Cubic critical

Average Error: 34.3 → 6.5

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.86885461377155539 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\frac{3}{1.5 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}\\ \mathbf{elif}\;b \le -7.08643729785537625 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 2.4235533085097852 \cdot 10^{67}:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{3}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

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 temp;
	if ((b <= -2.8688546137715554e+112)) {
		temp = (1.0 / (3.0 / ((1.5 * (c / b)) - (2.0 * (b / a)))));
	} else {
		double temp_1;
		if ((b <= -7.086437297855376e-264)) {
			temp_1 = (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / 3.0) / a);
		} else {
			double temp_2;
			if ((b <= 2.4235533085097852e+67)) {
				temp_2 = (1.0 / (3.0 / (3.0 / (1.0 * ((-b - sqrt(((b * b) - ((3.0 * a) * c)))) / c)))));
			} else {
				temp_2 = (-0.5 * (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 < -2.8688546137715554e+112

   1. Initial program 50.4

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

   3. Applied flip-+ 63.4

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

   4. Simplified 62.4

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

   6. Applied *-un-lft-identity 62.4

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

   7. Applied *-un-lft-identity 62.4

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

   8. Applied times-frac 62.4

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

   9. Simplified 62.4

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

   10. Simplified 62.4

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

   12. Applied clear-num 62.4

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

   13. Simplified 62.4

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

   14. Taylor expanded around -inf 4.4

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

   if -2.8688546137715554e+112 < b < -7.086437297855376e-264

   1. Initial program 8.2

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

   3. Applied associate-/r* 8.2

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

   if -7.086437297855376e-264 < b < 2.4235533085097852e+67

   1. Initial program 29.1

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

   3. Applied flip-+ 29.2

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

   4. Simplified 16.2

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

   6. Applied *-un-lft-identity 16.2

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

   7. Applied *-un-lft-identity 16.2

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

   8. Applied times-frac 16.2

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

   9. Simplified 16.2

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

   10. Simplified 16.4

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

   12. Applied clear-num 16.5

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

   13. Simplified 16.5

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

   15. Applied div-inv 16.5

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

   16. Applied associate-/l* 16.5

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

   17. Simplified 9.3

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

   if 2.4235533085097852e+67 < b

   1. Initial program 58.0

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

   2. Taylor expanded around inf 3.0

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.86885461377155539 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\frac{3}{1.5 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}\\ \mathbf{elif}\;b \le -7.08643729785537625 \cdot 10^{-264}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 2.4235533085097852 \cdot 10^{67}:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{3}{1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))