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

Average Error: 26.6 → 7.6

Time: 19.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -9.886517996672513 \cdot 10^{+181}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq -3.539568470212589 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot \frac{y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \left(a \cdot \frac{t}{y + \left(t + x\right)} + \frac{y \cdot z}{y + \left(t + x\right)}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \mathbf{elif}\;y \leq -1.7214341601461616 \cdot 10^{+96} \lor \neg \left(y \leq 1.7733936525202268 \cdot 10^{+46}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{y + \left(t + x\right)} + \left(\left(a \cdot \frac{t}{y + \left(t + x\right)} + \frac{y \cdot z}{y + \left(t + x\right)}\right) + z \cdot \frac{x}{y + \left(t + x\right)}\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \end{array}\]

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 -9.886517996672513e+181)
   (- (+ a z) b)
   (if (<= y -3.539568470212589e+134)
     (-
      (+
       (* a (/ y (+ y (+ t x))))
       (+
        (/ (* z x) (+ y (+ t x)))
        (+ (* a (/ t (+ y (+ t x)))) (/ (* y z) (+ y (+ t x))))))
      (/ (* y b) (+ y (+ t x))))
     (if (or (<= y -1.7214341601461616e+96)
             (not (<= y 1.7733936525202268e+46)))
       (- (+ a z) b)
       (-
        (+
         (/ (* y a) (+ y (+ t x)))
         (+
          (+ (* a (/ t (+ y (+ t x)))) (/ (* y z) (+ y (+ t x))))
          (* z (/ x (+ y (+ t x))))))
        (/ (* y b) (+ y (+ t x))))))))

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 <= -9.886517996672513e+181) {
		tmp = (a + z) - b;
	} else if (y <= -3.539568470212589e+134) {
		tmp = ((a * (y / (y + (t + x)))) + (((z * x) / (y + (t + x))) + ((a * (t / (y + (t + x)))) + ((y * z) / (y + (t + x)))))) - ((y * b) / (y + (t + x)));
	} else if ((y <= -1.7214341601461616e+96) || !(y <= 1.7733936525202268e+46)) {
		tmp = (a + z) - b;
	} else {
		tmp = (((y * a) / (y + (t + x))) + (((a * (t / (y + (t + x)))) + ((y * z) / (y + (t + x)))) + (z * (x / (y + (t + x)))))) - ((y * b) / (y + (t + x)));
	}
	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	26.6
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 3 regimes

2. if y < -9.88651799667251276e181 or -3.539568470212589e134 < y < -1.72143416014616156e96 or 1.77339365252022683e46 < y

   1. Initial program 43.2

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

   2. Taylor expanded around inf 14.7

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

   if -9.88651799667251276e181 < y < -3.539568470212589e134

   1. Initial program 39.6

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

   2. Taylor expanded around 0 39.6

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

   3. Simplified 39.6

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

   5. Applied *-un-lft-identity_binary64_23266 39.6

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

   6. Applied times-frac_binary64_23272 38.9

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

   7. Simplified 38.9

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

   9. Applied *-un-lft-identity_binary64_23266 38.9

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

   10. Applied times-frac_binary64_23272 30.5

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

   if -1.72143416014616156e96 < y < 1.77339365252022683e46

   1. Initial program 16.5

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

   2. Taylor expanded around 0 16.5

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

   3. Simplified 16.5

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

   5. Applied *-un-lft-identity_binary64_23266 16.5

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

   6. Applied times-frac_binary64_23272 9.7

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

   7. Simplified 9.7

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

   9. Applied *-un-lft-identity_binary64_23266 9.7

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

   10. Applied times-frac_binary64_23272 2.2

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

   11. Simplified 2.2

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.886517996672513 \cdot 10^{+181}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq -3.539568470212589 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot \frac{y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \left(a \cdot \frac{t}{y + \left(t + x\right)} + \frac{y \cdot z}{y + \left(t + x\right)}\right)\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \mathbf{elif}\;y \leq -1.7214341601461616 \cdot 10^{+96} \lor \neg \left(y \leq 1.7733936525202268 \cdot 10^{+46}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{y + \left(t + x\right)} + \left(\left(a \cdot \frac{t}{y + \left(t + x\right)} + \frac{y \cdot z}{y + \left(t + x\right)}\right) + z \cdot \frac{x}{y + \left(t + x\right)}\right)\right) - \frac{y \cdot b}{y + \left(t + x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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)))