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

Average Error: 27.0 → 20.9

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.404135654519083 \cdot 10^{+266}:\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq -1.4762398414904788 \cdot 10^{-41}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq 5.151723195463842 \cdot 10^{+116}:\\ \;\;\;\;\frac{z \cdot \left(y + x\right) + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \left(y \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \frac{\sqrt[3]{b}}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \end{array}\]

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 (<= y -5.404135654519083e+266)
   (- z (* y (/ b (+ x (+ y t)))))
   (if (<= y -1.4762398414904788e-41)
     (- a (* y (/ b (+ x (+ y t)))))
     (if (<= y 5.151723195463842e+116)
       (-
        (/ (+ (* z (+ y x)) (* (+ y t) a)) (+ x (+ y t)))
        (* (* y (* (cbrt b) (cbrt b))) (/ (cbrt b) (+ x (+ y t)))))
       (- z (* 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 (y <= -5.404135654519083e+266) {
		tmp = z - (y * (b / (x + (y + t))));
	} else if (y <= -1.4762398414904788e-41) {
		tmp = a - (y * (b / (x + (y + t))));
	} else if (y <= 5.151723195463842e+116) {
		tmp = (((z * (y + x)) + ((y + t) * a)) / (x + (y + t))) - ((y * (cbrt(b) * cbrt(b))) * (cbrt(b) / (x + (y + t))));
	} else {
		tmp = z - (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	27.0
Target	11.6
Herbie	20.9

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

Derivation

1. Split input into 3 regimes

2. if y < -5.4041356545190832e266 or 5.15172319546384202e116 < y

   1. Initial program 48.1

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

   3. Applied div-sub_binary64 48.1

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

   4. Simplified 48.1

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

   5. Simplified 48.1

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

   7. Applied *-un-lft-identity_binary64 48.1

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

   8. Applied times-frac_binary64 41.2

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

   9. Simplified 41.2

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

   10. Taylor expanded around inf 25.4

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

   if -5.4041356545190832e266 < y < -1.47623984149047883e-41

   1. Initial program 33.2

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

   3. Applied div-sub_binary64 33.2

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

   4. Simplified 33.2

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

   5. Simplified 33.2

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

   7. Applied *-un-lft-identity_binary64 33.2

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

   8. Applied times-frac_binary64 28.0

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

   9. Simplified 28.0

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

   10. Taylor expanded around 0 27.8

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

   if -1.47623984149047883e-41 < y < 5.15172319546384202e116

   1. Initial program 16.9

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

   3. Applied div-sub_binary64 16.9

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

   4. Simplified 16.9

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

   5. Simplified 16.9

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

   7. Applied *-un-lft-identity_binary64 16.9

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

   8. Applied times-frac_binary64 18.0

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

   9. Simplified 18.0

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

   11. Applied *-un-lft-identity_binary64 18.0

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

   12. Applied add-cube-cbrt_binary64 18.2

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - y \cdot \frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{1 \cdot \left(x + \left(y + t\right)\right)}\]

   13. Applied times-frac_binary64 18.2

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - y \cdot \color{blue}{\left(\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{1} \cdot \frac{\sqrt[3]{b}}{x + \left(y + t\right)}\right)}\]

   14. Applied associate-*r*_binary64 16.2

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \color{blue}{\left(y \cdot \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{1}\right) \cdot \frac{\sqrt[3]{b}}{x + \left(y + t\right)}}\]

   15. Simplified 16.2

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

4. Final simplification 20.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.404135654519083 \cdot 10^{+266}:\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq -1.4762398414904788 \cdot 10^{-41}:\\ \;\;\;\;a - y \cdot \frac{b}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq 5.151723195463842 \cdot 10^{+116}:\\ \;\;\;\;\frac{z \cdot \left(y + x\right) + \left(y + t\right) \cdot a}{x + \left(y + t\right)} - \left(y \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \frac{\sqrt[3]{b}}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot \frac{b}{x + \left(y + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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)))