Average Error: 12.1 → 0.5
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{\frac{2}{r}}{r} - 1.5\right) - \frac{w \cdot r}{\frac{\sqrt[3]{1 - v}}{0.125}} \cdot \frac{w \cdot r}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{3 - v \cdot 2}}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

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Your Program's Arguments

Results

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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. Initial simplification6.4

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

  4. Applied associate-/l*0.4

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

  5. Taylor expanded around -inf 0.4

    \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - 1.5\right)} - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{\left(3 - 2 \cdot v\right) \cdot 0.125}}\]

  6. Simplified0.4

    \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} - 1.5\right)} - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{\left(3 - 2 \cdot v\right) \cdot 0.125}}\]
  7. Using strategy rm

  8. Applied associate-/r*0.4

    \[\leadsto \left(\color{blue}{\frac{\frac{2}{r}}{r}} - 1.5\right) - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{\left(3 - 2 \cdot v\right) \cdot 0.125}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.5

    \[\leadsto \left(\frac{\frac{2}{r}}{r} - 1.5\right) - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\frac{\color{blue}{\left(\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}\right) \cdot \sqrt[3]{1 - v}}}{\left(3 - 2 \cdot v\right) \cdot 0.125}}\]

  11. Applied times-frac0.5

    \[\leadsto \left(\frac{\frac{2}{r}}{r} - 1.5\right) - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\color{blue}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{3 - 2 \cdot v} \cdot \frac{\sqrt[3]{1 - v}}{0.125}}}\]

  12. Applied times-frac0.5

    \[\leadsto \left(\frac{\frac{2}{r}}{r} - 1.5\right) - \color{blue}{\frac{w \cdot r}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{3 - 2 \cdot v}} \cdot \frac{w \cdot r}{\frac{\sqrt[3]{1 - v}}{0.125}}}\]

  13. Final simplification0.5

    \[\leadsto \left(\frac{\frac{2}{r}}{r} - 1.5\right) - \frac{w \cdot r}{\frac{\sqrt[3]{1 - v}}{0.125}} \cdot \frac{w \cdot r}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{3 - v \cdot 2}}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018234 
(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))