Average Error: 13.0 → 0.4
Time: 19.8s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
\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(3 + \frac{\frac{2}{r}}{r}\right) - \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)
double f(double v, double w, double r) {
        double r24867 = 3.0;
        double r24868 = 2.0;
        double r24869 = r;
        double r24870 = r24869 * r24869;
        double r24871 = r24868 / r24870;
        double r24872 = r24867 + r24871;
        double r24873 = 0.125;
        double r24874 = v;
        double r24875 = r24868 * r24874;
        double r24876 = r24867 - r24875;
        double r24877 = r24873 * r24876;
        double r24878 = w;
        double r24879 = r24878 * r24878;
        double r24880 = r24879 * r24869;
        double r24881 = r24880 * r24869;
        double r24882 = r24877 * r24881;
        double r24883 = 1.0;
        double r24884 = r24883 - r24874;
        double r24885 = r24882 / r24884;
        double r24886 = r24872 - r24885;
        double r24887 = 4.5;
        double r24888 = r24886 - r24887;
        return r24888;
}
↓
double f(double v, double w, double r) {
        double r24889 = 3.0;
        double r24890 = 2.0;
        double r24891 = r;
        double r24892 = r24890 / r24891;
        double r24893 = r24892 / r24891;
        double r24894 = r24889 + r24893;
        double r24895 = 0.125;
        double r24896 = v;
        double r24897 = r24890 * r24896;
        double r24898 = r24889 - r24897;
        double r24899 = r24895 * r24898;
        double r24900 = 1.0;
        double r24901 = r24900 - r24896;
        double r24902 = r24899 / r24901;
        double r24903 = w;
        double r24904 = r24903 * r24891;
        double r24905 = fabs(r24904);
        double r24906 = r24905 * r24905;
        double r24907 = 4.5;
        double r24908 = fma(r24902, r24906, r24907);
        double r24909 = r24894 - r24908;
        return r24909;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 13.0

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

  2. Simplified8.7

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

  4. Applied add-sqr-sqrt8.7

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

  5. Simplified8.7

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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))