Average Error: 12.9 → 0.4
Time: 8.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{2}{r \cdot r} - \left(\left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \cdot \left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) + \left(4.5 - 3\right)\right)\]
\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
↓
\frac{2}{r \cdot r} - \left(\left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \cdot \left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) + \left(4.5 - 3\right)\right)
double code(double v, double w, double r) {
	return ((double) (((double) (((double) (3.0 + ((double) (2.0 / ((double) (r * r)))))) - ((double) (((double) (((double) (0.125 * ((double) (3.0 - ((double) (2.0 * v)))))) * ((double) (((double) (((double) (w * w)) * r)) * r)))) / ((double) (1.0 - v)))))) - 4.5));
}
↓
double code(double v, double w, double r) {
	return ((double) (((double) (2.0 / ((double) (r * r)))) - ((double) (((double) (((double) (((double) fabs(((double) (r * w)))) * ((double) sqrt(((double) (((double) (0.125 * ((double) (3.0 - ((double) (2.0 * v)))))) / ((double) (1.0 - v)))))))) * ((double) (((double) fabs(((double) (r * w)))) * ((double) sqrt(((double) (((double) (0.125 * ((double) (3.0 - ((double) (2.0 * v)))))) / ((double) (1.0 - v)))))))))) + ((double) (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 12.9

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

  2. Simplified8.6

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

  4. Applied add-sqr-sqrt36.0

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

  5. Applied add-sqr-sqrt36.0

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

  6. Applied unswap-sqr33.0

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

  7. Applied unswap-sqr31.8

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

  8. Simplified31.8

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

  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 fma-udef0.3

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

  12. Applied associate--l+0.3

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

  14. Applied add-sqr-sqrt0.3

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

  15. Applied add-sqr-sqrt0.4

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

  16. Applied unswap-sqr0.4

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

  17. Simplified0.4

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

  18. Simplified0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2020114 +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))