Average Error: 52.7 → 0.2
Time: 8.1s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1 \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1 \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}
double f(double a, double b, double c) {
        double r35863 = b;
        double r35864 = -r35863;
        double r35865 = r35863 * r35863;
        double r35866 = 4.0;
        double r35867 = a;
        double r35868 = r35866 * r35867;
        double r35869 = c;
        double r35870 = r35868 * r35869;
        double r35871 = r35865 - r35870;
        double r35872 = sqrt(r35871);
        double r35873 = r35864 + r35872;
        double r35874 = 2.0;
        double r35875 = r35874 * r35867;
        double r35876 = r35873 / r35875;
        return r35876;
}
↓
double f(double a, double b, double c) {
        double r35877 = 1.0;
        double r35878 = 0.0;
        double r35879 = 4.0;
        double r35880 = a;
        double r35881 = c;
        double r35882 = r35880 * r35881;
        double r35883 = r35879 * r35882;
        double r35884 = r35878 + r35883;
        double r35885 = r35884 / r35880;
        double r35886 = b;
        double r35887 = -r35886;
        double r35888 = r35886 * r35886;
        double r35889 = r35879 * r35880;
        double r35890 = r35889 * r35881;
        double r35891 = r35888 - r35890;
        double r35892 = sqrt(r35891);
        double r35893 = r35887 - r35892;
        double r35894 = r35885 / r35893;
        double r35895 = r35877 * r35894;
        double r35896 = 2.0;
        double r35897 = r35895 / r35896;
        return r35897;
}

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 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.4

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

  6. Applied flip-+0.4

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

  7. Applied associate-/l/0.5

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

  9. Applied flip--0.5

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

  10. Applied associate-*r/0.5

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

  11. Applied associate-/r/0.5

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  15. Applied associate-*l/0.4

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2020035 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))