Average Error: 28.8 → 0.5
Time: 11.1s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{2}
double f(double a, double b, double c) {
        double r38781 = b;
        double r38782 = -r38781;
        double r38783 = r38781 * r38781;
        double r38784 = 4.0;
        double r38785 = a;
        double r38786 = r38784 * r38785;
        double r38787 = c;
        double r38788 = r38786 * r38787;
        double r38789 = r38783 - r38788;
        double r38790 = sqrt(r38789);
        double r38791 = r38782 + r38790;
        double r38792 = 2.0;
        double r38793 = r38792 * r38785;
        double r38794 = r38791 / r38793;
        return r38794;
}
↓
double f(double a, double b, double c) {
        double r38795 = 1.0;
        double r38796 = b;
        double r38797 = -r38796;
        double r38798 = r38796 * r38796;
        double r38799 = r38798 * r38798;
        double r38800 = 4.0;
        double r38801 = a;
        double r38802 = r38800 * r38801;
        double r38803 = c;
        double r38804 = r38802 * r38803;
        double r38805 = r38804 * r38804;
        double r38806 = r38799 - r38805;
        double r38807 = r38798 + r38804;
        double r38808 = r38806 / r38807;
        double r38809 = sqrt(r38808);
        double r38810 = r38797 - r38809;
        double r38811 = r38795 / r38810;
        double r38812 = r38811 / r38801;
        double r38813 = r38801 * r38803;
        double r38814 = r38813 * r38800;
        double r38815 = 2.0;
        double r38816 = r38814 / r38815;
        double r38817 = r38812 * r38816;
        return r38817;
}

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 28.8

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.8

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  7. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}}\]

  8. Simplified0.5

    \[\leadsto \color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{2}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\]
  9. Using strategy rm

  10. Applied flip--0.5

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{a}\]

  11. Final simplification0.5

    \[\leadsto \frac{\frac{1}{\left(-b\right) - \sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\left(a \cdot c\right) \cdot 4}{2}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))