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Derivation

1. Split input into 2 regimes

2. if v < -3.358397834266214e+70 or 2.747403245033089e-06 < v

   1. Initial program 17.4

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

   3. Applied *-un-lft-identity17.4

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

   4. Applied times-frac8.3

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

   5. Simplified8.3

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

   6. Taylor expanded around inf 16.8

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

   7. Simplified0.6

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

   if -3.358397834266214e+70 < v < 2.747403245033089e-06

   1. Initial program 7.9

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

   2. Taylor expanded around inf 16.8

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375 \cdot \left({r}^{2} \cdot {w}^{2}\right) - 0.25 \cdot \left({r}^{2} \cdot \left(v \cdot {w}^{2}\right)\right)}}{1 - v}\right) - 4.5\]

   3. Simplified0.5

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(v \cdot \left(-0.25\right) + 0.375\right)}}{1 - v}\right) - 4.5\]
3. Recombined 2 regimes into one program.

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \le -3.358397834266214 \cdot 10^{+70} \lor \neg \left(v \le 2.747403245033089 \cdot 10^{-06}\right):\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(0.25 - \frac{0.125}{v}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{0.125}{v \cdot v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(v \cdot \left(-0.25\right) + 0.375\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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

Time bar (total: 33.6s)Debug log