Average Error: 12.4 → 0.5
Time: 6.4s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\sqrt{\frac{2}{r \cdot r}} \cdot \frac{\sqrt{\frac{2}{\sqrt{r \cdot r}}}}{\sqrt{\sqrt{r \cdot r}}} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(r \cdot w\right) \cdot \left(r \cdot w\right), 4.5\right) - 3\right)\]
\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
↓
\sqrt{\frac{2}{r \cdot r}} \cdot \frac{\sqrt{\frac{2}{\sqrt{r \cdot r}}}}{\sqrt{\sqrt{r \cdot r}}} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(r \cdot w\right) \cdot \left(r \cdot w\right), 4.5\right) - 3\right)
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 ((sqrt((2.0 / (r * r))) * (sqrt((2.0 / sqrt((r * r)))) / sqrt(sqrt((r * r))))) - (fma(((0.125 * (3.0 - (2.0 * v))) / (1.0 - v)), ((r * w) * (r * w)), 4.5) - 3.0));
}

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. Simplified8.2

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

  4. Applied add-sqr-sqrt36.4

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

  5. Applied add-sqr-sqrt36.4

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

  6. Applied unswap-sqr33.8

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

  7. Applied unswap-sqr32.8

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

  8. Simplified32.7

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

  9. Simplified0.4

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

  11. Applied add-sqr-sqrt0.5

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

  13. Applied add-sqr-sqrt0.5

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

  14. Applied associate-/r*0.5

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

  15. Applied sqrt-div0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))