Quadratic roots, medium range

Average Error: 43.7 → 0.4

Time: 5.9s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r32006 = b;
        double r32007 = -r32006;
        double r32008 = r32006 * r32006;
        double r32009 = 4.0;
        double r32010 = a;
        double r32011 = r32009 * r32010;
        double r32012 = c;
        double r32013 = r32011 * r32012;
        double r32014 = r32008 - r32013;
        double r32015 = sqrt(r32014);
        double r32016 = r32007 + r32015;
        double r32017 = 2.0;
        double r32018 = r32017 * r32010;
        double r32019 = r32016 / r32018;
        return r32019;
}

↓

double f(double a, double b, double c) {
        double r32020 = 1.0;
        double r32021 = 2.0;
        double r32022 = r32020 / r32021;
        double r32023 = 4.0;
        double r32024 = a;
        double r32025 = c;
        double r32026 = r32024 * r32025;
        double r32027 = r32023 * r32026;
        double r32028 = b;
        double r32029 = r32028 * r32028;
        double r32030 = r32023 * r32024;
        double r32031 = r32030 * r32025;
        double r32032 = r32029 - r32031;
        double r32033 = sqrt(r32032);
        double r32034 = r32028 + r32033;
        double r32035 = r32024 * r32034;
        double r32036 = -r32035;
        double r32037 = r32027 / r32036;
        double r32038 = r32022 * r32037;
        return r32038;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

3. Applied flip-+ 43.6

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

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 times-frac 0.4

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

10. Simplified 0.4

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

11. Simplified 0.4

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

13. Applied sub-neg 0.4

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

14. Applied distribute-lft-in 0.4

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

15. Simplified 0.4

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

17. Applied distribute-rgt-neg-out 0.4

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

18. Applied mul-1-neg 0.4

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

19. Applied distribute-neg-out 0.4

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

20. Simplified 0.4

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

21. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019344 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))