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

Average Error: 27.1 → 7.9

Time: 15.3s

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 r959263 = x;
        double r959264 = y;
        double r959265 = r959263 + r959264;
        double r959266 = z;
        double r959267 = r959265 * r959266;
        double r959268 = t;
        double r959269 = r959268 + r959264;
        double r959270 = a;
        double r959271 = r959269 * r959270;
        double r959272 = r959267 + r959271;
        double r959273 = b;
        double r959274 = r959264 * r959273;
        double r959275 = r959272 - r959274;
        double r959276 = r959263 + r959268;
        double r959277 = r959276 + r959264;
        double r959278 = r959275 / r959277;
        return r959278;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r959279 = x;
        double r959280 = y;
        double r959281 = r959279 + r959280;
        double r959282 = z;
        double r959283 = r959281 * r959282;
        double r959284 = t;
        double r959285 = r959284 + r959280;
        double r959286 = a;
        double r959287 = r959285 * r959286;
        double r959288 = r959283 + r959287;
        double r959289 = b;
        double r959290 = r959280 * r959289;
        double r959291 = r959288 - r959290;
        double r959292 = r959279 + r959284;
        double r959293 = r959292 + r959280;
        double r959294 = r959291 / r959293;
        double r959295 = -inf.0;
        bool r959296 = r959294 <= r959295;
        double r959297 = 2.2836126721623562e+241;
        bool r959298 = r959294 <= r959297;
        double r959299 = !r959298;
        bool r959300 = r959296 || r959299;
        double r959301 = r959286 + r959282;
        double r959302 = r959301 - r959289;
        double r959303 = r959300 ? r959302 : r959294;
        return r959303;
}

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)))