Average Error: 28.7 → 0.3
Time: 8.7s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \cdot \left(3 \cdot a\right)\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \cdot \left(3 \cdot a\right)
double f(double a, double b, double c) {
        double r127060 = b;
        double r127061 = -r127060;
        double r127062 = r127060 * r127060;
        double r127063 = 3.0;
        double r127064 = a;
        double r127065 = r127063 * r127064;
        double r127066 = c;
        double r127067 = r127065 * r127066;
        double r127068 = r127062 - r127067;
        double r127069 = sqrt(r127068);
        double r127070 = r127061 + r127069;
        double r127071 = r127070 / r127065;
        return r127071;
}
↓
double f(double a, double b, double c) {
        double r127072 = c;
        double r127073 = b;
        double r127074 = -r127073;
        double r127075 = r127073 * r127073;
        double r127076 = 3.0;
        double r127077 = a;
        double r127078 = r127076 * r127077;
        double r127079 = r127078 * r127072;
        double r127080 = r127075 - r127079;
        double r127081 = sqrt(r127080);
        double r127082 = r127074 - r127081;
        double r127083 = r127072 / r127082;
        double r127084 = r127083 / r127078;
        double r127085 = r127084 * r127078;
        return r127085;
}

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 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-rgt-neg-in0.4

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

  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

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

Reproduce

herbie shell --seed 2020034 
(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)))