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Derivation

1. Split input into 4 regimes

2. if re < -3.4685742075803927e+145

   1. Initial program 57.8

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

   2. Taylor expanded around -inf 8.4

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

   3. Simplified8.4

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

   if -3.4685742075803927e+145 < re < -4.227035670334825e-245 or 7.77224422089703e-199 < re < 3.5211960401582444e+167

   1. Initial program 17.1

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

   if -4.227035670334825e-245 < re < 7.77224422089703e-199

   1. Initial program 29.6

      \[\sqrt{re \cdot re + im \cdot im}\]
   2. Using strategy rm

   3. Applied add-exp-log32.0

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   4. Taylor expanded around 0 32.4

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

   if 3.5211960401582444e+167 < re

   1. Initial program 59.3

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

   2. Taylor expanded around inf 6.7

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

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.4685742075803927 \cdot 10^{+145}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.227035670334825 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 7.77224422089703 \cdot 10^{-199}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.5211960401582444 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Runtime

Time bar (total: 6.3s)Debug logProfile

herbie shell --seed 2018352 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))