Average Error: 12.3 → 0.4
Time: 2.6m
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left(\left(\frac{3}{\frac{1 - v \cdot v}{0.125}} - \frac{2 \cdot 0.125}{\frac{1}{v} - v}\right) \cdot \left(1 + v\right)\right)\right) - 4.5\]

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. Using strategy rm

  3. Applied flip--13.3

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

  4. Applied associate-/r/13.3

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

  5. Applied simplify7.8

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

  7. Applied *-un-lft-identity7.8

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

  8. Applied times-frac5.6

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

  9. Applied simplify5.6

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

  11. Applied div-sub5.6

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

  12. Applied simplify0.4

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

  14. Applied associate-*l*0.4

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))