Average Error: 11.9 → 0.3
Time: 21.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, v, 3\right)}{1 - v}, \left(w \cdot r\right) \cdot \left(0.125 \cdot \left(w \cdot r\right)\right), 4.5\right) - 3\right)\]
\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{\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)
double f(double v, double w, double r) {
        double r1166982 = 3.0;
        double r1166983 = 2.0;
        double r1166984 = r;
        double r1166985 = r1166984 * r1166984;
        double r1166986 = r1166983 / r1166985;
        double r1166987 = r1166982 + r1166986;
        double r1166988 = 0.125;
        double r1166989 = v;
        double r1166990 = r1166983 * r1166989;
        double r1166991 = r1166982 - r1166990;
        double r1166992 = r1166988 * r1166991;
        double r1166993 = w;
        double r1166994 = r1166993 * r1166993;
        double r1166995 = r1166994 * r1166984;
        double r1166996 = r1166995 * r1166984;
        double r1166997 = r1166992 * r1166996;
        double r1166998 = 1.0;
        double r1166999 = r1166998 - r1166989;
        double r1167000 = r1166997 / r1166999;
        double r1167001 = r1166987 - r1167000;
        double r1167002 = 4.5;
        double r1167003 = r1167001 - r1167002;
        return r1167003;
}
↓
double f(double v, double w, double r) {
        double r1167004 = 2.0;
        double r1167005 = r;
        double r1167006 = r1167004 / r1167005;
        double r1167007 = r1167006 / r1167005;
        double r1167008 = -2.0;
        double r1167009 = v;
        double r1167010 = 3.0;
        double r1167011 = fma(r1167008, r1167009, r1167010);
        double r1167012 = 1.0;
        double r1167013 = r1167012 - r1167009;
        double r1167014 = r1167011 / r1167013;
        double r1167015 = w;
        double r1167016 = r1167015 * r1167005;
        double r1167017 = 0.125;
        double r1167018 = r1167017 * r1167016;
        double r1167019 = r1167016 * r1167018;
        double r1167020 = 4.5;
        double r1167021 = fma(r1167014, r1167019, r1167020);
        double r1167022 = r1167021 - r1167010;
        double r1167023 = r1167007 - r1167022;
        return r1167023;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 11.9

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

  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 associate-/r*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +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))