Rosa's TurbineBenchmark

Average Error: 12.1 → 1.3

Time: 6.3s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

↓

(FPCore (v w r)
 :precision binary64
 (+
  3.0
  (+
   (/ 2.0 (* r r))
   (-
    (*
     (* (cbrt r) (* (cbrt r) (* r w)))
     (*
      (* w (* (/ 0.125 (- 1.0 v)) (- (* 2.0 v) 3.0)))
      (* (cbrt (* (cbrt r) (cbrt r))) (cbrt (cbrt r)))))
    4.5))))

C

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) (((double) cbrt(r)) * ((double) (((double) cbrt(r)) * ((double) (r * w)))))) * ((double) (((double) (w * ((double) ((0.125 / ((double) (1.0 - v))) * ((double) (((double) (2.0 * v)) - 3.0)))))) * ((double) (((double) cbrt(((double) (((double) cbrt(r)) * ((double) cbrt(r)))))) * ((double) cbrt(((double) cbrt(r)))))))))) - 4.5))))));
}

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. Simplified 7.8

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

4. Applied associate-*r* 2.1

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

6. Applied add-cube-cbrt 2.4

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

7. Applied associate-*l* 2.4

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

8. Simplified 1.8

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

10. Applied add-cube-cbrt 1.9

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

11. Applied cbrt-prod 1.9

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

13. Applied associate-*r* 1.3

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

14. Simplified 1.3

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

15. Final simplification 1.3

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

Reproduce

herbie shell --seed 2020198 
(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))