Cubic critical, wide range

Average Error: 52.3 → 0.1

Time: 5.5s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r88064 = b;
        double r88065 = -r88064;
        double r88066 = r88064 * r88064;
        double r88067 = 3.0;
        double r88068 = a;
        double r88069 = r88067 * r88068;
        double r88070 = c;
        double r88071 = r88069 * r88070;
        double r88072 = r88066 - r88071;
        double r88073 = sqrt(r88072);
        double r88074 = r88065 + r88073;
        double r88075 = r88074 / r88069;
        return r88075;
}

↓

double f(double a, double b, double c) {
        double r88076 = c;
        double r88077 = 1.0;
        double r88078 = r88076 / r88077;
        double r88079 = b;
        double r88080 = -r88079;
        double r88081 = r88079 * r88079;
        double r88082 = 3.0;
        double r88083 = a;
        double r88084 = r88082 * r88083;
        double r88085 = r88084 * r88076;
        double r88086 = r88081 - r88085;
        double r88087 = sqrt(r88086);
        double r88088 = r88080 - r88087;
        double r88089 = r88078 / r88088;
        return r88089;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

3. Applied flip-+ 52.3

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

4. Simplified 0.5

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

6. Applied *-un-lft-identity 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied times-frac 0.5

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

9. Applied times-frac 0.6

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

10. Simplified 0.6

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

11. Simplified 0.6

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

13. Applied associate-*r/ 0.5

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

14. Simplified 0.2

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

16. Applied *-un-lft-identity 0.2

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

17. Applied associate-/r* 0.2

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

18. Simplified 0.1

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

19. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020060 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))