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

Average Error: 27.1 → 21.3

Time: 5.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.2192263232079549 \cdot 10^{244}:\\ \;\;\;\;a - \frac{b}{\frac{\left(x + t\right) + y}{y}}\\ \mathbf{elif}\;a \le -2.1529316069315559 \cdot 10^{134}:\\ \;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -2.4075371916051229 \cdot 10^{-5}:\\ \;\;\;\;a - \frac{b}{\frac{\left(x + t\right) + y}{y}}\\ \mathbf{elif}\;a \le -1.76681565249545171 \cdot 10^{-85}:\\ \;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -1.27994229446098491 \cdot 10^{-129}:\\ \;\;\;\;z - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -1.6811233784297703 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 2.92090303831041133 \cdot 10^{-291}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \le 9247920914101468920000:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;a - \frac{b}{\frac{\left(x + t\right) + y}{y}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((a <= -1.2192263232079549e+244)) {
		VAR = (a - (b / (((x + t) + y) / y)));
	} else {
		double VAR_1;
		if ((a <= -2.152931606931556e+134)) {
			VAR_1 = (((((x + y) * z) + ((t + y) * a)) * (1.0 / ((x + t) + y))) - (b * (y / ((x + t) + y))));
		} else {
			double VAR_2;
			if ((a <= -2.407537191605123e-05)) {
				VAR_2 = (a - (b / (((x + t) + y) / y)));
			} else {
				double VAR_3;
				if ((a <= -1.7668156524954517e-85)) {
					VAR_3 = (((((x + y) * z) + ((t + y) * a)) * (1.0 / ((x + t) + y))) - (b * (y / ((x + t) + y))));
				} else {
					double VAR_4;
					if ((a <= -1.2799422944609849e-129)) {
						VAR_4 = (z - (b * (y / ((x + t) + y))));
					} else {
						double VAR_5;
						if ((a <= -1.6811233784297703e-292)) {
							VAR_5 = ((1.0 / (((x + t) + y) / (((x + y) * z) + ((t + y) * a)))) - (b * (y / ((x + t) + y))));
						} else {
							double VAR_6;
							if ((a <= 2.9209030383104113e-291)) {
								VAR_6 = z;
							} else {
								double VAR_7;
								if ((a <= 9.247920914101469e+21)) {
									VAR_7 = ((1.0 / (((x + t) + y) / (((x + y) * z) + ((t + y) * a)))) - (b * (y / ((x + t) + y))));
								} else {
									VAR_7 = (a - (b / (((x + t) + y) / y)));
								}
								VAR_6 = VAR_7;
							}
							VAR_5 = VAR_6;
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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	27.1
Target	11.3
Herbie	21.3

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

Derivation

1. Split input into 5 regimes

2. if a < -1.2192263232079549e+244 or -2.152931606931556e+134 < a < -2.407537191605123e-05 or 9.247920914101469e+21 < a

   1. Initial program 34.9

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

   3. Applied div-sub 34.9

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

   5. Applied *-commutative 34.9

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

   6. Applied associate-/l* 33.2

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

   7. Taylor expanded around 0 25.4

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

   if -1.2192263232079549e+244 < a < -2.152931606931556e+134 or -2.407537191605123e-05 < a < -1.7668156524954517e-85

   1. Initial program 29.6

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

   3. Applied div-sub 29.6

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

   5. Applied *-commutative 29.6

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

   6. Applied associate-/l* 27.4

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

   8. Applied div-inv 27.4

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{b \cdot \frac{1}{\frac{\left(x + t\right) + y}{y}}}\]

   9. Simplified 27.4

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

   11. Applied div-inv 27.5

       \[\leadsto \color{blue}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y}} - b \cdot \frac{y}{\left(x + t\right) + y}\]

   if -1.7668156524954517e-85 < a < -1.2799422944609849e-129

   1. Initial program 20.5

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

   3. Applied div-sub 20.5

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

   5. Applied *-commutative 20.5

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

   6. Applied associate-/l* 14.1

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

   8. Applied div-inv 14.1

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{b \cdot \frac{1}{\frac{\left(x + t\right) + y}{y}}}\]

   9. Simplified 14.1

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

   10. Taylor expanded around inf 20.0

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

   if -1.2799422944609849e-129 < a < -1.6811233784297703e-292 or 2.9209030383104113e-291 < a < 9.247920914101469e+21

   1. Initial program 18.9

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

   3. Applied div-sub 18.9

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

   5. Applied *-commutative 18.9

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

   6. Applied associate-/l* 14.1

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

   8. Applied div-inv 14.1

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{b \cdot \frac{1}{\frac{\left(x + t\right) + y}{y}}}\]

   9. Simplified 14.0

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

   11. Applied clear-num 14.1

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}} - b \cdot \frac{y}{\left(x + t\right) + y}\]

   if -1.6811233784297703e-292 < a < 2.9209030383104113e-291

   1. Initial program 24.9

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

   2. Taylor expanded around inf 31.2

      \[\leadsto \color{blue}{z}\]
3. Recombined 5 regimes into one program.

4. Final simplification 21.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.2192263232079549 \cdot 10^{244}:\\ \;\;\;\;a - \frac{b}{\frac{\left(x + t\right) + y}{y}}\\ \mathbf{elif}\;a \le -2.1529316069315559 \cdot 10^{134}:\\ \;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -2.4075371916051229 \cdot 10^{-5}:\\ \;\;\;\;a - \frac{b}{\frac{\left(x + t\right) + y}{y}}\\ \mathbf{elif}\;a \le -1.76681565249545171 \cdot 10^{-85}:\\ \;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -1.27994229446098491 \cdot 10^{-129}:\\ \;\;\;\;z - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le -1.6811233784297703 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 2.92090303831041133 \cdot 10^{-291}:\\ \;\;\;\;z\\ \mathbf{elif}\;a \le 9247920914101468920000:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;a - \frac{b}{\frac{\left(x + t\right) + y}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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)))