Average Error: 12.6 → 0.6
Time: 3.0m
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{\frac{2}{r}}{r} + \left(3 - 4.5\right)\right) - \frac{\left(\sqrt[3]{\left|w \cdot r\right| \cdot \left|w \cdot r\right|} \cdot \sqrt[3]{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}\right) \cdot \sqrt[3]{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}}{\frac{1 - v}{0.375 - v \cdot 0.25}}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.6

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

  3. Applied add-sqr-sqrt12.6

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

  4. Applied simplify12.6

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

  5. Applied simplify6.6

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

  6. Taylor expanded around 0 6.6

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375 \cdot {\left(\left|w \cdot r\right|\right)}^{2} - 0.25 \cdot \left({\left(\left|w \cdot r\right|\right)}^{2} \cdot v\right)}}{1 - v}\right) - 4.5\]

  7. Applied simplify0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{2}{r}}{r} + \left(3 - 4.5\right)\right) - \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{\frac{1 - v}{0.375 - v \cdot 0.25}}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.6

    \[\leadsto \left(\frac{\frac{2}{r}}{r} + \left(3 - 4.5\right)\right) - \frac{\color{blue}{\left(\sqrt[3]{\left|w \cdot r\right| \cdot \left|w \cdot r\right|} \cdot \sqrt[3]{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}\right) \cdot \sqrt[3]{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}}}{\frac{1 - v}{0.375 - v \cdot 0.25}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' 
(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))