Average Error: 12.3 → 0.8
Time: 8.6s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{r} \cdot \frac{\sqrt{\sqrt{2}}}{r}\right) + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]
\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(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{r} \cdot \frac{\sqrt{\sqrt{2}}}{r}\right) + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}
(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
 (-
  (+
   (*
    (* (* (cbrt (sqrt 2.0)) (cbrt (sqrt 2.0))) (sqrt (sqrt 2.0)))
    (* (/ (cbrt (sqrt 2.0)) r) (/ (sqrt (sqrt 2.0)) r)))
   -1.5)
  (*
   (* (/ (* r w) (* (cbrt (- 1.0 v)) (cbrt (- 1.0 v)))) (+ 0.375 (* v -0.25)))
   (/ r (/ (cbrt (- 1.0 v)) w)))))
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) (((double) (((double) (((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) sqrt(((double) sqrt(2.0)))))) * ((double) ((((double) cbrt(((double) sqrt(2.0)))) / r) * (((double) sqrt(((double) sqrt(2.0)))) / r))))) + -1.5)) - ((double) (((double) ((((double) (r * w)) / ((double) (((double) cbrt(((double) (1.0 - v)))) * ((double) cbrt(((double) (1.0 - v))))))) * ((double) (0.375 + ((double) (v * -0.25)))))) * (r / (((double) cbrt(((double) (1.0 - v)))) / w))))));
}

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 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. Simplified8.8

    \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{r \cdot \left(w \cdot w\right)}}}\]
  3. Using strategy rm

  4. Applied associate-*r*_binary644.0

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot w}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt_binary644.1

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{\color{blue}{\left(\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}\right) \cdot \sqrt[3]{1 - v}}}{\left(r \cdot w\right) \cdot w}}\]

  7. Applied times-frac_binary643.0

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\color{blue}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{r \cdot w} \cdot \frac{\sqrt[3]{1 - v}}{w}}}\]

  8. Applied *-un-lft-identity_binary643.0

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{\color{blue}{1 \cdot r}}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{r \cdot w} \cdot \frac{\sqrt[3]{1 - v}}{w}}\]

  9. Applied times-frac_binary640.8

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{r \cdot w}} \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\right)}\]

  10. Applied associate-*r*_binary640.7

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{1}{\frac{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}{r \cdot w}}\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}}\]

  11. Simplified0.7

    \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right)} \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt_binary641.1

    \[\leadsto \left(\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{r \cdot r} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  14. Applied associate-/l*_binary640.9

    \[\leadsto \left(\color{blue}{\frac{\sqrt{2}}{\frac{r \cdot r}{\sqrt{2}}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  15. Simplified1.0

    \[\leadsto \left(\frac{\sqrt{2}}{\color{blue}{\frac{r}{\frac{\sqrt{2}}{r}}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]
  16. Using strategy rm

  17. Applied *-un-lft-identity_binary641.0

    \[\leadsto \left(\frac{\sqrt{2}}{\frac{r}{\frac{\sqrt{2}}{\color{blue}{1 \cdot r}}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  18. Applied add-sqr-sqrt_binary641.0

    \[\leadsto \left(\frac{\sqrt{2}}{\frac{r}{\frac{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}{1 \cdot r}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  19. Applied sqrt-prod_binary640.8

    \[\leadsto \left(\frac{\sqrt{2}}{\frac{r}{\frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{1 \cdot r}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  20. Applied times-frac_binary640.8

    \[\leadsto \left(\frac{\sqrt{2}}{\frac{r}{\color{blue}{\frac{\sqrt{\sqrt{2}}}{1} \cdot \frac{\sqrt{\sqrt{2}}}{r}}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  21. Applied *-un-lft-identity_binary640.8

    \[\leadsto \left(\frac{\sqrt{2}}{\frac{\color{blue}{1 \cdot r}}{\frac{\sqrt{\sqrt{2}}}{1} \cdot \frac{\sqrt{\sqrt{2}}}{r}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  22. Applied times-frac_binary640.8

    \[\leadsto \left(\frac{\sqrt{2}}{\color{blue}{\frac{1}{\frac{\sqrt{\sqrt{2}}}{1}} \cdot \frac{r}{\frac{\sqrt{\sqrt{2}}}{r}}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  23. Applied add-cube-cbrt_binary640.8

    \[\leadsto \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}{\frac{1}{\frac{\sqrt{\sqrt{2}}}{1}} \cdot \frac{r}{\frac{\sqrt{\sqrt{2}}}{r}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  24. Applied times-frac_binary640.9

    \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{1}{\frac{\sqrt{\sqrt{2}}}{1}}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{r}{\frac{\sqrt{\sqrt{2}}}{r}}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  25. Simplified0.8

    \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right)} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{r}{\frac{\sqrt{\sqrt{2}}}{r}}} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  26. Simplified0.8

    \[\leadsto \left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt{2}}}{r} \cdot \frac{\sqrt{\sqrt{2}}}{r}\right)} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

  27. Final simplification0.8

    \[\leadsto \left(\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{r} \cdot \frac{\sqrt{\sqrt{2}}}{r}\right) + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]

Reproduce

herbie shell --seed 2020210 
(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))