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

Average Error: 26.4 → 7.4

Time: 16.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 4.742473172711981 \cdot 10^{+237}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{\left(x + y\right) + t} + \left(\frac{x \cdot z}{\left(x + y\right) + t} + \left(\frac{t \cdot a}{\left(x + y\right) + t} + \frac{y \cdot z}{\left(x + y\right) + t}\right)\right)\right) - \frac{y \cdot b}{\left(x + y\right) + t}\\ \end{array}\]

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 (or (<=
          (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) (+ y (+ x t)))
          (- INFINITY))
         (not
          (<=
           (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) (+ y (+ x t)))
           4.742473172711981e+237)))
   (- (+ z a) b)
   (-
    (+
     (/ (* y a) (+ (+ x y) t))
     (+
      (/ (* x z) (+ (+ x y) t))
      (+ (/ (* t a) (+ (+ x y) t)) (/ (* y z) (+ (+ x y) t)))))
    (/ (* y b) (+ (+ x y) t)))))

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 tmp;
	if (((((((x + y) * z) + ((y + t) * a)) - (y * b)) / (y + (x + t))) <= -((double) INFINITY)) || !((((((x + y) * z) + ((y + t) * a)) - (y * b)) / (y + (x + t))) <= 4.742473172711981e+237)) {
		tmp = (z + a) - b;
	} else {
		tmp = (((y * a) / ((x + y) + t)) + (((x * z) / ((x + y) + t)) + (((t * a) / ((x + y) + t)) + ((y * z) / ((x + y) + t))))) - ((y * b) / ((x + y) + t));
	}
	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	26.4
Target	11.1
Herbie	7.4

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

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)) < -inf.0 or 4.742473172711981e237 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.7

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

   2. Taylor expanded around inf 17.1

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.742473172711981e237

   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 around 0 0.3

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 4.742473172711981 \cdot 10^{+237}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{\left(x + y\right) + t} + \left(\frac{x \cdot z}{\left(x + y\right) + t} + \left(\frac{t \cdot a}{\left(x + y\right) + t} + \frac{y \cdot z}{\left(x + y\right) + t}\right)\right)\right) - \frac{y \cdot b}{\left(x + y\right) + t}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(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)))