Average Error: 13.3 → 0.6
Time: 5.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left({\left(\sqrt{\left|w \cdot r\right|}\right)}^{3} \cdot \sqrt{\left|w \cdot r\right|}\right) + \left(4.5 - 3\right)\right)\]
\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{\frac{2}{r}}{r} - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left({\left(\sqrt{\left|w \cdot r\right|}\right)}^{3} \cdot \sqrt{\left|w \cdot r\right|}\right) + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r18596 = 3.0;
        double r18597 = 2.0;
        double r18598 = r;
        double r18599 = r18598 * r18598;
        double r18600 = r18597 / r18599;
        double r18601 = r18596 + r18600;
        double r18602 = 0.125;
        double r18603 = v;
        double r18604 = r18597 * r18603;
        double r18605 = r18596 - r18604;
        double r18606 = r18602 * r18605;
        double r18607 = w;
        double r18608 = r18607 * r18607;
        double r18609 = r18608 * r18598;
        double r18610 = r18609 * r18598;
        double r18611 = r18606 * r18610;
        double r18612 = 1.0;
        double r18613 = r18612 - r18603;
        double r18614 = r18611 / r18613;
        double r18615 = r18601 - r18614;
        double r18616 = 4.5;
        double r18617 = r18615 - r18616;
        return r18617;
}
↓
double f(double v, double w, double r) {
        double r18618 = 2.0;
        double r18619 = r;
        double r18620 = r18618 / r18619;
        double r18621 = r18620 / r18619;
        double r18622 = 0.125;
        double r18623 = 3.0;
        double r18624 = v;
        double r18625 = r18618 * r18624;
        double r18626 = r18623 - r18625;
        double r18627 = r18622 * r18626;
        double r18628 = 1.0;
        double r18629 = r18628 - r18624;
        double r18630 = r18627 / r18629;
        double r18631 = w;
        double r18632 = r18631 * r18619;
        double r18633 = fabs(r18632);
        double r18634 = sqrt(r18633);
        double r18635 = 3.0;
        double r18636 = pow(r18634, r18635);
        double r18637 = r18636 * r18634;
        double r18638 = r18630 * r18637;
        double r18639 = 4.5;
        double r18640 = r18639 - r18623;
        double r18641 = r18638 + r18640;
        double r18642 = r18621 - r18641;
        return r18642;
}

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 13.3

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

  2. Simplified9.1

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

  4. Applied add-sqr-sqrt9.1

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

  5. Simplified9.1

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

  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 fma-udef0.4

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

  11. Applied associate--l+0.4

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

  13. Applied add-sqr-sqrt0.5

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

  14. Applied associate-*r*0.5

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

  15. Simplified0.6

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

  16. Final simplification0.6

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

Reproduce

herbie shell --seed 2020018 +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))