Average Error: 12.3 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(\frac{2}{r \cdot r} + 3\right) - (\left(\frac{3 - v \cdot 2}{1 - v}\right) \cdot \left(\sqrt{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)} \cdot \left(\sqrt{\sqrt{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}} \cdot \left(\sqrt{\sqrt{\sqrt{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}} \cdot \sqrt{\sqrt{\sqrt{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}}\right)\right)\right) + 4.5)_*\right) + 0\]

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 simplify0.4

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

  4. Applied add-cube-cbrt0.7

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

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

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

  6. Applied prod-diff0.7

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

  7. Applied simplify0.4

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

  8. Applied simplify0.3

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

  10. Applied add-sqr-sqrt0.4

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

  12. Applied add-sqr-sqrt0.4

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

  14. Applied add-sqr-sqrt0.4

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

  15. Applied sqrt-prod0.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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))