Cubic critical, narrow range

Average Error: 28.7 → 0.3

Time: 12.6s

Precision: 64

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

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

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r118682 = b;
        double r118683 = -r118682;
        double r118684 = r118682 * r118682;
        double r118685 = 3.0;
        double r118686 = a;
        double r118687 = r118685 * r118686;
        double r118688 = c;
        double r118689 = r118687 * r118688;
        double r118690 = r118684 - r118689;
        double r118691 = sqrt(r118690);
        double r118692 = r118683 + r118691;
        double r118693 = r118692 / r118687;
        return r118693;
}

↓

double f(double a, double b, double c) {
        double r118694 = 1.0;
        double r118695 = sqrt(r118694);
        double r118696 = c;
        double r118697 = b;
        double r118698 = -r118697;
        double r118699 = r118697 * r118697;
        double r118700 = 3.0;
        double r118701 = a;
        double r118702 = r118700 * r118701;
        double r118703 = r118702 * r118696;
        double r118704 = r118699 - r118703;
        double r118705 = sqrt(r118704);
        double r118706 = r118698 - r118705;
        double r118707 = r118696 / r118706;
        double r118708 = r118695 * r118707;
        double r118709 = r118695 * r118708;
        return r118709;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.7

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

3. Applied flip-+ 28.7

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

4. Simplified 0.6

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

6. Applied *-un-lft-identity 0.6

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

7. Applied *-un-lft-identity 0.6

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

8. Applied times-frac 0.6

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

9. Applied associate-/l* 0.6

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

10. Simplified 0.5

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

12. Applied *-un-lft-identity 0.5

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

13. Applied times-frac 0.4

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

14. Applied associate-/r* 0.4

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

15. Simplified 0.4

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

17. Applied *-un-lft-identity 0.4

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

18. Applied *-un-lft-identity 0.4

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

19. Applied times-frac 0.4

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

20. Applied times-frac 0.4

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

21. Applied times-frac 0.4

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

22. Applied add-sqr-sqrt 0.4

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

23. Applied times-frac 0.4

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

24. Simplified 0.4

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

25. Simplified 0.3

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

26. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019351 
(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)))