Average Error: 28.4 → 0.6
Time: 8.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\frac{a \cdot c}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a} \cdot 3\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{\frac{a \cdot c}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a} \cdot 3
double f(double a, double b, double c) {
        double r119913 = b;
        double r119914 = -r119913;
        double r119915 = r119913 * r119913;
        double r119916 = 3.0;
        double r119917 = a;
        double r119918 = r119916 * r119917;
        double r119919 = c;
        double r119920 = r119918 * r119919;
        double r119921 = r119915 - r119920;
        double r119922 = sqrt(r119921);
        double r119923 = r119914 + r119922;
        double r119924 = r119923 / r119918;
        return r119924;
}
↓
double f(double a, double b, double c) {
        double r119925 = a;
        double r119926 = c;
        double r119927 = r119925 * r119926;
        double r119928 = 3.0;
        double r119929 = r119927 / r119928;
        double r119930 = b;
        double r119931 = -r119930;
        double r119932 = r119930 * r119930;
        double r119933 = r119928 * r119925;
        double r119934 = r119933 * r119926;
        double r119935 = r119932 - r119934;
        double r119936 = sqrt(r119935);
        double r119937 = r119931 - r119936;
        double r119938 = r119929 / r119937;
        double r119939 = r119938 / r119925;
        double r119940 = r119939 * r119928;
        return r119940;
}

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

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.4

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  4. Simplified0.6

    \[\leadsto \frac{\frac{\color{blue}{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied div-inv0.6

    \[\leadsto \frac{\color{blue}{\left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  7. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}}\]

  8. Simplified0.6

    \[\leadsto \color{blue}{\frac{3 \cdot \left(a \cdot c\right)}{3}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity0.6

    \[\leadsto \frac{3 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot 3}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]

  11. Applied times-frac0.5

    \[\leadsto \color{blue}{\left(\frac{3}{1} \cdot \frac{a \cdot c}{3}\right)} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]

  12. Applied associate-*l*0.6

    \[\leadsto \color{blue}{\frac{3}{1} \cdot \left(\frac{a \cdot c}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\right)}\]
  13. Using strategy rm

  14. Applied associate-*r/0.6

    \[\leadsto \frac{3}{1} \cdot \color{blue}{\frac{\frac{a \cdot c}{3} \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}}\]

  15. Simplified0.6

    \[\leadsto \frac{3}{1} \cdot \frac{\color{blue}{\frac{\frac{a \cdot c}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{a}\]

  16. Final simplification0.6

    \[\leadsto \frac{\frac{\frac{a \cdot c}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a} \cdot 3\]

Reproduce

herbie shell --seed 2019318 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))