Average Error: 12.1 → 0.4
Time: 26.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\mathsf{fma}\left(1, \frac{2}{r \cdot r}, \frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.125\right)\right)\right)\right) + \mathsf{fma}\left(\sqrt{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot w\right)\right)} \cdot \left(-\sqrt{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot w\right)\right)}\right), \frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v}, \left(\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v}\right)\right) - \left(4.5 - 3\right)\]
\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(\mathsf{fma}\left(1, \frac{2}{r \cdot r}, \frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.125\right)\right)\right)\right) + \mathsf{fma}\left(\sqrt{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot w\right)\right)} \cdot \left(-\sqrt{\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot w\right)\right)}\right), \frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v}, \left(\left(r \cdot w\right) \cdot \left(0.125 \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v}\right)\right) - \left(4.5 - 3\right)
double f(double v, double w, double r) {
        double r1565759 = 3.0;
        double r1565760 = 2.0;
        double r1565761 = r;
        double r1565762 = r1565761 * r1565761;
        double r1565763 = r1565760 / r1565762;
        double r1565764 = r1565759 + r1565763;
        double r1565765 = 0.125;
        double r1565766 = v;
        double r1565767 = r1565760 * r1565766;
        double r1565768 = r1565759 - r1565767;
        double r1565769 = r1565765 * r1565768;
        double r1565770 = w;
        double r1565771 = r1565770 * r1565770;
        double r1565772 = r1565771 * r1565761;
        double r1565773 = r1565772 * r1565761;
        double r1565774 = r1565769 * r1565773;
        double r1565775 = 1.0;
        double r1565776 = r1565775 - r1565766;
        double r1565777 = r1565774 / r1565776;
        double r1565778 = r1565764 - r1565777;
        double r1565779 = 4.5;
        double r1565780 = r1565778 - r1565779;
        return r1565780;
}
↓
double f(double v, double w, double r) {
        double r1565781 = 1.0;
        double r1565782 = 2.0;
        double r1565783 = r;
        double r1565784 = r1565783 * r1565783;
        double r1565785 = r1565782 / r1565784;
        double r1565786 = -2.0;
        double r1565787 = v;
        double r1565788 = 3.0;
        double r1565789 = fma(r1565786, r1565787, r1565788);
        double r1565790 = r1565781 - r1565787;
        double r1565791 = r1565789 / r1565790;
        double r1565792 = w;
        double r1565793 = r1565783 * r1565792;
        double r1565794 = 0.125;
        double r1565795 = -r1565794;
        double r1565796 = r1565793 * r1565795;
        double r1565797 = r1565793 * r1565796;
        double r1565798 = r1565791 * r1565797;
        double r1565799 = fma(r1565781, r1565785, r1565798);
        double r1565800 = r1565794 * r1565793;
        double r1565801 = r1565793 * r1565800;
        double r1565802 = sqrt(r1565801);
        double r1565803 = -r1565802;
        double r1565804 = r1565802 * r1565803;
        double r1565805 = r1565801 * r1565791;
        double r1565806 = fma(r1565804, r1565791, r1565805);
        double r1565807 = r1565799 + r1565806;
        double r1565808 = 4.5;
        double r1565809 = r1565808 - r1565788;
        double r1565810 = r1565807 - r1565809;
        return r1565810;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.1

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

  2. Simplified0.3

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

  4. Applied fma-udef0.3

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

  5. Applied associate--l+0.3

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

  6. Applied associate--r+0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied prod-diff0.3

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

  11. Applied add-sqr-sqrt0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))