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Derivation

1. Split input into 4 regimes

2. if re < -1.1439498147443434e+154

   1. Initial program 59.4

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

   2. Taylor expanded around -inf 7.5

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

   3. Simplified7.5

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

   if -1.1439498147443434e+154 < re < 4.181280593835928e-218 or 8.782846735010695e-123 < re < 1.585583854766267e+148

   1. Initial program 19.4

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

   if 4.181280593835928e-218 < re < 8.782846735010695e-123

   1. Initial program 25.2

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

   2. Taylor expanded around 0 36.8

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

   if 1.585583854766267e+148 < re

   1. Initial program 58.2

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

   2. Taylor expanded around inf 8.1

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

4. Final simplification17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1439498147443434 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.181280593835928 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 8.782846735010695 \cdot 10^{-123}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.585583854766267 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 1.9s)Debug logProfile

herbie shell --seed 2018339 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))