Cubic critical, medium range

Average Error: 43.6 → 0.3

Time: 18.8s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r67640 = b;
        double r67641 = -r67640;
        double r67642 = r67640 * r67640;
        double r67643 = 3.0;
        double r67644 = a;
        double r67645 = r67643 * r67644;
        double r67646 = c;
        double r67647 = r67645 * r67646;
        double r67648 = r67642 - r67647;
        double r67649 = sqrt(r67648);
        double r67650 = r67641 + r67649;
        double r67651 = r67650 / r67645;
        return r67651;
}

↓

double f(double a, double b, double c) {
        double r67652 = 3.0;
        double r67653 = a;
        double r67654 = r67652 * r67653;
        double r67655 = r67654 / r67652;
        double r67656 = c;
        double r67657 = b;
        double r67658 = -r67657;
        double r67659 = r67657 * r67657;
        double r67660 = r67654 * r67656;
        double r67661 = r67659 - r67660;
        double r67662 = sqrt(r67661);
        double r67663 = r67658 - r67662;
        double r67664 = r67656 / r67663;
        double r67665 = r67664 / r67653;
        double r67666 = r67655 * r67665;
        return r67666;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.6

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

3. Applied flip-+ 43.6

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

4. Simplified 0.5

   \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
5. Using strategy rm

6. Applied clear-num 0.6

   \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{0 + 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]

7. Simplified 0.5

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

9. Applied *-un-lft-identity 0.5

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

10. Applied times-frac 0.5

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

11. Applied add-cube-cbrt 0.5

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

12. Applied times-frac 0.4

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

13. Simplified 0.4

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

14. Simplified 0.4

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

16. Applied times-frac 0.5

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019323 
(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)))