Average Error: 12.7 → 0.3
Time: 5.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
\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
↓
\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{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 (((2.0 / r) / r) - (fma(((0.375 - (0.25 * 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 12.7

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

  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-sqrt8.6

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

  5. Simplified8.6

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

  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. Taylor expanded around 0 0.3

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{\color{blue}{0.375 - 0.25 \cdot v}}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
  8. Using strategy rm

  9. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} - \left(\mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]

  10. Final simplification0.3

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]

Reproduce

herbie shell --seed 2020106 +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))