Average Error: 43.7 → 0.5
Time: 6.5s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a \cdot c}} \cdot \frac{\frac{3}{\sqrt[3]{1}}}{3}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a \cdot c}} \cdot \frac{\frac{3}{\sqrt[3]{1}}}{3}}{a}
double f(double a, double b, double c) {
        double r119909 = b;
        double r119910 = -r119909;
        double r119911 = r119909 * r119909;
        double r119912 = 3.0;
        double r119913 = a;
        double r119914 = r119912 * r119913;
        double r119915 = c;
        double r119916 = r119914 * r119915;
        double r119917 = r119911 - r119916;
        double r119918 = sqrt(r119917);
        double r119919 = r119910 + r119918;
        double r119920 = r119919 / r119914;
        return r119920;
}
↓
double f(double a, double b, double c) {
        double r119921 = 1.0;
        double r119922 = cbrt(r119921);
        double r119923 = r119922 * r119922;
        double r119924 = r119921 / r119923;
        double r119925 = b;
        double r119926 = -r119925;
        double r119927 = r119925 * r119925;
        double r119928 = 3.0;
        double r119929 = a;
        double r119930 = r119928 * r119929;
        double r119931 = c;
        double r119932 = r119930 * r119931;
        double r119933 = r119927 - r119932;
        double r119934 = sqrt(r119933);
        double r119935 = r119926 - r119934;
        double r119936 = r119929 * r119931;
        double r119937 = r119935 / r119936;
        double r119938 = r119924 / r119937;
        double r119939 = r119928 / r119922;
        double r119940 = r119939 / r119928;
        double r119941 = r119938 * r119940;
        double r119942 = r119941 / r119929;
        return r119942;
}

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 43.7

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

  3. Applied flip-+43.7

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

  4. Simplified0.5

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

  6. Applied associate-/r*0.5

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

  7. Simplified0.5

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

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

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

  10. Applied times-frac0.5

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

  11. Applied associate-/l*0.5

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

  13. Applied associate-/r/0.5

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

  14. Applied add-cube-cbrt0.5

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

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

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

  16. Applied times-frac0.5

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

  17. Applied times-frac0.5

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

  18. Final simplification0.5

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

Reproduce

herbie shell --seed 2020062 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))