Average Error: 12.5 → 0.4
Time: 20.9s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{1}{1 - v}, \left|r \cdot w\right| \cdot \left|r \cdot w\right|, 4.5\right)\]
\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(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{1}{1 - v}, \left|r \cdot w\right| \cdot \left|r \cdot w\right|, 4.5\right)
double f(double v, double w, double r) {
        double r26874 = 3.0;
        double r26875 = 2.0;
        double r26876 = r;
        double r26877 = r26876 * r26876;
        double r26878 = r26875 / r26877;
        double r26879 = r26874 + r26878;
        double r26880 = 0.125;
        double r26881 = v;
        double r26882 = r26875 * r26881;
        double r26883 = r26874 - r26882;
        double r26884 = r26880 * r26883;
        double r26885 = w;
        double r26886 = r26885 * r26885;
        double r26887 = r26886 * r26876;
        double r26888 = r26887 * r26876;
        double r26889 = r26884 * r26888;
        double r26890 = 1.0;
        double r26891 = r26890 - r26881;
        double r26892 = r26889 / r26891;
        double r26893 = r26879 - r26892;
        double r26894 = 4.5;
        double r26895 = r26893 - r26894;
        return r26895;
}
↓
double f(double v, double w, double r) {
        double r26896 = 3.0;
        double r26897 = 2.0;
        double r26898 = r;
        double r26899 = r26897 / r26898;
        double r26900 = r26899 / r26898;
        double r26901 = r26896 + r26900;
        double r26902 = 0.125;
        double r26903 = v;
        double r26904 = r26897 * r26903;
        double r26905 = r26896 - r26904;
        double r26906 = r26902 * r26905;
        double r26907 = 1.0;
        double r26908 = 1.0;
        double r26909 = r26908 - r26903;
        double r26910 = r26907 / r26909;
        double r26911 = r26906 * r26910;
        double r26912 = w;
        double r26913 = r26898 * r26912;
        double r26914 = fabs(r26913);
        double r26915 = r26914 * r26914;
        double r26916 = 4.5;
        double r26917 = fma(r26911, r26915, r26916);
        double r26918 = r26901 - r26917;
        return r26918;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.5

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

  2. Simplified8.0

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

  4. Applied add-sqr-sqrt8.1

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

  5. Simplified8.0

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

  6. Simplified0.3

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

  8. Applied div-inv0.4

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

  10. Applied associate-/r*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019208 +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))