Error

Derivation

1. Split input into 4 regimes

2. if re < -1.183566067701958e+165

   1. Initial program 59.3

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

   2. Taylor expanded around -inf 7.9

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

   3. Simplified7.9

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

   if -1.183566067701958e+165 < re < 6.197016014496118e-262 or 2.6429575845296566e-179 < re < 7.188021498228873e+127

   1. Initial program 19.0

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

   if 6.197016014496118e-262 < re < 2.6429575845296566e-179

   1. Initial program 29.0

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

   2. Taylor expanded around 0 35.9

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

   if 7.188021498228873e+127 < re

   1. Initial program 52.5

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

   2. Taylor expanded around inf 8.1

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

4. Final simplification17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.183566067701958 \cdot 10^{+165}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 6.197016014496118 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.6429575845296566 \cdot 10^{-179}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 7.188021498228873 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))