Average Error: 28.7 → 0.4
Time: 5.7s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(3 \cdot a\right) \cdot \left(-b\right) + \left(3 \cdot a\right) \cdot \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(3 \cdot a\right) \cdot \left(-b\right) + \left(3 \cdot a\right) \cdot \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r96062 = b;
        double r96063 = -r96062;
        double r96064 = r96062 * r96062;
        double r96065 = 3.0;
        double r96066 = a;
        double r96067 = r96065 * r96066;
        double r96068 = c;
        double r96069 = r96067 * r96068;
        double r96070 = r96064 - r96069;
        double r96071 = sqrt(r96070);
        double r96072 = r96063 + r96071;
        double r96073 = r96072 / r96067;
        return r96073;
}
↓
double f(double a, double b, double c) {
        double r96074 = b;
        double r96075 = 2.0;
        double r96076 = pow(r96074, r96075);
        double r96077 = r96076 - r96076;
        double r96078 = 3.0;
        double r96079 = a;
        double r96080 = r96078 * r96079;
        double r96081 = c;
        double r96082 = r96080 * r96081;
        double r96083 = r96077 + r96082;
        double r96084 = -r96074;
        double r96085 = r96080 * r96084;
        double r96086 = r96074 * r96074;
        double r96087 = r96086 - r96082;
        double r96088 = sqrt(r96087);
        double r96089 = -r96088;
        double r96090 = r96080 * r96089;
        double r96091 = r96085 + r96090;
        double r96092 = r96083 / r96091;
        return r96092;
}

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 28.7

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  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. 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 associate-*r*0.5

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

  8. Applied div-inv0.5

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

  9. Applied associate-/l*0.5

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

  10. Simplified0.5

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

  12. Applied sub-neg0.5

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

  13. Applied distribute-lft-in0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020020 
(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)))