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

Average Error: 26.3 → 19.9

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.3465238493089848 \cdot 10^{92}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;a \le -1.36117654171489388 \cdot 10^{-293}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z + a \cdot \left(y + t\right)}{x + \left(y + t\right)} - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{elif}\;a \le 1.8149584907349981 \cdot 10^{-236}:\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;a \le 699.680100954575096:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z + a \cdot \left(y + t\right)}{x + \left(y + t\right)} - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{else}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \end{array}\]

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 ((a <= -2.3465238493089848e+92)) {
		VAR = ((double) (a - ((double) (y * ((double) (b / ((double) (x + ((double) (y + t))))))))));
	} else {
		double VAR_1;
		if ((a <= -1.361176541714894e-293)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (y + x)) * z)) + ((double) (a * ((double) (y + t)))))) / ((double) (x + ((double) (y + t)))))) - ((double) (((double) (y / ((double) (((double) cbrt(((double) (x + ((double) (y + t)))))) * ((double) cbrt(((double) (x + ((double) (y + t)))))))))) * ((double) (b / ((double) cbrt(((double) (x + ((double) (y + t))))))))))));
		} else {
			double VAR_2;
			if ((a <= 1.814958490734998e-236)) {
				VAR_2 = ((double) (z - ((double) (y * ((double) (b / ((double) (x + ((double) (y + t))))))))));
			} else {
				double VAR_3;
				if ((a <= 699.6801009545751)) {
					VAR_3 = ((double) (((double) (((double) (((double) (((double) (y + x)) * z)) + ((double) (a * ((double) (y + t)))))) / ((double) (x + ((double) (y + t)))))) - ((double) (((double) (y / ((double) (((double) cbrt(((double) (x + ((double) (y + t)))))) * ((double) cbrt(((double) (x + ((double) (y + t)))))))))) * ((double) (b / ((double) cbrt(((double) (x + ((double) (y + t))))))))))));
				} else {
					VAR_3 = ((double) (a - ((double) (y * ((double) (b / ((double) (x + ((double) (y + t))))))))));
				}
				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	11.4
Herbie	19.9

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

Derivation

1. Split input into 3 regimes

2. if a < -2.3465238493089848e92 or 699.680100954575096 < a

   1. Initial program 36.0

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

   3. Applied div-sub 36.0

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

   4. Simplified 36.0

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

   5. Simplified 35.6

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

   6. Taylor expanded around 0 26.0

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

   if -2.3465238493089848e92 < a < -1.36117654171489388e-293 or 1.8149584907349981e-236 < a < 699.680100954575096

   1. Initial program 19.4

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

   3. Applied div-sub 19.3

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

   4. Simplified 19.3

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

   5. Simplified 16.1

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

   7. Applied add-cube-cbrt 16.3

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

   8. Applied *-un-lft-identity 16.3

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - y \cdot \frac{\color{blue}{1 \cdot b}}{\left(\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}\right) \cdot \sqrt[3]{x + \left(y + t\right)}}\]

   9. Applied times-frac 16.3

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\right)}\]

   10. Applied associate-*r* 15.2

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}}\right) \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}}\]

   11. Simplified 15.2

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

   if -1.36117654171489388e-293 < a < 1.8149584907349981e-236

   1. Initial program 19.5

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

   3. Applied div-sub 19.5

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

   4. Simplified 19.5

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

   5. Simplified 14.6

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

   6. Taylor expanded around inf 17.4

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

4. Final simplification 19.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.3465238493089848 \cdot 10^{92}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;a \le -1.36117654171489388 \cdot 10^{-293}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z + a \cdot \left(y + t\right)}{x + \left(y + t\right)} - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{elif}\;a \le 1.8149584907349981 \cdot 10^{-236}:\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;a \le 699.680100954575096:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z + a \cdot \left(y + t\right)}{x + \left(y + t\right)} - \frac{y}{\sqrt[3]{x + \left(y + t\right)} \cdot \sqrt[3]{x + \left(y + t\right)}} \cdot \frac{b}{\sqrt[3]{x + \left(y + t\right)}}\\ \mathbf{else}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))