Average Error: 12.4 → 0.4
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\sqrt{\sqrt{2}}}{\frac{r}{\sqrt{2}}} \cdot \frac{\sqrt{\sqrt{2}}}{r}\right) - (\left(\frac{(v \cdot -2 + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.4

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

  2. 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 add-sqr-sqrt0.8

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

  5. Applied associate-/l*0.6

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

  7. Applied *-un-lft-identity0.6

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

  8. Applied times-frac0.6

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019090 +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))