-exp-log54.6

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

   5. Applied pow154.6

      \[\leadsto e^{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}\]

   6. Applied log-pow54.6

      \[\leadsto e^{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   7. Applied exp-prod54.7

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\]

   8. Simplified54.7

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

   9. Taylor expanded around inf 8.2

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

4. Final simplification16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2572156992507207 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.585496216458394 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 3.8s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))