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Derivation

1. Split input into 2 regimes

2. if v < -2.675614305948926e+38 or 1.8489521301429376e-10 < v

   1. Initial program 17.3

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
   2. Using strategy rm

   3. Applied associate-*l*13.9

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5\]
   4. Using strategy rm

   5. Applied associate-/l*2.7

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(w \cdot \left(w \cdot r\right)\right) \cdot r}}}\right) - 4.5\]

   6. Taylor expanded around inf 17.0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right) - \left(0.125 \cdot \frac{{r}^{2} \cdot {w}^{2}}{v} + 0.125 \cdot \frac{{r}^{2} \cdot {w}^{2}}{{v}^{2}}\right)\right)}\right) - 4.5\]

   7. Simplified0.8

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.25 - \frac{0.125}{v}\right) - \frac{0.125}{v \cdot v} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)}\right) - 4.5\]

   if -2.675614305948926e+38 < v < 1.8489521301429376e-10

   1. Initial program 7.8

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
   2. Using strategy rm

   3. Applied associate-*l*2.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5\]
   4. Using strategy rm

   5. Applied flip--2.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \color{blue}{\frac{3 \cdot 3 - \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)}{3 + 2 \cdot v}}\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\]

   6. Applied associate-*r/2.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{0.125 \cdot \left(3 \cdot 3 - \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)}{3 + 2 \cdot v}} \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\]

   7. Applied associate-*l/2.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{\left(0.125 \cdot \left(3 \cdot 3 - \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{3 + 2 \cdot v}}}{1 - v}\right) - 4.5\]

   8. Applied associate-/l/2.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(0.125 \cdot \left(3 \cdot 3 - \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{\left(1 - v\right) \cdot \left(3 + 2 \cdot v\right)}}\right) - 4.5\]

   9. Simplified0.5

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(9 - \left(v \cdot v\right) \cdot 4\right) \cdot \left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.125\right)}}{\left(1 - v\right) \cdot \left(3 + 2 \cdot v\right)}\right) - 4.5\]
3. Recombined 2 regimes into one program.

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \le -2.675614305948926 \cdot 10^{+38} \lor \neg \left(v \le 1.8489521301429376 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(0.25 - \frac{0.125}{v}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{0.125}{v \cdot v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(9 - 4 \cdot \left(v \cdot v\right)\right) \cdot \left(0.125 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)}{\left(1 - v\right) \cdot \left(2 \cdot v + 3\right)}\right) - 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))

Details

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