Quadratic roots, wide range

Average Error: 52.6 → 0.1

Time: 4.9s

Precision: binary64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

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

C

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

2. Simplified 52.6

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

4. Applied flip-- 52.6

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

5. Simplified 0.4

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

6. Simplified 0.4

   \[\leadsto \frac{\frac{0 - 4 \cdot \left(a \cdot c\right)}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2}\]
7. Using strategy rm

8. Applied div-inv 0.4

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

9. Applied associate-/l* 0.5

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

10. Simplified 0.4

    \[\leadsto \frac{0 - 4 \cdot \left(a \cdot c\right)}{\color{blue}{a \cdot \left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot 2\right)}}\]
11. Using strategy rm

12. Applied sub0-neg 0.4

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

13. Applied distribute-frac-neg 0.4

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

14. Simplified 0.3

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

16. Applied div-inv 0.3

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))