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

Average Error: 26.9 → 7.6

Time: 17.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + \left(x + t\right)\\ \mathbf{if}\;\begin{array}{l} t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\ t_2 \leq -3.4176847541715013 \cdot 10^{+258} \lor \neg \left(t_2 \leq 8.754816949282347 \cdot 10^{+270}\right) \end{array}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z + \left(\mathsf{fma}\left(a, y + t, x \cdot z\right) - y \cdot b\right)}{t_1}\\ \end{array} \]

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 (+ x t))))
   (if (let* ((t_2 (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) t_1)))
         (or (<= t_2 -3.4176847541715013e+258)
             (not (<= t_2 8.754816949282347e+270))))
     (- (+ z a) b)
     (/ (+ (* y z) (- (fma a (+ y t) (* x z)) (* 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 + (x + t);
	double t_2 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1;
	double tmp;
	if ((t_2 <= -3.4176847541715013e+258) || !(t_2 <= 8.754816949282347e+270)) {
		tmp = (z + a) - b;
	} else {
		tmp = ((y * z) + (fma(a, (y + t), (x * z)) - (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	26.9
Target	11.3
Herbie	7.6

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -3.41768475417150134e258 or 8.75481694928234717e270 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 60.9

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

   2. Taylor expanded in y around inf 16.9

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

   if -3.41768475417150134e258 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 8.75481694928234717e270

   1. Initial program 0.3

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

   2. Taylor expanded in x around 0 0.3

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

   3. Applied associate--l+_binary64 0.3

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

   4. Simplified 0.3

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -3.4176847541715013 \cdot 10^{+258} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 8.754816949282347 \cdot 10^{+270}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot z + \left(\mathsf{fma}\left(a, y + t, x \cdot z\right) - y \cdot b\right)}{y + \left(x + t\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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)))