Average Error: 52.7 → 0.5
Time: 7.3s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1 \cdot \frac{4}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1 \cdot \frac{4}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r32886 = b;
        double r32887 = -r32886;
        double r32888 = r32886 * r32886;
        double r32889 = 4.0;
        double r32890 = a;
        double r32891 = r32889 * r32890;
        double r32892 = c;
        double r32893 = r32891 * r32892;
        double r32894 = r32888 - r32893;
        double r32895 = sqrt(r32894);
        double r32896 = r32887 + r32895;
        double r32897 = 2.0;
        double r32898 = r32897 * r32890;
        double r32899 = r32896 / r32898;
        return r32899;
}
↓
double f(double a, double b, double c) {
        double r32900 = 1.0;
        double r32901 = 4.0;
        double r32902 = b;
        double r32903 = -r32902;
        double r32904 = r32902 * r32902;
        double r32905 = a;
        double r32906 = r32901 * r32905;
        double r32907 = c;
        double r32908 = r32906 * r32907;
        double r32909 = r32904 - r32908;
        double r32910 = sqrt(r32909);
        double r32911 = r32903 - r32910;
        double r32912 = r32905 * r32907;
        double r32913 = r32911 / r32912;
        double r32914 = r32901 / r32913;
        double r32915 = r32900 * r32914;
        double r32916 = 2.0;
        double r32917 = r32916 * r32905;
        double r32918 = r32915 / r32917;
        return r32918;
}

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 *-un-lft-identity0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.4

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

  9. Simplified0.4

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)))