Average Error: 12.6 → 0.3
Time: 5.1s
Precision: binary64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{2}{r \cdot r} + -1.5\right) - \sqrt{{\left(r \cdot w\right)}^{2} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}} \cdot \sqrt{{\left(r \cdot w\right)}^{2} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}}\]
\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) - \sqrt{{\left(r \cdot w\right)}^{2} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}} \cdot \sqrt{{\left(r \cdot w\right)}^{2} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}}
(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)
  (*
   (sqrt (* (pow (* r w) 2.0) (/ (+ 0.375 (* v -0.25)) (- 1.0 v))))
   (sqrt (* (pow (* r w) 2.0) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))))
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) - (sqrt(pow((r * w), 2.0) * ((0.375 + (v * -0.25)) / (1.0 - v))) * sqrt(pow((r * w), 2.0) * ((0.375 + (v * -0.25)) / (1.0 - v))));
}

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 12.6

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

  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 div-inv_binary643.7

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

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

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

  8. Applied times-frac_binary642.4

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

  9. Applied associate-*r*_binary642.4

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

  10. Simplified2.4

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

  11. Taylor expanded around 0 16.7

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

  12. Simplified0.3

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

  14. Applied add-sqr-sqrt_binary640.3

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

  15. Simplified0.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2020224 
(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))