Cubic critical, narrow range

Average Error: 28.5 → 0.3

Time: 4.9s

Precision: binary64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

\[\frac{-c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{3 \cdot \left(c \cdot a\right) + b \cdot b}}}\]

C

double code(double a, double b, double c) {
	return ((double) (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a))));
}

↓

double code(double a, double b, double c) {
	return ((double) (((double) -(c)) / ((double) (b + ((double) sqrt(((double) (((double) (((double) pow(b, 4.0)) - ((double) (3.0 * ((double) (3.0 * ((double) (a * ((double) (a * ((double) (c * c)))))))))))) / ((double) (((double) (3.0 * ((double) (c * a)))) + ((double) (b * b))))))))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.5

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

2. Simplified 28.5

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

4. Applied flip-- 28.5

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

5. Simplified 0.6

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

6. Simplified 0.6

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

8. Applied sub0-neg 0.6

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

9. Applied distribute-frac-neg 0.6

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

10. Applied distribute-frac-neg 0.6

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

11. Simplified 0.3

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

13. Applied flip-- 0.3

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

14. Simplified 0.3

    \[\leadsto -1 \cdot \frac{c}{b + \sqrt{\frac{\color{blue}{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}}{b \cdot b + 3 \cdot \left(a \cdot c\right)}}}\]

15. Simplified 0.3

    \[\leadsto -1 \cdot \frac{c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\color{blue}{3 \cdot \left(a \cdot c\right) + b \cdot b}}}}\]

16. Final simplification 0.3

    \[\leadsto \frac{-c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{3 \cdot \left(c \cdot a\right) + b \cdot b}}}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))