Quadratic roots, wide range

Average Error: 52.9 → 0.3

Time: 6.9s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

\[\frac{\frac{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}}}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

C

double f(double a, double b, double c) {
        double r47859 = b;
        double r47860 = -r47859;
        double r47861 = r47859 * r47859;
        double r47862 = 4.0;
        double r47863 = a;
        double r47864 = r47862 * r47863;
        double r47865 = c;
        double r47866 = r47864 * r47865;
        double r47867 = r47861 - r47866;
        double r47868 = sqrt(r47867);
        double r47869 = r47860 + r47868;
        double r47870 = 2.0;
        double r47871 = r47870 * r47863;
        double r47872 = r47869 / r47871;
        return r47872;
}

↓

double f(double a, double b, double c) {
        double r47873 = 1.0;
        double r47874 = cbrt(r47873);
        double r47875 = r47874 * r47874;
        double r47876 = r47873 / r47875;
        double r47877 = 2.0;
        double r47878 = a;
        double r47879 = r47877 * r47878;
        double r47880 = 4.0;
        double r47881 = c;
        double r47882 = r47878 * r47881;
        double r47883 = r47880 * r47882;
        double r47884 = r47879 / r47883;
        double r47885 = r47876 / r47884;
        double r47886 = r47885 / r47874;
        double r47887 = b;
        double r47888 = -r47887;
        double r47889 = r47887 * r47887;
        double r47890 = r47880 * r47878;
        double r47891 = r47890 * r47881;
        double r47892 = r47889 - r47891;
        double r47893 = sqrt(r47892);
        double r47894 = r47888 - r47893;
        double r47895 = r47886 / r47894;
        return r47895;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.9

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

3. Applied flip-+ 52.9

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

4. Simplified 0.4

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

6. Applied *-un-lft-identity 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied times-frac 0.4

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

9. Applied associate-/l* 0.4

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

10. Simplified 0.5

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

12. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

13. Applied add-sqr-sqrt 0.5

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

14. Applied times-frac 0.5

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{\sqrt{1}}{\sqrt[3]{1}}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

15. Applied times-frac 0.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

16. Simplified 0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

17. Simplified 0.5

    \[\leadsto \frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}} \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]
18. Using strategy rm

19. Applied associate-*r/ 0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

20. Simplified 0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}}}{\sqrt[3]{1}}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

21. Final simplification 0.3

    \[\leadsto \frac{\frac{\frac{\frac{1}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)}}}{\sqrt[3]{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

Reproduce

herbie shell --seed 2019346 
(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)))