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

Average Error: 27.0 → 7.8

Time: 9.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -2.502165254925273188924039774576570078499 \cdot 10^{287} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 9.56574528958459179758686821776355245136 \cdot 10^{256}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r519442 = x;
        double r519443 = y;
        double r519444 = r519442 + r519443;
        double r519445 = z;
        double r519446 = r519444 * r519445;
        double r519447 = t;
        double r519448 = r519447 + r519443;
        double r519449 = a;
        double r519450 = r519448 * r519449;
        double r519451 = r519446 + r519450;
        double r519452 = b;
        double r519453 = r519443 * r519452;
        double r519454 = r519451 - r519453;
        double r519455 = r519442 + r519447;
        double r519456 = r519455 + r519443;
        double r519457 = r519454 / r519456;
        return r519457;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r519458 = x;
        double r519459 = y;
        double r519460 = r519458 + r519459;
        double r519461 = z;
        double r519462 = r519460 * r519461;
        double r519463 = t;
        double r519464 = r519463 + r519459;
        double r519465 = a;
        double r519466 = r519464 * r519465;
        double r519467 = r519462 + r519466;
        double r519468 = b;
        double r519469 = r519459 * r519468;
        double r519470 = r519467 - r519469;
        double r519471 = r519458 + r519463;
        double r519472 = r519471 + r519459;
        double r519473 = r519470 / r519472;
        double r519474 = -2.5021652549252732e+287;
        bool r519475 = r519473 <= r519474;
        double r519476 = 9.565745289584592e+256;
        bool r519477 = r519473 <= r519476;
        double r519478 = !r519477;
        bool r519479 = r519475 || r519478;
        double r519480 = r519465 + r519461;
        double r519481 = r519480 - r519468;
        double r519482 = 1.0;
        double r519483 = r519472 / r519470;
        double r519484 = r519482 / r519483;
        double r519485 = r519479 ? r519481 : r519484;
        return r519485;
}

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	27.0
Target	11.3
Herbie	7.8

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -2.5021652549252732e+287 or 9.565745289584592e+256 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 61.6

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

   3. Applied clear-num 61.6

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

   4. Taylor expanded around 0 17.3

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

   if -2.5021652549252732e+287 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 9.565745289584592e+256

   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. Using strategy rm

   3. Applied clear-num 0.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -2.502165254925273188924039774576570078499 \cdot 10^{287} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 9.56574528958459179758686821776355245136 \cdot 10^{256}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)) -3.5813117084150564e153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e82) (/ 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)))