jeff quadratic root 1

Average Error: 19.5 → 7.7

Time: 5.6s

Precision: 64

Math

\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.2226693456193597 \cdot 10^{93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right) - 2 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \le 7.0423614716856544 \cdot 10^{82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \end{array}\]

C

double code(double a, double b, double c) {
	double VAR;
	if ((b >= 0.0)) {
		VAR = ((-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
	} else {
		VAR = ((2.0 * c) / (-b + sqrt(((b * b) - ((4.0 * a) * c)))));
	}
	return VAR;
}

↓

double code(double a, double b, double c) {
	double VAR;
	if ((b <= -1.2226693456193597e+93)) {
		double VAR_1;
		if ((b >= 0.0)) {
			VAR_1 = ((-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
		} else {
			VAR_1 = ((2.0 * c) / ((2.0 * ((cbrt((a / (cbrt(b) * cbrt(b)))) * cbrt((a / (cbrt(b) * cbrt(b))))) * (cbrt((a / (cbrt(b) * cbrt(b)))) * (c / cbrt(b))))) - (2.0 * b)));
		}
		VAR = VAR_1;
	} else {
		double VAR_2;
		if ((b <= 7.042361471685654e+82)) {
			double VAR_3;
			if ((b >= 0.0)) {
				VAR_3 = ((-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
			} else {
				VAR_3 = ((2.0 * c) / ((cbrt((-b + sqrt(((b * b) - ((4.0 * a) * c))))) * cbrt((-b + sqrt(((b * b) - ((4.0 * a) * c)))))) * cbrt((-b + sqrt(((b * b) - ((4.0 * a) * c)))))));
			}
			VAR_2 = VAR_3;
		} else {
			double VAR_4;
			if ((b >= 0.0)) {
				VAR_4 = ((-b - (b - (2.0 * ((a * c) / b)))) / (2.0 * a));
			} else {
				VAR_4 = ((2.0 * c) / ((2.0 * ((a * c) / b)) - (2.0 * b)));
			}
			VAR_2 = VAR_4;
		}
		VAR = VAR_2;
	}
	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 < -1.2226693456193597e+93

   1. Initial program 29.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Taylor expanded around -inf 6.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 6.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}} - 2 \cdot b}\\ \end{array}\]

   5. Applied times-frac 2.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \color{blue}{c}}{2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - 2 \cdot b}\\ \end{array}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 2.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \frac{c}{\sqrt[3]{b}}\right) - 2 \cdot b}\\ \end{array}\]

   8. Applied associate-*l* 2.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \color{blue}{c}}{2 \cdot \left(\left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right) - 2 \cdot b}\\ \end{array}\]

   if -1.2226693456193597e+93 < b < 7.042361471685654e+82

   1. Initial program 8.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 9.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

   if 7.042361471685654e+82 < b

   1. Initial program 41.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Taylor expanded around -inf 41.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

   3. Taylor expanded around inf 9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2226693456193597 \cdot 10^{93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \left(\sqrt[3]{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right) - 2 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \le 7.0423614716856544 \cdot 10^{82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(\sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))