Average Error: 12.3 → 0.5
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[(\left(\sqrt{\frac{3 - 2 \cdot v}{1 - v}} \cdot \sqrt{\frac{3 - 2 \cdot v}{1 - v}}\right) \cdot \left(\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(-w \cdot r\right)\right) + \left(\left(\left(-4.5\right) + 3\right) + \frac{2}{r \cdot r}\right))_* + 0\]

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. Applied simplify0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied add-sqr-sqrt0.9

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

  6. Applied prod-diff0.9

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

  7. Applied simplify0.3

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

  8. Applied simplify0.3

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

  10. Applied add-sqr-sqrt0.5

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2019053 +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))