Average Error: 44.1 → 0.5
Time: 15.2s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3 \cdot \left(a \cdot c\right)}{\left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot 3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3 \cdot \left(a \cdot c\right)}{\left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot 3}
double f(double a, double b, double c) {
        double r84783 = b;
        double r84784 = -r84783;
        double r84785 = r84783 * r84783;
        double r84786 = 3.0;
        double r84787 = a;
        double r84788 = r84786 * r84787;
        double r84789 = c;
        double r84790 = r84788 * r84789;
        double r84791 = r84785 - r84790;
        double r84792 = sqrt(r84791);
        double r84793 = r84784 + r84792;
        double r84794 = r84793 / r84788;
        return r84794;
}
↓
double f(double a, double b, double c) {
        double r84795 = 3.0;
        double r84796 = a;
        double r84797 = c;
        double r84798 = r84796 * r84797;
        double r84799 = r84795 * r84798;
        double r84800 = b;
        double r84801 = -r84800;
        double r84802 = r84800 * r84800;
        double r84803 = r84795 * r84796;
        double r84804 = r84803 * r84797;
        double r84805 = r84802 - r84804;
        double r84806 = sqrt(r84805);
        double r84807 = r84801 - r84806;
        double r84808 = r84796 * r84807;
        double r84809 = r84808 * r84795;
        double r84810 = r84799 / r84809;
        return r84810;
}

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 44.1

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

  3. Applied flip-+44.1

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied div-inv0.6

    \[\leadsto \frac{\color{blue}{\left(0 + 3 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  7. Applied associate-/l*0.6

    \[\leadsto \color{blue}{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\frac{3 \cdot a}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

  8. Simplified0.5

    \[\leadsto \frac{0 + 3 \cdot \left(a \cdot c\right)}{\color{blue}{\left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot 3}}\]

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))