Average Error: 28.4 → 0.4
Time: 7.7s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\left(-b\right) - \frac{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\left(-b\right) - \frac{\sqrt{\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)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}
double f(double a, double b, double c) {
        double r36900 = b;
        double r36901 = -r36900;
        double r36902 = r36900 * r36900;
        double r36903 = 4.0;
        double r36904 = a;
        double r36905 = r36903 * r36904;
        double r36906 = c;
        double r36907 = r36905 * r36906;
        double r36908 = r36902 - r36907;
        double r36909 = sqrt(r36908);
        double r36910 = r36901 + r36909;
        double r36911 = 2.0;
        double r36912 = r36911 * r36904;
        double r36913 = r36910 / r36912;
        return r36913;
}
↓
double f(double a, double b, double c) {
        double r36914 = 4.0;
        double r36915 = a;
        double r36916 = c;
        double r36917 = r36915 * r36916;
        double r36918 = r36914 * r36917;
        double r36919 = 2.0;
        double r36920 = r36918 / r36919;
        double r36921 = r36920 / r36915;
        double r36922 = b;
        double r36923 = -r36922;
        double r36924 = r36922 * r36922;
        double r36925 = r36924 * r36924;
        double r36926 = r36914 * r36915;
        double r36927 = r36926 * r36916;
        double r36928 = r36927 * r36927;
        double r36929 = r36925 - r36928;
        double r36930 = sqrt(r36929);
        double r36931 = r36924 + r36927;
        double r36932 = sqrt(r36931);
        double r36933 = r36930 / r36932;
        double r36934 = r36923 - r36933;
        double r36935 = r36921 / r36934;
        return r36935;
}

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 28.4

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

  3. Applied flip-+28.4

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

  4. Simplified0.5

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.5

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

  10. Applied associate-/r*0.4

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

  11. Simplified0.4

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

  13. Applied flip--0.4

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

  14. Applied sqrt-div0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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)))