Average Error: 43.6 → 0.5
Time: 5.5s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \frac{1}{3 \cdot a}\]
\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(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \frac{1}{3 \cdot a}
double f(double a, double b, double c) {
        double r109768 = b;
        double r109769 = -r109768;
        double r109770 = r109768 * r109768;
        double r109771 = 3.0;
        double r109772 = a;
        double r109773 = r109771 * r109772;
        double r109774 = c;
        double r109775 = r109773 * r109774;
        double r109776 = r109770 - r109775;
        double r109777 = sqrt(r109776);
        double r109778 = r109769 + r109777;
        double r109779 = r109778 / r109773;
        return r109779;
}
↓
double f(double a, double b, double c) {
        double r109780 = b;
        double r109781 = 2.0;
        double r109782 = pow(r109780, r109781);
        double r109783 = r109782 - r109782;
        double r109784 = 3.0;
        double r109785 = a;
        double r109786 = r109784 * r109785;
        double r109787 = c;
        double r109788 = r109786 * r109787;
        double r109789 = r109783 + r109788;
        double r109790 = -r109780;
        double r109791 = r109780 * r109780;
        double r109792 = r109791 - r109788;
        double r109793 = sqrt(r109792);
        double r109794 = r109790 - r109793;
        double r109795 = r109789 / r109794;
        double r109796 = 1.0;
        double r109797 = r109796 / r109786;
        double r109798 = r109795 * r109797;
        return r109798;
}

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 43.6

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

  3. Applied flip-+43.7

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

  4. Simplified0.5

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

  6. Applied associate-*r*0.4

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

  8. Applied div-inv0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020059 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))