Error

Derivation

1. Split input into 4 regimes

2. if re < -2.7939645872999884e+159

   1. Initial program 59.4

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

   2. Taylor expanded around -inf 6.8

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

   3. Simplified6.8

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

   if -2.7939645872999884e+159 < re < -1.189594430540218e-265 or 2.1843411654809065e-268 < re < 1.133910525523642e+118

   1. Initial program 19.2

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

   if -1.189594430540218e-265 < re < 2.1843411654809065e-268

   1. Initial program 29.8

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

   2. Taylor expanded around 0 30.3

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

   if 1.133910525523642e+118 < re

   1. Initial program 50.0

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

   2. Taylor expanded around inf 9.4

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

4. Final simplification17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.7939645872999884 \cdot 10^{+159}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.189594430540218 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.1843411654809065 \cdot 10^{-268}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.133910525523642 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))