Average Error: 12.3 → 0.3
Time: 2.4m
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} + \left(3 - (\left(\frac{(-2 \cdot v + 3)_* \cdot 0.125}{1 - v}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\right)\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.3

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

  2. Simplified0.4

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

  4. Applied associate-/r*0.4

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

  6. Applied fma-udef0.4

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

  8. Applied *-un-lft-identity0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Applied distribute-lft-out--0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019091 +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))