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

Average Error: 26.3 → 21.5

Time: 18.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.0677618729601775 \cdot 10^{258}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \le -6821795244285730230000 \lor \neg \left(y \le 4.32234753106425839 \cdot 10^{-38} \lor \neg \left(y \le 2.8904966271832427 \cdot 10^{109}\right) \land y \le 1.8982219138742053 \cdot 10^{145}\right):\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot \left(y + x\right) + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\ \end{array}\]

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 ((y <= -1.0677618729601775e+258)) {
		VAR = ((double) (a - ((double) (y * ((double) (b / ((double) (x + ((double) (y + t))))))))));
	} else {
		double VAR_1;
		if (((y <= -6.82179524428573e+21) || !((y <= 4.3223475310642584e-38) || (!(y <= 2.8904966271832427e+109) && (y <= 1.8982219138742053e+145))))) {
			VAR_1 = ((double) (z - ((double) (y * ((double) (b / ((double) (x + ((double) (y + t))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (z * ((double) (y + x)))) + ((double) (a * ((double) (y + t)))))) - ((double) (y * b)))) * ((double) (1.0 / ((double) (x + ((double) (y + t))))))));
		}
		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	11.0
Herbie	21.5

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

Derivation

1. Split input into 3 regimes

2. if y < -1.0677618729601775e258

   1. Initial program 54.0

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

   3. Applied div-sub 54.0

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

   4. Simplified 54.0

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

   5. Simplified 45.4

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

   6. Taylor expanded around 0 22.1

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

   if -1.0677618729601775e258 < y < -6821795244285730230000 or 4.32234753106425839e-38 < y < 2.8904966271832427e109 or 1.8982219138742053e145 < y

   1. Initial program 35.4

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

   3. Applied div-sub 35.4

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

   4. Simplified 35.4

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

   5. Simplified 29.8

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

   6. Taylor expanded around inf 27.6

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

   if -6821795244285730230000 < y < 4.32234753106425839e-38 or 2.8904966271832427e109 < y < 1.8982219138742053e145

   1. Initial program 16.0

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

   3. Applied div-inv 16.1

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

   4. Simplified 16.1

      \[\leadsto \left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \color{blue}{\frac{1}{x + \left(y + t\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 21.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.0677618729601775 \cdot 10^{258}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \le -6821795244285730230000 \lor \neg \left(y \le 4.32234753106425839 \cdot 10^{-38} \lor \neg \left(y \le 2.8904966271832427 \cdot 10^{109}\right) \land y \le 1.8982219138742053 \cdot 10^{145}\right):\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot \left(y + x\right) + a \cdot \left(y + t\right)\right) - y \cdot b\right) \cdot \frac{1}{x + \left(y + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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) (/ 1.0 (/ (+ (+ 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)))