Average Error: 28.4 → 0.5
Time: 7.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1 \cdot \frac{4}{\frac{\frac{-b}{a}}{c} - \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1 \cdot \frac{4}{\frac{\frac{-b}{a}}{c} - \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r49913 = b;
        double r49914 = -r49913;
        double r49915 = r49913 * r49913;
        double r49916 = 4.0;
        double r49917 = a;
        double r49918 = r49916 * r49917;
        double r49919 = c;
        double r49920 = r49918 * r49919;
        double r49921 = r49915 - r49920;
        double r49922 = sqrt(r49921);
        double r49923 = r49914 + r49922;
        double r49924 = 2.0;
        double r49925 = r49924 * r49917;
        double r49926 = r49923 / r49925;
        return r49926;
}
↓
double f(double a, double b, double c) {
        double r49927 = 1.0;
        double r49928 = 4.0;
        double r49929 = b;
        double r49930 = -r49929;
        double r49931 = a;
        double r49932 = r49930 / r49931;
        double r49933 = c;
        double r49934 = r49932 / r49933;
        double r49935 = r49929 * r49929;
        double r49936 = r49928 * r49931;
        double r49937 = r49936 * r49933;
        double r49938 = r49935 - r49937;
        double r49939 = sqrt(r49938);
        double r49940 = r49939 / r49931;
        double r49941 = r49940 / r49933;
        double r49942 = r49934 - r49941;
        double r49943 = r49928 / r49942;
        double r49944 = r49927 * r49943;
        double r49945 = 2.0;
        double r49946 = r49945 * r49931;
        double r49947 = r49944 / r49946;
        return r49947;
}

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 *-un-lft-identity0.5

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

  7. Applied *-un-lft-identity0.5

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

  8. Applied times-frac0.5

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

  9. Simplified0.5

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

  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. Using strategy rm

  12. Applied associate-/r*0.5

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

  14. Applied div-sub0.5

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

  15. Applied div-sub0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020062 +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)))