Average Error: 28.7 → 0.3
Time: 9.5s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{3 \cdot a}{-1}}{\frac{3 \cdot a}{\frac{-c}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{3 \cdot a}{-1}}{\frac{3 \cdot a}{\frac{-c}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}}}
double f(double a, double b, double c) {
        double r150118 = b;
        double r150119 = -r150118;
        double r150120 = r150118 * r150118;
        double r150121 = 3.0;
        double r150122 = a;
        double r150123 = r150121 * r150122;
        double r150124 = c;
        double r150125 = r150123 * r150124;
        double r150126 = r150120 - r150125;
        double r150127 = sqrt(r150126);
        double r150128 = r150119 + r150127;
        double r150129 = r150128 / r150123;
        return r150129;
}
↓
double f(double a, double b, double c) {
        double r150130 = 3.0;
        double r150131 = a;
        double r150132 = r150130 * r150131;
        double r150133 = 1.0;
        double r150134 = -r150133;
        double r150135 = r150132 / r150134;
        double r150136 = c;
        double r150137 = -r150136;
        double r150138 = b;
        double r150139 = -r150138;
        double r150140 = r150138 * r150138;
        double r150141 = r150140 * r150140;
        double r150142 = r150132 * r150136;
        double r150143 = r150142 * r150142;
        double r150144 = r150141 - r150143;
        double r150145 = r150140 + r150142;
        double r150146 = r150144 / r150145;
        double r150147 = sqrt(r150146);
        double r150148 = r150139 - r150147;
        double r150149 = r150137 / r150148;
        double r150150 = r150132 / r150149;
        double r150151 = r150135 / r150150;
        return r150151;
}

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

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

  7. Simplified0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied distribute-lft-neg-in0.5

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

  11. Applied distribute-rgt-neg-in0.5

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

  12. Applied times-frac0.3

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

  13. Applied associate-/l*0.3

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

  15. Applied flip--0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020059 
(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)))