Error

Derivation

1. Initial program 12.9

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

2. Initial simplification0.4

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

4. Applied add-sqr-sqrt1.0

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

5. Applied *-un-lft-identity1.0

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

6. Applied prod-diff1.0

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

7. Simplified0.4

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

8. Simplified0.3

   \[\leadsto (\left(0.125 \cdot \frac{(-2 \cdot v + 3)_*}{1 - v}\right) \cdot \left(-\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + \left(\left(\left(-4.5\right) + 3\right) + \frac{\frac{2}{r}}{r}\right))_* + \color{blue}{0}\]
9. Using strategy rm

10. Applied fma-udef0.3

    \[\leadsto \color{blue}{\left(\left(0.125 \cdot \frac{(-2 \cdot v + 3)_*}{1 - v}\right) \cdot \left(-\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + \left(\left(\left(-4.5\right) + 3\right) + \frac{\frac{2}{r}}{r}\right)\right)} + 0\]
11. Using strategy rm

12. Applied distribute-rgt-neg-in0.3

    \[\leadsto \left(\left(0.125 \cdot \frac{(-2 \cdot v + 3)_*}{1 - v}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(-w \cdot r\right)\right)} + \left(\left(\left(-4.5\right) + 3\right) + \frac{\frac{2}{r}}{r}\right)\right) + 0\]

13. Applied associate-*r*0.3

    \[\leadsto \left(\color{blue}{\left(\left(0.125 \cdot \frac{(-2 \cdot v + 3)_*}{1 - v}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(-w \cdot r\right)} + \left(\left(\left(-4.5\right) + 3\right) + \frac{\frac{2}{r}}{r}\right)\right) + 0\]

14. Final simplification0.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018336 +o rules:numerics
(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))