Cubic critical, wide range

Average Error: 52.7 → 0.1

Time: 4.8s

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

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

↓

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

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) (c / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * a))))))))))));
}

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

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

5. Simplified 0.5

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

7. Applied clear-num 0.6

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

8. Simplified 0.3

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

10. Applied div-inv 0.4

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

11. Applied add-sqr-sqrt 0.4

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

12. Applied times-frac 0.4

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

13. Simplified 0.4

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

14. Simplified 0.3

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

16. Applied associate-*l/ 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))