Average Error: 12.7 → 0.7
Time: 4.6min
Precision: binary64
Cost: 2304
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\left(r \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]
\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(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\left(r \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}}\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
 (-
  (+ (/ (/ 2.0 r) r) -1.5)
  (*
   (* (* r w) (/ (+ 0.375 (* v -0.25)) (* (cbrt (- 1.0 v)) (cbrt (- 1.0 v)))))
   (/ r (/ (cbrt (- 1.0 v)) w)))))
double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}
↓
double code(double v, double w, double r) {
	return (((2.0 / r) / r) + -1.5) - (((r * w) * ((0.375 + (v * -0.25)) / (cbrt(1.0 - v) * cbrt(1.0 - v)))) * (r / (cbrt(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
Alternative 1
Accuracy0.7
Cost2496
\[\left(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1} \cdot \left(\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}\right)} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r}{\frac{\sqrt[3]{1 - v}}{w}}\]
Alternative 2
Accuracy0.6
Cost2944
\[\left(\frac{\frac{2}{r}}{r} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{\frac{r}{\frac{\sqrt[3]{1}}{\sqrt[3]{w} \cdot \sqrt[3]{w}}}}{\frac{\sqrt[3]{1 - v}}{\sqrt[3]{w}}}\]
Alternative 3
Accuracy0.9
Cost2944
\[\left(\frac{2}{r \cdot r} + -1.5\right) - \left(\frac{r \cdot w}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \left(\frac{\sqrt[3]{r} \cdot \sqrt[3]{r}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{r}}{\frac{\sqrt[3]{1 - v}}{w}}\right)\]
Alternative 4
Accuracy3.7
Cost1600
\[\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{w \cdot \left(r \cdot w\right)}}\]

Derivation

  1. Initial program 12.7

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

  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*_binary643.7

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

  6. Applied add-cube-cbrt_binary643.8

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

  7. Applied times-frac_binary642.8

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

  8. Applied *-un-lft-identity_binary642.8

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

  9. Applied times-frac_binary640.7

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

  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 associate-/r*_binary640.7

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

  15. Applied div-inv_binary640.7

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

  16. Applied associate-*l*_binary640.7

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

  17. Simplified0.7

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

  18. Final simplification0.7

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

Reproduce

herbie shell --seed 2020322 
(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))