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

Average Error: 26.6 → 21.3

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.788086194814728 \cdot 10^{+257}:\\ \;\;\;\;z + b \cdot \frac{-1}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{elif}\;y \leq -1.0435287910912604 \cdot 10^{+224}:\\ \;\;\;\;a - b \cdot \frac{y}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq -2.3432962437726123 \cdot 10^{+57} \lor \neg \left(y \leq 1.499046436222361 \cdot 10^{-23}\right):\\ \;\;\;\;z + b \cdot \frac{-1}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.788086194814728e+257)
   (+ z (* b (/ -1.0 (/ (+ y (+ x t)) y))))
   (if (<= y -1.0435287910912604e+224)
     (- a (* b (/ y (+ x (+ y t)))))
     (if (or (<= y -2.3432962437726123e+57) (not (<= y 1.499046436222361e-23)))
       (+ z (* b (/ -1.0 (/ (+ y (+ x t)) y))))
       (/
        1.0
        (/ (+ x (+ y t)) (+ (* z (+ y x)) (+ (* t a) (* y (- a b))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((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 tmp;
	if ((y <= -1.788086194814728e+257)) {
		tmp = ((double) (z + ((double) (b * (-1.0 / (((double) (y + ((double) (x + t)))) / y))))));
	} else {
		double tmp_1;
		if ((y <= -1.0435287910912604e+224)) {
			tmp_1 = ((double) (a - ((double) (b * (y / ((double) (x + ((double) (y + t)))))))));
		} else {
			double tmp_2;
			if (((y <= -2.3432962437726123e+57) || !(y <= 1.499046436222361e-23))) {
				tmp_2 = ((double) (z + ((double) (b * (-1.0 / (((double) (y + ((double) (x + t)))) / y))))));
			} else {
				tmp_2 = (1.0 / (((double) (x + ((double) (y + t)))) / ((double) (((double) (z * ((double) (y + x)))) + ((double) (((double) (t * a)) + ((double) (y * ((double) (a - b))))))))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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	26.6
Target	11.5
Herbie	21.3

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

Derivation

1. Split input into 3 regimes

2. if y < -1.78808619481472796e257 or -1.0435287910912604e224 < y < -2.3432962437726123e57 or 1.4990464362223609e-23 < y

   1. Initial program Error: 38.0 bits

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

   3. Applied div-sub Error: 38.0 bits

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

   4. Simplified Error: 38.0 bits

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

   5. Simplified Error: 31.8 bits

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

   6. Taylor expanded around inf Error: 27.5 bits

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

   8. Applied clear-num Error: 27.6 bits

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

   9. Simplified Error: 27.6 bits

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

   if -1.78808619481472796e257 < y < -1.0435287910912604e224

   1. Initial program Error: 49.6 bits

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

   3. Applied div-sub Error: 49.6 bits

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

   4. Simplified Error: 49.6 bits

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

   5. Simplified Error: 38.0 bits

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

   6. Taylor expanded around 0 Error: 22.2 bits

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

   if -2.3432962437726123e57 < y < 1.4990464362223609e-23

   1. Initial program Error: 15.8 bits

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

   3. Applied clear-num Error: 15.9 bits

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

   4. Simplified Error: 15.9 bits

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

4. Final simplification Error: 21.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.788086194814728 \cdot 10^{+257}:\\ \;\;\;\;z + b \cdot \frac{-1}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{elif}\;y \leq -1.0435287910912604 \cdot 10^{+224}:\\ \;\;\;\;a - b \cdot \frac{y}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq -2.3432962437726123 \cdot 10^{+57} \lor \neg \left(y \leq 1.499046436222361 \cdot 10^{-23}\right):\\ \;\;\;\;z + b \cdot \frac{-1}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))