Cubic critical

Average Error: 34.3 → 7.4

Time: 6.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -8.0615263523114905 \cdot 10^{142}:\\ \;\;\;\;\frac{1}{\frac{3}{1.5 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}\\ \mathbf{elif}\;b \le -3.7607628574451261 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 1.44430318372460742:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \left(c \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

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

↓

double code(double a, double b, double c) {
	double VAR;
	if ((b <= -8.06152635231149e+142)) {
		VAR = (1.0 / (3.0 / ((1.5 * (c / b)) - (2.0 * (b / a)))));
	} else {
		double VAR_1;
		if ((b <= -3.760762857445126e-174)) {
			VAR_1 = (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / 3.0) / a);
		} else {
			double VAR_2;
			if ((b <= 1.4443031837246074)) {
				VAR_2 = (1.0 / (3.0 / ((1.0 / (-b - sqrt(((b * b) - ((3.0 * a) * c))))) * (c * 3.0))));
			} else {
				VAR_2 = (-0.5 * (c / b));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -8.06152635231149e+142

   1. Initial program 59.5

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

   3. Applied flip-+ 63.9

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

   4. Simplified 62.7

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

   6. Applied *-un-lft-identity 62.7

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

   7. Applied *-un-lft-identity 62.7

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

   8. Applied times-frac 62.7

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

   9. Simplified 62.7

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

   10. Simplified 62.7

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

   12. Applied clear-num 62.7

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

   13. Simplified 62.7

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

   14. Taylor expanded around -inf 3.3

       \[\leadsto \frac{1}{\frac{3}{\color{blue}{1.5 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}}\]

   if -8.06152635231149e+142 < b < -3.760762857445126e-174

   1. Initial program 6.3

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

   3. Applied associate-/r* 6.3

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

   if -3.760762857445126e-174 < b < 1.4443031837246074

   1. Initial program 23.1

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

   3. Applied flip-+ 23.4

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

   4. Simplified 17.0

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

   6. Applied *-un-lft-identity 17.0

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

   7. Applied *-un-lft-identity 17.0

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

   8. Applied times-frac 17.0

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

   9. Simplified 17.0

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

   10. Simplified 17.0

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

   12. Applied clear-num 17.1

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

   13. Simplified 17.0

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

   15. Applied *-un-lft-identity 17.0

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

   16. Applied div-inv 17.6

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

   17. Applied *-un-lft-identity 17.6

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

   18. Applied times-frac 17.6

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

   19. Applied times-frac 17.6

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

   20. Simplified 17.6

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

   21. Simplified 11.9

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

   if 1.4443031837246074 < b

   1. Initial program 56.0

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

   2. Taylor expanded around inf 5.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.0615263523114905 \cdot 10^{142}:\\ \;\;\;\;\frac{1}{\frac{3}{1.5 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}\\ \mathbf{elif}\;b \le -3.7607628574451261 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 1.44430318372460742:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \left(c \cdot 3\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))