Average Error: 12.8 → 0.4
Time: 23.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \cdot \left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right)\right) - 4.5\]
\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 \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \cdot \left(\left|r \cdot w\right| \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right)\right) - 4.5
double f(double v, double w, double r) {
        double r26876 = 3.0;
        double r26877 = 2.0;
        double r26878 = r;
        double r26879 = r26878 * r26878;
        double r26880 = r26877 / r26879;
        double r26881 = r26876 + r26880;
        double r26882 = 0.125;
        double r26883 = v;
        double r26884 = r26877 * r26883;
        double r26885 = r26876 - r26884;
        double r26886 = r26882 * r26885;
        double r26887 = w;
        double r26888 = r26887 * r26887;
        double r26889 = r26888 * r26878;
        double r26890 = r26889 * r26878;
        double r26891 = r26886 * r26890;
        double r26892 = 1.0;
        double r26893 = r26892 - r26883;
        double r26894 = r26891 / r26893;
        double r26895 = r26881 - r26894;
        double r26896 = 4.5;
        double r26897 = r26895 - r26896;
        return r26897;
}
↓
double f(double v, double w, double r) {
        double r26898 = 3.0;
        double r26899 = 2.0;
        double r26900 = r;
        double r26901 = r26899 / r26900;
        double r26902 = r26901 / r26900;
        double r26903 = r26898 + r26902;
        double r26904 = w;
        double r26905 = r26900 * r26904;
        double r26906 = fabs(r26905);
        double r26907 = 0.125;
        double r26908 = v;
        double r26909 = r26899 * r26908;
        double r26910 = r26898 - r26909;
        double r26911 = r26907 * r26910;
        double r26912 = 1.0;
        double r26913 = r26912 - r26908;
        double r26914 = r26911 / r26913;
        double r26915 = sqrt(r26914);
        double r26916 = r26906 * r26915;
        double r26917 = r26916 * r26916;
        double r26918 = r26903 - r26917;
        double r26919 = 4.5;
        double r26920 = r26918 - r26919;
        return r26920;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.8

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

  2. Simplified8.6

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

  4. Applied add-sqr-sqrt8.6

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

  5. Simplified8.6

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

  6. Simplified0.3

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

  8. Applied associate-/r*0.3

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

  10. Applied fma-udef0.3

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

  11. Applied associate--r+0.3

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

  12. Simplified0.3

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

  14. Applied add-sqr-sqrt0.5

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

  15. Applied unpow-prod-down0.5

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

  16. Applied add-sqr-sqrt0.5

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

  17. Applied unswap-sqr0.5

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

  18. Simplified0.4

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

  19. Simplified0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))