Cubic critical

Average Error: 34.1 → 11.5

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.57925205560055901 \cdot 10^{-75}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le 8.47915506480201008 \cdot 10^{53}:\\ \;\;\;\;1 \cdot \frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 \cdot c}{-6 \cdot b}\\ \end{array}\]

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 <= -3.579252055600559e-75)) {
		VAR = (((1.5 * ((a * c) / b)) - (2.0 * b)) / (3.0 * a));
	} else {
		double VAR_1;
		if ((b <= 8.47915506480201e+53)) {
			VAR_1 = (1.0 * (c / (a * ((-b - sqrt(((b * b) - (3.0 * (a * c))))) / a))));
		} else {
			VAR_1 = ((3.0 * c) / (-6.0 * b));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -3.579252055600559e-75

   1. Initial program 27.1

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

   2. Taylor expanded around -inf 15.5

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

   if -3.579252055600559e-75 < b < 8.47915506480201e+53

   1. Initial program 23.2

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

   3. Applied associate-*l* 23.2

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

   5. Applied flip-+ 26.5

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

   6. Simplified 18.5

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

   8. Applied *-commutative 18.5

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

   9. Applied associate-*r* 18.6

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

   10. Applied associate-/l* 17.2

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

   11. Applied associate-/l/ 14.3

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

   13. Applied associate-*l* 14.2

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

   14. Applied times-frac 14.0

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

   15. Simplified 14.0

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

   if 8.47915506480201e+53 < b

   1. Initial program 57.2

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

   3. Applied associate-*l* 57.2

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

   5. Applied flip-+ 57.2

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

   6. Simplified 28.1

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

   8. Applied *-commutative 28.1

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

   9. Applied associate-*r* 28.1

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

   10. Applied associate-/l* 30.3

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

   11. Applied associate-/l/ 29.0

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

   12. Taylor expanded around 0 3.8

       \[\leadsto \frac{3 \cdot c}{\color{blue}{-6 \cdot b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.57925205560055901 \cdot 10^{-75}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le 8.47915506480201008 \cdot 10^{53}:\\ \;\;\;\;1 \cdot \frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{3 \cdot c}{-6 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))