Cubic critical, medium range

Average Error: 44.0 → 0.3

Time: 19.9s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\frac{\frac{c}{\left(-\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}\right) - b} \cdot 3}{3}\]

C

double f(double a, double b, double c) {
        double r135839 = b;
        double r135840 = -r135839;
        double r135841 = r135839 * r135839;
        double r135842 = 3.0;
        double r135843 = a;
        double r135844 = r135842 * r135843;
        double r135845 = c;
        double r135846 = r135844 * r135845;
        double r135847 = r135841 - r135846;
        double r135848 = sqrt(r135847);
        double r135849 = r135840 + r135848;
        double r135850 = r135849 / r135844;
        return r135850;
}

↓

double f(double a, double b, double c) {
        double r135851 = c;
        double r135852 = b;
        double r135853 = r135852 * r135852;
        double r135854 = a;
        double r135855 = 3.0;
        double r135856 = r135855 * r135851;
        double r135857 = r135854 * r135856;
        double r135858 = r135853 - r135857;
        double r135859 = sqrt(r135858);
        double r135860 = -r135859;
        double r135861 = r135860 - r135852;
        double r135862 = r135851 / r135861;
        double r135863 = r135862 * r135855;
        double r135864 = r135863 / r135855;
        return r135864;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.0

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
2. Using strategy rm

3. Applied flip-+ 44.0

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

4. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 3\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

5. Simplified 0.4

   \[\leadsto \frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 3\right)}{\color{blue}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}}{3 \cdot a}\]
6. Using strategy rm

7. Applied associate-/r* 0.5

   \[\leadsto \color{blue}{\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}{3}}{a}}\]

8. Simplified 0.5

   \[\leadsto \frac{\color{blue}{\frac{\frac{3 \cdot \left(c \cdot a\right)}{3}}{-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)}}}{a}\]
9. Using strategy rm

10. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\frac{\frac{3 \cdot \left(c \cdot a\right)}{3}}{-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)}}{\color{blue}{1 \cdot a}}\]

11. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\frac{\frac{3 \cdot \left(c \cdot a\right)}{3}}{\color{blue}{1 \cdot \left(-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)\right)}}}{1 \cdot a}\]

12. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\frac{\frac{3 \cdot \left(c \cdot a\right)}{\color{blue}{1 \cdot 3}}}{1 \cdot \left(-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)\right)}}{1 \cdot a}\]

13. Applied times-frac 0.5

    \[\leadsto \frac{\frac{\color{blue}{\frac{3}{1} \cdot \frac{c \cdot a}{3}}}{1 \cdot \left(-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)\right)}}{1 \cdot a}\]

14. Applied times-frac 0.5

    \[\leadsto \frac{\color{blue}{\frac{\frac{3}{1}}{1} \cdot \frac{\frac{c \cdot a}{3}}{-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)}}}{1 \cdot a}\]

15. Applied times-frac 0.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{3}{1}}{1}}{1} \cdot \frac{\frac{\frac{c \cdot a}{3}}{-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)}}{a}}\]

16. Simplified 0.5

    \[\leadsto \color{blue}{3} \cdot \frac{\frac{\frac{c \cdot a}{3}}{-\left(b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}\right)}}{a}\]

17. Simplified 0.5

    \[\leadsto 3 \cdot \color{blue}{\frac{\frac{c}{-\left(\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} + b\right)} \cdot \frac{a}{3}}{a}}\]
18. Using strategy rm

19. Applied pow1 0.5

    \[\leadsto 3 \cdot \color{blue}{{\left(\frac{\frac{c}{-\left(\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} + b\right)} \cdot \frac{a}{3}}{a}\right)}^{1}}\]

20. Applied pow1 0.5

    \[\leadsto \color{blue}{{3}^{1}} \cdot {\left(\frac{\frac{c}{-\left(\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} + b\right)} \cdot \frac{a}{3}}{a}\right)}^{1}\]

21. Applied pow-prod-down 0.5

    \[\leadsto \color{blue}{{\left(3 \cdot \frac{\frac{c}{-\left(\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} + b\right)} \cdot \frac{a}{3}}{a}\right)}^{1}}\]

22. Simplified 0.3

    \[\leadsto {\color{blue}{\left(\frac{3 \cdot \frac{c}{\left(-\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)}\right) - b}}{3}\right)}}^{1}\]

23. Final simplification 0.3

    \[\leadsto \frac{\frac{c}{\left(-\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}\right) - b} \cdot 3}{3}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))