Average Error: 13.0 → 0.4
Time: 47.1s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\right) - \frac{0.125 \cdot \left(3 - v \cdot 2\right)}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 13.0

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

  2. Simplified0.4

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

  4. Applied associate-/r*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019050 
(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))