Cubic critical

Average Error: 34.0 → 11.2

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -34801737241079.82:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -1.4167290353563065 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.3695280426588302 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \left(b - 1.5 \cdot \frac{a \cdot c}{b}\right)\right)}\\ \end{array}\]

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 temp;
	if ((b <= -34801737241079.82)) {
		temp = ((-b + ((1.5 * ((a * c) / b)) - b)) / (3.0 * a));
	} else {
		double temp_1;
		if ((b <= -1.4167290353563065e-64)) {
			temp_1 = (1.0 / (((3.0 * a) / (3.0 * (a * c))) * (((pow(b, 2.0) - pow(b, 2.0)) + (3.0 * (a * c))) / (sqrt(((b * b) - ((3.0 * a) * c))) - b))));
		} else {
			double temp_2;
			if ((b <= 1.3695280426588302e+154)) {
				temp_2 = (1.0 * (c / (-b - sqrt(((b * b) - ((3.0 * a) * c))))));
			} else {
				temp_2 = (1.0 / ((1.0 / (c / 1.0)) * (-b - (b - (1.5 * ((a * c) / b))))));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -34801737241079.82

   1. Initial program 34.0

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

   2. Taylor expanded around -inf 12.9

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a}\]

   if -34801737241079.82 < b < -1.4167290353563065e-64

   1. Initial program 4.5

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

   3. Applied flip-+ 40.3

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

   4. Simplified 40.4

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

   6. Applied clear-num 40.4

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

   7. Simplified 40.4

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

   9. Applied flip-- 40.5

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

   10. Simplified 14.6

       \[\leadsto \frac{1}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \frac{\color{blue}{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\]

   11. Simplified 14.6

       \[\leadsto \frac{1}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}}\]

   if -1.4167290353563065e-64 < b < 1.3695280426588302e+154

   1. Initial program 27.4

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

   3. Applied flip-+ 30.2

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

   4. Simplified 16.8

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

   6. Applied clear-num 17.0

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

   7. Simplified 16.0

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

   9. Applied clear-num 16.0

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

   10. Simplified 11.8

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

   12. Applied div-inv 11.8

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

   13. Simplified 11.4

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

   if 1.3695280426588302e+154 < b

   1. Initial program 64.0

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

   3. Applied flip-+ 64.0

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

   4. Simplified 38.1

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

   6. Applied clear-num 38.1

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

   7. Simplified 38.1

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

   9. Applied clear-num 38.1

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

   10. Simplified 38.0

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

   11. Taylor expanded around inf 7.5

       \[\leadsto \frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \color{blue}{\left(b - 1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -34801737241079.82:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -1.4167290353563065 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.3695280426588302 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \left(b - 1.5 \cdot \frac{a \cdot c}{b}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))