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

Average Error: 27.2 → 7.3

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + \left(t + x\right)\\ \mathbf{if}\;y \leq -4.942401878015299 \cdot 10^{+169} \lor \neg \left(y \leq 5.335083905977998 \cdot 10^{+118}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot z}{t_1} + \left(z \cdot \frac{x}{t_1} + \left(\frac{a}{\frac{t_1}{t}} + \frac{y}{\frac{t_1}{a}}\right)\right)\right) - \frac{y \cdot b}{t_1}\\ \end{array} \]

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
 (let* ((t_1 (+ y (+ t x))))
   (if (or (<= y -4.942401878015299e+169) (not (<= y 5.335083905977998e+118)))
     (- (+ a z) b)
     (-
      (+
       (/ (* y z) t_1)
       (+ (* z (/ x t_1)) (+ (/ a (/ t_1 t)) (/ y (/ t_1 a)))))
      (/ (* y b) t_1)))))

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 t_1 = y + (t + x);
	double tmp;
	if ((y <= -4.942401878015299e+169) || !(y <= 5.335083905977998e+118)) {
		tmp = (a + z) - b;
	} else {
		tmp = (((y * z) / t_1) + ((z * (x / t_1)) + ((a / (t_1 / t)) + (y / (t_1 / a))))) - ((y * b) / t_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	27.2
Target	11.0
Herbie	7.3

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

Derivation

1. Split input into 2 regimes

2. if y < -4.942401878015299e169 or 5.33508390597799842e118 < y

   1. Initial program 47.9

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

   2. Simplified 47.9

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

   3. Taylor expanded in y around inf 10.7

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   if -4.942401878015299e169 < y < 5.33508390597799842e118

   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. Simplified 19.4

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

   3. Taylor expanded in a around 0 19.4

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

   4. Applied *-un-lft-identity_binary64 19.4

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

   5. Applied times-frac_binary64 13.3

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

   6. Simplified 13.3

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

   7. Simplified 13.3

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

   8. Applied associate-/l*_binary64 6.7

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

   9. Simplified 6.7

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

   10. Applied associate-/l*_binary64 6.0

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

   11. Simplified 6.0

       \[\leadsto \left(\frac{y \cdot z}{y + \left(t + x\right)} + \left(z \cdot \frac{x}{y + \left(x + t\right)} + \left(\frac{a}{\frac{y + \left(x + t\right)}{t}} + \frac{y}{\color{blue}{\frac{y + \left(x + t\right)}{a}}}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.942401878015299 \cdot 10^{+169} \lor \neg \left(y \leq 5.335083905977998 \cdot 10^{+118}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot z}{y + \left(t + x\right)} + \left(z \cdot \frac{x}{y + \left(t + x\right)} + \left(\frac{a}{\frac{y + \left(t + x\right)}{t}} + \frac{y}{\frac{y + \left(t + x\right)}{a}}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))