AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Average Error: 26.3 → 24.0

Time: 6.6s

Precision: 64

Math

\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.6666664544296116 \cdot 10^{139}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \le -2.448519852454424 \cdot 10^{-201}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(\left(t + y\right) \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \sqrt[3]{a}\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -7.34397192678170121 \cdot 10^{-280}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \le 1.2083114452226689 \cdot 10^{172}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(\left(t + y\right) \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \sqrt[3]{a}\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (((double) (((double) (((double) (x + y)) * z)) + ((double) (((double) (t + y)) * a)))) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y))));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((a <= -6.666666454429612e+139)) {
		VAR = a;
	} else {
		double VAR_1;
		if ((a <= -2.448519852454424e-201)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (x + y)) * z)) + ((double) (((double) (((double) (t + y)) * ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) cbrt(a)))))) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y))));
		} else {
			double VAR_2;
			if ((a <= -7.343971926781701e-280)) {
				VAR_2 = z;
			} else {
				double VAR_3;
				if ((a <= 1.2083114452226689e+172)) {
					VAR_3 = ((double) (((double) (((double) (((double) (((double) (x + y)) * z)) + ((double) (((double) (((double) (t + y)) * ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) cbrt(a)))))) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y))));
				} else {
					VAR_3 = a;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	26.3
Target	11.5
Herbie	24.0

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -6.666666454429612e+139 or 1.2083114452226689e+172 < a

   1. Initial program 43.2

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

   2. Taylor expanded around 0 28.1

      \[\leadsto \color{blue}{a}\]

   if -6.666666454429612e+139 < a < -2.448519852454424e-201 or -7.343971926781701e-280 < a < 1.2083114452226689e+172

   1. Initial program 21.2

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 21.5

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

   4. Applied associate-*r* 21.5

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

   if -2.448519852454424e-201 < a < -7.343971926781701e-280

   1. Initial program 16.7

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

   2. Taylor expanded around inf 36.1

      \[\leadsto \color{blue}{z}\]
3. Recombined 3 regimes into one program.

4. Final simplification 24.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.6666664544296116 \cdot 10^{139}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \le -2.448519852454424 \cdot 10^{-201}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(\left(t + y\right) \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \sqrt[3]{a}\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -7.34397192678170121 \cdot 10^{-280}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \le 1.2083114452226689 \cdot 10^{172}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(\left(t + y\right) \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \sqrt[3]{a}\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))