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

Average Error: 26.9 → 7.9

Time: 10.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + \left(x + t\right)\\ \mathbf{if}\;\begin{array}{l} t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\ t_2 \leq -1.9455832036881786 \cdot 10^{+295} \lor \neg \left(t_2 \leq 5.14014547158814 \cdot 10^{+247}\right) \end{array}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot z}{t_1} + \left(z \cdot \frac{x}{t_1} + \left(\frac{t \cdot a}{t_1} + \frac{y \cdot a}{t_1}\right)\right)\right) - \frac{y \cdot b}{t_1}\\ \end{array} \]

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 -1.9455832036881786e+295)
             (not (<= t_2 5.14014547158814e+247))))
     (- (+ z a) b)
     (-
      (+
       (/ (* y z) t_1)
       (+ (* z (/ x t_1)) (+ (/ (* t a) t_1) (/ (* y a) t_1))))
      (/ (* 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 <= -1.9455832036881786e+295) || !(t_2 <= 5.14014547158814e+247)) {
		tmp = (z + a) - b;
	} else {
		tmp = (((y * z) / t_1) + ((z * (x / t_1)) + (((t * a) / t_1) + ((y * a) / t_1)))) - ((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	11.8
Herbie	7.9

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

Derivation

1. Split input into 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)) < -1.9455832036881786e295 or 5.1401454715881399e247 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.5

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

   2. Simplified 61.5

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

   3. Taylor expanded in y around inf 17.5

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

   if -1.9455832036881786e295 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.1401454715881399e247

   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. Simplified 0.3

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

   3. Applied div-inv_binary64 0.5

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

   4. Taylor expanded in a around 0 0.3

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

   5. Simplified 0.3

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

   6. Applied *-un-lft-identity_binary64 0.3

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

   7. Applied times-frac_binary64 0.4

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

   8. Simplified 0.4

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

   9. Simplified 0.4

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -1.9455832036881786 \cdot 10^{+295} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 5.14014547158814 \cdot 10^{+247}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot z}{y + \left(x + t\right)} + \left(z \cdot \frac{x}{y + \left(x + t\right)} + \left(\frac{t \cdot a}{y + \left(x + t\right)} + \frac{y \cdot a}{y + \left(x + t\right)}\right)\right)\right) - \frac{y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))