Average Error: 12.1 → 0.4
Time: 8.0m
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(4.5 + \left(-4.5\right)\right) + \left(\left(-4.5\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{(v \cdot -2 + 3)_*}{\frac{\frac{1}{\frac{r \cdot w}{\frac{1 - v}{0.125}}}}{r \cdot w}}\right)\right)\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.1

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

  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 fma-udef0.4

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

  5. Applied associate--r+0.4

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

  7. Applied add-cube-cbrt0.4

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

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

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

  9. Applied prod-diff0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied clear-num0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019094 +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))