Quadratic roots, wide range

Average Error: 52.7 → 0.1

Time: 4.5s

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

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

↓

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

C

double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
}

↓

double code(double a, double b, double c) {
	return ((4.0 * c) / ((1.0 * 2.0) * (-b - sqrt(((b * b) - ((4.0 * a) * c))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

3. Applied flip-+ 52.7

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

4. Simplified 0.4

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

6. Applied *-commutative 0.4

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

7. Applied associate-*r* 0.4

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

8. Applied associate-/l* 0.4

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

9. Applied associate-/l/ 0.2

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

11. Applied div-sub 0.2

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

13. Applied div-inv 0.3

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

14. Applied div-inv 0.3

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

15. Applied distribute-rgt-out-- 0.3

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

16. Applied associate-*r* 0.2

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020071 
(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)))