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

Average Error: 26.9 → 7.4

Time: 20.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2.3144902186190993 \cdot 10^{+297}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot a + \left(\left(x + y\right) \cdot z - y \cdot b\right)}{y + \left(x + t\right)}\\ \end{array}\]

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 (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)))
           2.3144902186190993e+297)))
   (- (+ z a) b)
   (/ (+ (* (+ y t) a) (- (* (+ x y) z) (* y b))) (+ y (+ x 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))) <= 2.3144902186190993e+297)) {
		tmp = (z + a) - b;
	} else {
		tmp = (((y + t) * a) + (((x + y) * z) - (y * b))) / (y + (x + 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	26.9
Target	11.3
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 2.31449021861909935e297 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 63.8

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

   3. Applied clear-num_binary64 63.8

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

   4. Simplified 63.8

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

   5. Taylor expanded around inf 17.3

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

   6. Simplified 17.3

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.31449021861909935e297

   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. Using strategy rm

   3. Applied *-un-lft-identity_binary64 0.3

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

   4. Applied *-un-lft-identity_binary64 0.3

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

   5. Applied times-frac_binary64 0.3

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

   6. Simplified 0.3

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

   7. Simplified 0.3

      \[\leadsto 1 \cdot \color{blue}{\frac{a \cdot \left(y + t\right) + \left(z \cdot \left(y + x\right) - y \cdot b\right)}{y + \left(t + x\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2.3144902186190993 \cdot 10^{+297}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + t\right) \cdot a + \left(\left(x + y\right) \cdot z - y \cdot b\right)}{y + \left(x + t\right)}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))