Average Error: 12.7 → 0.5
Time: 5.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, {\left(\sqrt{\left|w \cdot r\right|}\right)}^{3} \cdot \left(\left|\sqrt[3]{\left|w \cdot r\right|}\right| \cdot \sqrt{\sqrt[3]{\left|w \cdot r\right|}}\right), 4.5\right) - 3\right)\]
\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{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, {\left(\sqrt{\left|w \cdot r\right|}\right)}^{3} \cdot \left(\left|\sqrt[3]{\left|w \cdot r\right|}\right| \cdot \sqrt{\sqrt[3]{\left|w \cdot r\right|}}\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(((0.125 * (3.0 - (2.0 * v))) / (1.0 - v)), (pow(sqrt(fabs((w * r))), 3.0) * (fabs(cbrt(fabs((w * r)))) * sqrt(cbrt(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. Simplified8.4

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

  4. Applied add-sqr-sqrt8.5

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

  5. Simplified8.4

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

  6. Simplified0.3

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-*r*0.4

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

  10. Simplified0.5

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

  12. Applied add-cube-cbrt0.5

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

  13. Applied sqrt-prod0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020105 +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))