Cubic critical, wide range

Average Error: 52.1 → 0.1

Time: 10.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

\[\frac{c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\]

C

double f(double a, double b, double c) {
        double r101972 = b;
        double r101973 = -r101972;
        double r101974 = r101972 * r101972;
        double r101975 = 3.0;
        double r101976 = a;
        double r101977 = r101975 * r101976;
        double r101978 = c;
        double r101979 = r101977 * r101978;
        double r101980 = r101974 - r101979;
        double r101981 = sqrt(r101980);
        double r101982 = r101973 + r101981;
        double r101983 = r101982 / r101977;
        return r101983;
}

↓

double f(double a, double b, double c) {
        double r101984 = c;
        double r101985 = b;
        double r101986 = -r101985;
        double r101987 = 2.0;
        double r101988 = pow(r101985, r101987);
        double r101989 = 3.0;
        double r101990 = a;
        double r101991 = r101990 * r101984;
        double r101992 = r101989 * r101991;
        double r101993 = r101988 - r101992;
        double r101994 = sqrt(r101993);
        double r101995 = r101986 - r101994;
        double r101996 = r101984 / r101995;
        return r101996;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.1

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

3. Applied flip-+ 52.1

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

4. Simplified 0.5

   \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

5. Simplified 0.5

   \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 3}{\color{blue}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]
6. Using strategy rm

7. Applied div-inv 0.6

   \[\leadsto \frac{\color{blue}{\left(0 + \left(a \cdot c\right) \cdot 3\right) \cdot \frac{1}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]

8. Applied times-frac 0.5

   \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 3}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{a}}\]

9. Simplified 0.5

   \[\leadsto \color{blue}{\frac{3 \cdot \left(a \cdot c\right)}{3}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{a}\]
10. Using strategy rm

11. Applied pow1 0.5

    \[\leadsto \frac{3 \cdot \left(a \cdot c\right)}{3} \cdot \color{blue}{{\left(\frac{\frac{1}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{a}\right)}^{1}}\]

12. Applied pow1 0.5

    \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right)}{3}\right)}^{1}} \cdot {\left(\frac{\frac{1}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{a}\right)}^{1}\]

13. Applied pow-prod-down 0.5

    \[\leadsto \color{blue}{{\left(\frac{3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{a}\right)}^{1}}\]

14. Simplified 0.4

    \[\leadsto {\color{blue}{\left(\frac{\frac{a \cdot c}{1}}{a \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}}^{1}\]
15. Using strategy rm

16. Applied add-cube-cbrt 0.4

    \[\leadsto {\left(\frac{\frac{a \cdot c}{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}}{a \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}^{1}\]

17. Applied times-frac 0.4

    \[\leadsto {\left(\frac{\color{blue}{\frac{a}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{c}{\sqrt[3]{1}}}}{a \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}^{1}\]

18. Applied times-frac 0.1

    \[\leadsto {\color{blue}{\left(\frac{\frac{a}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{a} \cdot \frac{\frac{c}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\right)}}^{1}\]

19. Simplified 0.1

    \[\leadsto {\left(\color{blue}{1} \cdot \frac{\frac{c}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\right)}^{1}\]

20. Simplified 0.1

    \[\leadsto {\left(1 \cdot \color{blue}{\frac{c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}\right)}^{1}\]

21. Final simplification 0.1

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))