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Derivation

1. Split input into 3 regimes

2. if re < -1.3993449472415332e+160

   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. Taylor expanded around -inf 7.1

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

   4. Simplified7.1

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

   if -1.3993449472415332e+160 < re < 1.5721373574251358e+104

   1. Initial program 20.2

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

   2. Initial simplification20.2

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

   if 1.5721373574251358e+104 < re

   1. Initial program 48.3

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

   2. Initial simplification48.3

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

   3. Taylor expanded around inf 9.7

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3993449472415332 \cdot 10^{+160}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.5721373574251358 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 7.1s)Debug logProfile

herbie shell --seed 2018277 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))