Derivation

Split input into 3 regimes
if re < -1.3450563247296257e+154
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. Taylor expanded around -inf 12.9
  \[\leadsto \color{blue}{e^{-\log \left(\frac{-1}{re}\right)}}\]
6. Simplified7.0
  \[\leadsto \color{blue}{-re}\]
if -1.3450563247296257e+154 < re < 5.2137866734293315e+151
1. Initial program 19.6
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Initial simplification19.6
  \[\leadsto \sqrt{re \cdot re + im \cdot im}\]
if 5.2137866734293315e+151 < re
1. Initial program 58.5
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Initial simplification58.5
  \[\leadsto \sqrt{re \cdot re + im \cdot im}\]
3. Taylor expanded around inf 8.2
  \[\leadsto \color{blue}{re}\]
Recombined 3 regimes into one program.
Final simplification16.6
\[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3450563247296257 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.2137866734293315 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

herbie shell --seed 2018248 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))

In
Out