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

Average Error: 26.8 → 23.1

Time: 8.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.51042017274667401 \cdot 10^{171}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.50708683770980189 \cdot 10^{134}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -485115.40011078131:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -6.6876587271108054 \cdot 10^{-170}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -2.860753366813772 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}, \frac{1}{\left(x + t\right) + y}, -\frac{y \cdot b}{\left(x + t\right) + y}\right)\\ \mathbf{elif}\;x \le 6.7642958276510964 \cdot 10^{-203}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 9.7109844173215245 \cdot 10^{230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

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 temp;
	if ((x <= -1.510420172746674e+171)) {
		temp = (z - (y / (((x + t) + y) / b)));
	} else {
		double temp_1;
		if ((x <= -1.5070868377098019e+134)) {
			temp_1 = (((fma((x + y), z, ((t + y) * a)) / 1.0) / ((x + t) + y)) - (y * (b / ((x + t) + y))));
		} else {
			double temp_2;
			if ((x <= -485115.4001107813)) {
				temp_2 = (z - (y / (((x + t) + y) / b)));
			} else {
				double temp_3;
				if ((x <= -6.687658727110805e-170)) {
					temp_3 = (a - (y * (b / ((x + t) + y))));
				} else {
					double temp_4;
					if ((x <= -2.8607533668137723e-271)) {
						temp_4 = fma((fma((x + y), z, ((t + y) * a)) / 1.0), (1.0 / ((x + t) + y)), -((y * b) / ((x + t) + y)));
					} else {
						double temp_5;
						if ((x <= 6.764295827651096e-203)) {
							temp_5 = (a - (y * (b / ((x + t) + y))));
						} else {
							double temp_6;
							if ((x <= 9.710984417321525e+230)) {
								temp_6 = (((fma((x + y), z, ((t + y) * a)) / 1.0) / ((x + t) + y)) - (y * (b / ((x + t) + y))));
							} else {
								temp_6 = (z - (y / (((x + t) + y) / b)));
							}
							temp_5 = temp_6;
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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	26.8
Target	10.8
Herbie	23.1

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

Derivation

1. Split input into 4 regimes

2. if x < -1.510420172746674e+171 or -1.5070868377098019e+134 < x < -485115.4001107813 or 9.710984417321525e+230 < x

   1. Initial program 33.1

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

   3. Applied div-sub 33.1

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

   4. Simplified 33.1

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied associate-/l* 30.2

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

   7. Taylor expanded around inf 23.8

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

   if -1.510420172746674e+171 < x < -1.5070868377098019e+134 or 6.764295827651096e-203 < x < 9.710984417321525e+230

   1. Initial program 25.6

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

   3. Applied div-sub 25.6

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

   4. Simplified 25.6

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 25.6

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

   7. Applied times-frac 23.4

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

   8. Simplified 23.4

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

   if -485115.4001107813 < x < -6.687658727110805e-170 or -2.8607533668137723e-271 < x < 6.764295827651096e-203

   1. Initial program 23.6

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

   3. Applied div-sub 23.6

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

   4. Simplified 23.6

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 23.6

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

   7. Applied times-frac 22.2

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

   8. Simplified 22.2

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

   9. Taylor expanded around 0 22.6

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

   if -6.687658727110805e-170 < x < -2.8607533668137723e-271

   1. Initial program 21.0

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

   3. Applied div-sub 21.0

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

   4. Simplified 21.0

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y}} - \frac{y \cdot b}{\left(x + t\right) + y}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 21.0

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

   7. Applied div-inv 21.0

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

   8. Applied times-frac 21.0

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

   9. Applied fma-neg 21.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}, \frac{\frac{1}{1}}{\left(x + t\right) + y}, -\frac{y \cdot b}{\left(x + t\right) + y}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.51042017274667401 \cdot 10^{171}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.50708683770980189 \cdot 10^{134}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -485115.40011078131:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -6.6876587271108054 \cdot 10^{-170}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -2.860753366813772 \cdot 10^{-271}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}, \frac{1}{\left(x + t\right) + y}, -\frac{y \cdot b}{\left(x + t\right) + y}\right)\\ \mathbf{elif}\;x \le 6.7642958276510964 \cdot 10^{-203}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 9.7109844173215245 \cdot 10^{230}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x + y, z, \left(t + y\right) \cdot a\right)}{1}}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))