Cubic critical, narrow range

Average Error: 28.5 → 0.3

Time: 11.0s

Precision: 64

\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r142661 = b;
        double r142662 = -r142661;
        double r142663 = r142661 * r142661;
        double r142664 = 3.0;
        double r142665 = a;
        double r142666 = r142664 * r142665;
        double r142667 = c;
        double r142668 = r142666 * r142667;
        double r142669 = r142663 - r142668;
        double r142670 = sqrt(r142669);
        double r142671 = r142662 + r142670;
        double r142672 = r142671 / r142666;
        return r142672;
}

↓

double f(double a, double b, double c) {
        double r142673 = a;
        double r142674 = r142673 / r142673;
        double r142675 = c;
        double r142676 = b;
        double r142677 = -r142676;
        double r142678 = r142676 * r142676;
        double r142679 = 3.0;
        double r142680 = r142679 * r142673;
        double r142681 = r142680 * r142675;
        double r142682 = r142678 - r142681;
        double r142683 = sqrt(r142682);
        double r142684 = r142677 - r142683;
        double r142685 = r142675 / r142684;
        double r142686 = r142674 * r142685;
        return r142686;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.5

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

3. Applied flip-+ 28.5

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

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 frac-2neg 0.6

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

7. Simplified 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied distribute-rgt-neg-in 0.5

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

11. Applied times-frac 0.3

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

12. Applied associate-/l* 0.3

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

14. Applied neg-mul-1 0.3

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

15. Applied neg-mul-1 0.3

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

16. Applied times-frac 0.3

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

17. Applied times-frac 0.5

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

18. Applied *-un-lft-identity 0.5

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

19. Applied times-frac 0.5

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

20. Applied times-frac 0.3

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

21. Simplified 0.3

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

22. Simplified 0.3

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

23. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020064 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))