Average Error: 13.5 → 1.4
Time: 6.1s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[3 + \left(\frac{2}{r \cdot r} - \left(\left(r \cdot \left(w \cdot \left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right)\right)\right) \cdot \left(\sqrt[3]{r} \cdot \left(w \cdot \left(\frac{0.125}{1 - v} \cdot \left(3 - 2 \cdot v\right)\right)\right)\right) + 4.5\right)\right)\]
\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
↓
3 + \left(\frac{2}{r \cdot r} - \left(\left(r \cdot \left(w \cdot \left(\sqrt[3]{r} \cdot \sqrt[3]{r}\right)\right)\right) \cdot \left(\sqrt[3]{r} \cdot \left(w \cdot \left(\frac{0.125}{1 - v} \cdot \left(3 - 2 \cdot v\right)\right)\right)\right) + 4.5\right)\right)
double code(double v, double w, double r) {
	return ((double) (((double) (((double) (3.0 + (2.0 / ((double) (r * r))))) - (((double) (((double) (0.125 * ((double) (3.0 - ((double) (2.0 * v)))))) * ((double) (((double) (((double) (w * w)) * r)) * r)))) / ((double) (1.0 - v))))) - 4.5));
}
↓
double code(double v, double w, double r) {
	return ((double) (3.0 + ((double) ((2.0 / ((double) (r * r))) - ((double) (((double) (((double) (r * ((double) (w * ((double) (((double) cbrt(r)) * ((double) cbrt(r)))))))) * ((double) (((double) cbrt(r)) * ((double) (w * ((double) ((0.125 / ((double) (1.0 - v))) * ((double) (3.0 - ((double) (2.0 * v)))))))))))) + 4.5))))));
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

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

  2. Simplified8.8

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

  4. Applied associate-*r*2.7

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

  6. Applied add-cube-cbrt2.9

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

  7. Applied associate-*l*2.9

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

  8. Simplified2.1

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

  10. Applied associate-*r*1.4

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

  11. Simplified1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))