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

Average Error: 27.1 → 7.9

Time: 12.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} = -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 2.28361267216235624 \cdot 10^{241}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r1041547 = x;
        double r1041548 = y;
        double r1041549 = r1041547 + r1041548;
        double r1041550 = z;
        double r1041551 = r1041549 * r1041550;
        double r1041552 = t;
        double r1041553 = r1041552 + r1041548;
        double r1041554 = a;
        double r1041555 = r1041553 * r1041554;
        double r1041556 = r1041551 + r1041555;
        double r1041557 = b;
        double r1041558 = r1041548 * r1041557;
        double r1041559 = r1041556 - r1041558;
        double r1041560 = r1041547 + r1041552;
        double r1041561 = r1041560 + r1041548;
        double r1041562 = r1041559 / r1041561;
        return r1041562;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r1041563 = x;
        double r1041564 = y;
        double r1041565 = r1041563 + r1041564;
        double r1041566 = z;
        double r1041567 = r1041565 * r1041566;
        double r1041568 = t;
        double r1041569 = r1041568 + r1041564;
        double r1041570 = a;
        double r1041571 = r1041569 * r1041570;
        double r1041572 = r1041567 + r1041571;
        double r1041573 = b;
        double r1041574 = r1041564 * r1041573;
        double r1041575 = r1041572 - r1041574;
        double r1041576 = r1041563 + r1041568;
        double r1041577 = r1041576 + r1041564;
        double r1041578 = r1041575 / r1041577;
        double r1041579 = -inf.0;
        bool r1041580 = r1041578 <= r1041579;
        double r1041581 = 2.2836126721623562e+241;
        bool r1041582 = r1041578 <= r1041581;
        double r1041583 = !r1041582;
        bool r1041584 = r1041580 || r1041583;
        double r1041585 = r1041570 + r1041566;
        double r1041586 = r1041585 - r1041573;
        double r1041587 = r1041584 ? r1041586 : r1041578;
        return r1041587;
}

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	27.1
Target	11.5
Herbie	7.9

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

Derivation

1. Split input into 2 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -inf.0 or 2.2836126721623562e+241 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 61.7

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

   3. Applied clear-num 61.7

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

   4. Simplified 61.7

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

   5. Taylor expanded around 0 17.8

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

   if -inf.0 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 2.2836126721623562e+241

   1. Initial program 0.3

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} = -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 2.28361267216235624 \cdot 10^{241}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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 (/ (+ (+ 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)))