Average Error: 12.1 → 0.4
Time: 3.2m
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{2}{r \cdot r} - 1.5\right) - \sqrt[3]{\left(\left(\frac{0.125}{1 - v} \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(\frac{0.125}{1 - v} \cdot \left(3 - v \cdot 2\right)\right)\right) \cdot \left(\frac{0.125}{1 - v} \cdot \left(3 - v \cdot 2\right)\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

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. Simplified0.4

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

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  6. Applied add-cbrt-cube7.2

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

  7. Applied add-cbrt-cube20.7

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

  8. Applied cbrt-undiv20.7

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019089 
(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))