Cubic critical, narrow range

Average Error: 28.6 → 0.5

Time: 8.1s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r132851 = b;
        double r132852 = -r132851;
        double r132853 = r132851 * r132851;
        double r132854 = 3.0;
        double r132855 = a;
        double r132856 = r132854 * r132855;
        double r132857 = c;
        double r132858 = r132856 * r132857;
        double r132859 = r132853 - r132858;
        double r132860 = sqrt(r132859);
        double r132861 = r132852 + r132860;
        double r132862 = r132861 / r132856;
        return r132862;
}

↓

double f(double a, double b, double c) {
        double r132863 = 1.0;
        double r132864 = a;
        double r132865 = r132863 / r132864;
        double r132866 = b;
        double r132867 = -r132866;
        double r132868 = r132866 * r132866;
        double r132869 = 3.0;
        double r132870 = r132869 * r132864;
        double r132871 = c;
        double r132872 = r132870 * r132871;
        double r132873 = r132868 - r132872;
        double r132874 = sqrt(r132873);
        double r132875 = r132867 - r132874;
        double r132876 = r132865 / r132875;
        double r132877 = r132864 * r132871;
        double r132878 = r132876 * r132877;
        return r132878;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.6

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

3. Applied flip-+ 28.6

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

4. Simplified 0.6

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

6. Applied *-un-lft-identity 0.6

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

7. Applied *-un-lft-identity 0.6

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

8. Applied times-frac 0.6

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

9. Applied times-frac 0.6

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

10. Simplified 0.6

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

11. Simplified 0.6

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

13. Applied *-un-lft-identity 0.6

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

14. Applied div-inv 0.7

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

15. Applied times-frac 0.7

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

16. Applied associate-*r* 0.6

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

17. Simplified 0.5

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

18. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019322 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))