Average Error: 52.4 → 0.5
Time: 6.7s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\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}
double f(double a, double b, double c) {
        double r116175 = b;
        double r116176 = -r116175;
        double r116177 = r116175 * r116175;
        double r116178 = 3.0;
        double r116179 = a;
        double r116180 = r116178 * r116179;
        double r116181 = c;
        double r116182 = r116180 * r116181;
        double r116183 = r116177 - r116182;
        double r116184 = sqrt(r116183);
        double r116185 = r116176 + r116184;
        double r116186 = r116185 / r116180;
        return r116186;
}
↓
double f(double a, double b, double c) {
        double r116187 = b;
        double r116188 = 2.0;
        double r116189 = pow(r116187, r116188);
        double r116190 = r116189 - r116189;
        double r116191 = 3.0;
        double r116192 = a;
        double r116193 = c;
        double r116194 = r116192 * r116193;
        double r116195 = r116191 * r116194;
        double r116196 = r116190 + r116195;
        double r116197 = -r116187;
        double r116198 = r116187 * r116187;
        double r116199 = r116191 * r116192;
        double r116200 = r116199 * r116193;
        double r116201 = r116198 - r116200;
        double r116202 = sqrt(r116201);
        double r116203 = r116197 - r116202;
        double r116204 = r116196 / r116203;
        double r116205 = r116204 / r116199;
        return r116205;
}

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 52.4

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

  3. Applied flip-+52.4

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020020 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))