Average Error: 12.4 → 0.4
Time: 6.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\frac{\frac{\sqrt{2}}{1}}{1}}\right) \cdot \sqrt{\frac{\frac{\sqrt{2}}{r}}{r}} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
\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
↓
\left(\sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\frac{\frac{\sqrt{2}}{1}}{1}}\right) \cdot \sqrt{\frac{\frac{\sqrt{2}}{r}}{r}} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\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(((sqrt(2.0) / 1.0) / 1.0))) * sqrt(((sqrt(2.0) / r) / r))) - (fma(((0.125 * (3.0 - (2.0 * v))) / (1.0 - v)), (fabs((w * r)) * fabs((w * r))), 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. Simplified8.1

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

  4. Applied add-sqr-sqrt8.1

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

  5. Simplified8.1

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.4

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

  10. Applied add-sqr-sqrt0.5

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

  12. Applied *-un-lft-identity0.5

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

  13. Applied *-un-lft-identity0.5

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

  14. Applied add-sqr-sqrt0.5

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

  15. Applied times-frac0.5

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

  16. Applied times-frac0.5

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

  17. Applied sqrt-prod0.5

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

  18. Applied associate-*r*0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2020075 +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))