Average Error: 11.7 → 0.4
Time: 34.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(\sqrt{\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}} \cdot \sqrt{\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}}\right)\right) - 4.5\]
\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(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(\sqrt{\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}} \cdot \sqrt{\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}}\right)\right) - 4.5
double f(double v, double w, double r) {
        double r672959 = 3.0;
        double r672960 = 2.0;
        double r672961 = r;
        double r672962 = r672961 * r672961;
        double r672963 = r672960 / r672962;
        double r672964 = r672959 + r672963;
        double r672965 = 0.125;
        double r672966 = v;
        double r672967 = r672960 * r672966;
        double r672968 = r672959 - r672967;
        double r672969 = r672965 * r672968;
        double r672970 = w;
        double r672971 = r672970 * r672970;
        double r672972 = r672971 * r672961;
        double r672973 = r672972 * r672961;
        double r672974 = r672969 * r672973;
        double r672975 = 1.0;
        double r672976 = r672975 - r672966;
        double r672977 = r672974 / r672976;
        double r672978 = r672964 - r672977;
        double r672979 = 4.5;
        double r672980 = r672978 - r672979;
        return r672980;
}
↓
double f(double v, double w, double r) {
        double r672981 = 3.0;
        double r672982 = 2.0;
        double r672983 = r;
        double r672984 = r672982 / r672983;
        double r672985 = r672984 / r672983;
        double r672986 = r672981 + r672985;
        double r672987 = w;
        double r672988 = r672983 * r672987;
        double r672989 = r672988 * r672988;
        double r672990 = 0.125;
        double r672991 = 1.0;
        double r672992 = v;
        double r672993 = r672991 - r672992;
        double r672994 = -2.0;
        double r672995 = fma(r672994, r672992, r672981);
        double r672996 = r672993 / r672995;
        double r672997 = r672990 / r672996;
        double r672998 = sqrt(r672997);
        double r672999 = r672998 * r672998;
        double r673000 = r672989 * r672999;
        double r673001 = r672986 - r673000;
        double r673002 = 4.5;
        double r673003 = r673001 - r673002;
        return r673003;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 11.7

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

  2. Simplified0.3

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

  4. Applied associate-/r*0.4

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

  6. Applied fma-udef0.4

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

  7. Applied associate--r+0.3

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

  9. Applied add-sqr-sqrt0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 +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))