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Derivation

1. Split input into 4 regimes

2. if re < -6.620693005917062e+153

   1. Initial program 59.4

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

   2. Initial simplification59.4

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

   4. Applied add-exp-log59.4

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

   6. Applied pow1/259.4

      \[\leadsto e^{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}\]

   7. Applied log-pow59.4

      \[\leadsto e^{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   8. Applied exp-prod59.4

      \[\leadsto \color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}\]

   9. Taylor expanded around -inf 7.1

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

   10. Simplified7.1

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

   if -6.620693005917062e+153 < re < -3.377774402336158e-177 or 8.625314473020069e-300 < re < 3.893950535769218e+152

   1. Initial program 18.5

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

   2. Initial simplification18.5

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

   if -3.377774402336158e-177 < re < 8.625314473020069e-300

   1. Initial program 28.1

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

   2. Initial simplification28.1

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

   4. Applied add-sqr-sqrt28.4

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

   5. Taylor expanded around 0 33.2

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

   if 3.893950535769218e+152 < re

   1. Initial program 59.0

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

   2. Initial simplification59.0

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

   4. Applied add-exp-log59.0

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

   6. Applied pow1/259.0

      \[\leadsto e^{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}\]

   7. Applied log-pow59.0

      \[\leadsto e^{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   8. Applied exp-prod59.0

      \[\leadsto \color{blue}{{\left(e^{\frac{1}{2}}\right)}^{\left(\log \left(re \cdot re + im \cdot im\right)\right)}}\]

   9. Taylor expanded around inf 8.5

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

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.620693005917062 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.377774402336158 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 8.625314473020069 \cdot 10^{-300}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.893950535769218 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 11.3s)Debug logProfile

herbie shell --seed 2018354 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))