Average Error: 12.7 → 0.4
Time: 27.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{0.125}{1 - v}\right)\right), \left(w \cdot r\right) \cdot \left(w \cdot r\right), 4.5\right) - 3\right)\]
\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(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{0.125}{1 - v}\right)\right), \left(w \cdot r\right) \cdot \left(w \cdot r\right), 4.5\right) - 3\right)
double f(double v, double w, double r) {
        double r727596 = 3.0;
        double r727597 = 2.0;
        double r727598 = r;
        double r727599 = r727598 * r727598;
        double r727600 = r727597 / r727599;
        double r727601 = r727596 + r727600;
        double r727602 = 0.125;
        double r727603 = v;
        double r727604 = r727597 * r727603;
        double r727605 = r727596 - r727604;
        double r727606 = r727602 * r727605;
        double r727607 = w;
        double r727608 = r727607 * r727607;
        double r727609 = r727608 * r727598;
        double r727610 = r727609 * r727598;
        double r727611 = r727606 * r727610;
        double r727612 = 1.0;
        double r727613 = r727612 - r727603;
        double r727614 = r727611 / r727613;
        double r727615 = r727601 - r727614;
        double r727616 = 4.5;
        double r727617 = r727615 - r727616;
        return r727617;
}
↓
double f(double v, double w, double r) {
        double r727618 = 2.0;
        double r727619 = r;
        double r727620 = r727618 / r727619;
        double r727621 = r727620 / r727619;
        double r727622 = -2.0;
        double r727623 = v;
        double r727624 = 3.0;
        double r727625 = fma(r727622, r727623, r727624);
        double r727626 = 0.125;
        double r727627 = 1.0;
        double r727628 = r727627 - r727623;
        double r727629 = r727626 / r727628;
        double r727630 = r727625 * r727629;
        double r727631 = log1p(r727630);
        double r727632 = expm1(r727631);
        double r727633 = w;
        double r727634 = r727633 * r727619;
        double r727635 = r727634 * r727634;
        double r727636 = 4.5;
        double r727637 = fma(r727632, r727635, r727636);
        double r727638 = r727637 - r727624;
        double r727639 = r727621 - r727638;
        return r727639;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

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. Simplified0.4

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

  4. Applied associate-/r*0.4

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

  6. Applied expm1-log1p-u0.4

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right)\right)\right)}, \left(w \cdot r\right) \cdot \left(w \cdot r\right), 4.5\right) - 3\right)\]

  7. Final simplification0.4

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{0.125}{1 - v}\right)\right), \left(w \cdot r\right) \cdot \left(w \cdot r\right), 4.5\right) - 3\right)\]

Reproduce

herbie shell --seed 2019138 +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))