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

Average Error: 26.9 → 8.4

Time: 10.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\ t_3 := \left(z + a\right) - b\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 6.302495604699535 \cdot 10^{+274}:\\ \;\;\;\;\left(\frac{y \cdot z}{t_1} + \left(\frac{a}{\frac{t_1}{y}} + \left(\frac{x \cdot z}{t_1} + \frac{t \cdot a}{t_1}\right)\right)\right) - b \cdot \frac{y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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)))
        (t_2 (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) t_1))
        (t_3 (- (+ z a) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 6.302495604699535e+274)
       (-
        (+
         (/ (* y z) t_1)
         (+ (/ a (/ t_1 y)) (+ (/ (* x z) t_1) (/ (* t a) t_1))))
        (* b (/ y t_1)))
       t_3))))

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 t_3 = (z + a) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 6.302495604699535e+274) {
		tmp = (((y * z) / t_1) + ((a / (t_1 / y)) + (((x * z) / t_1) + ((t * a) / t_1)))) - (b * (y / t_1));
	} else {
		tmp = t_3;
	}
	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	8.4

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

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 6.30249560469953504e274 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 63.1

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

   2. Taylor expanded in y around inf 18.4

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 6.30249560469953504e274

   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 in z around 0 0.3

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

   3. Applied associate-/l*_binary64 2.2

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

   4. Applied associate-/l*_binary64 2.3

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

   5. Applied associate-/r/_binary64 1.1

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 6.302495604699535 \cdot 10^{+274}:\\ \;\;\;\;\left(\frac{y \cdot z}{y + \left(x + t\right)} + \left(\frac{a}{\frac{y + \left(x + t\right)}{y}} + \left(\frac{x \cdot z}{y + \left(x + t\right)} + \frac{t \cdot a}{y + \left(x + t\right)}\right)\right)\right) - b \cdot \frac{y}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))