Average Error: 28.4 → 0.3
Time: 6.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{3 \cdot a}{-1} \cdot \frac{-c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\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}
double f(double a, double b, double c) {
        double r108153 = b;
        double r108154 = -r108153;
        double r108155 = r108153 * r108153;
        double r108156 = 3.0;
        double r108157 = a;
        double r108158 = r108156 * r108157;
        double r108159 = c;
        double r108160 = r108158 * r108159;
        double r108161 = r108155 - r108160;
        double r108162 = sqrt(r108161);
        double r108163 = r108154 + r108162;
        double r108164 = r108163 / r108158;
        return r108164;
}
↓
double f(double a, double b, double c) {
        double r108165 = 3.0;
        double r108166 = a;
        double r108167 = r108165 * r108166;
        double r108168 = -1.0;
        double r108169 = r108167 / r108168;
        double r108170 = c;
        double r108171 = -r108170;
        double r108172 = b;
        double r108173 = -r108172;
        double r108174 = r108172 * r108172;
        double r108175 = r108167 * r108170;
        double r108176 = r108174 - r108175;
        double r108177 = sqrt(r108176);
        double r108178 = r108173 - r108177;
        double r108179 = r108171 / r108178;
        double r108180 = r108169 * r108179;
        double r108181 = r108180 / r108167;
        return r108181;
}

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 28.4

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

  3. Applied flip-+28.3

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

  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 neg-mul-10.5

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

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

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

  11. Applied times-frac0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 
(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)))