Error

Derivation

1. Split input into 4 regimes

2. if re < -1.0449732451013338e+148

   1. Initial program 57.8

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

   2. Taylor expanded around -inf 8.6

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

   3. Simplified8.6

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

   if -1.0449732451013338e+148 < re < -8.830714642883144e-300 or 1.2167077094438113e-254 < re < 1.2288226746954848e+174

   1. Initial program 20.2

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

   if -8.830714642883144e-300 < re < 1.2167077094438113e-254

   1. Initial program 31.1

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

   2. Taylor expanded around 0 31.9

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

   if 1.2288226746954848e+174 < re

   1. Initial program 59.2

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

   2. Taylor expanded around inf 6.4

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

4. Final simplification17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0449732451013338 \cdot 10^{+148}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -8.830714642883144 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2167077094438113 \cdot 10^{-254}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.2288226746954848 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))