Average Error: 12.4 → 0.4
Time: 11.8s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\]
\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
↓
\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)
(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)
  (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* (* r w) (* r 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) - (((0.375 + (v * -0.25)) / (1.0 - v)) * ((r * w) * (r * 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 12.4

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

  2. Simplified8.7

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

  4. Applied associate-*r*_binary643.5

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

  6. Applied div-inv_binary643.5

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

  7. Applied *-un-lft-identity_binary643.5

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

  8. Applied times-frac_binary642.3

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

  9. Applied associate-*r*_binary642.3

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

  10. Simplified2.3

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

  12. Applied add-sqr-sqrt_binary642.3

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

  13. Applied times-frac_binary642.1

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

  14. Applied *-un-lft-identity_binary642.1

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

  15. Applied times-frac_binary640.4

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

  16. Simplified0.3

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2020356 
(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))