Average Error: 28.6 → 0.3
Time: 6.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(3 \cdot a\right) \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(3 \cdot a\right) \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}
double f(double a, double b, double c) {
        double r99843 = b;
        double r99844 = -r99843;
        double r99845 = r99843 * r99843;
        double r99846 = 3.0;
        double r99847 = a;
        double r99848 = r99846 * r99847;
        double r99849 = c;
        double r99850 = r99848 * r99849;
        double r99851 = r99845 - r99850;
        double r99852 = sqrt(r99851);
        double r99853 = r99844 + r99852;
        double r99854 = r99853 / r99848;
        return r99854;
}
↓
double f(double a, double b, double c) {
        double r99855 = 3.0;
        double r99856 = a;
        double r99857 = r99855 * r99856;
        double r99858 = c;
        double r99859 = b;
        double r99860 = -r99859;
        double r99861 = r99859 * r99859;
        double r99862 = r99857 * r99858;
        double r99863 = r99861 - r99862;
        double r99864 = sqrt(r99863);
        double r99865 = r99860 - r99864;
        double r99866 = r99858 / r99865;
        double r99867 = r99866 / r99857;
        double r99868 = r99857 * r99867;
        return r99868;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.6

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

  6. Applied frac-2neg0.6

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

  7. Simplified0.4

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

  9. Applied neg-mul-10.4

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

  10. Applied *-un-lft-identity0.4

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

  11. Applied distribute-lft-neg-in0.4

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

  12. Applied times-frac0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 
(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)))