Cubic critical, wide range

Average Error: 52.7 → 0.1

Time: 6.5s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r99551 = b;
        double r99552 = -r99551;
        double r99553 = r99551 * r99551;
        double r99554 = 3.0;
        double r99555 = a;
        double r99556 = r99554 * r99555;
        double r99557 = c;
        double r99558 = r99556 * r99557;
        double r99559 = r99553 - r99558;
        double r99560 = sqrt(r99559);
        double r99561 = r99552 + r99560;
        double r99562 = r99561 / r99556;
        return r99562;
}

↓

double f(double a, double b, double c) {
        double r99563 = c;
        double r99564 = b;
        double r99565 = -r99564;
        double r99566 = r99564 * r99564;
        double r99567 = 3.0;
        double r99568 = a;
        double r99569 = r99567 * r99568;
        double r99570 = r99569 * r99563;
        double r99571 = r99566 - r99570;
        double r99572 = sqrt(r99571);
        double r99573 = r99565 - r99572;
        double r99574 = r99563 / r99573;
        return r99574;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

3. Applied flip-+ 52.7

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

4. 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 associate-*r* 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied times-frac 0.4

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

11. Applied associate-/l* 0.4

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

12. Simplified 0.4

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

14. Applied div-inv 0.4

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020034 
(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)))