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

Average Error: 26.9 → 14.5

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.334876060238265610113965261028670807354 \cdot 10^{125} \lor \neg \left(y \le 4.716438578734007639407707901307795091714 \cdot 10^{80}\right):\\ \;\;\;\;\left(z + a\right) - \frac{1}{\frac{\frac{\left(y + t\right) + x}{y}}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(y + t\right) + x}{a \cdot \left(y + t\right) + z \cdot \left(y + x\right)}} - \frac{y}{\frac{t + \left(y + x\right)}{b}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r596588 = x;
        double r596589 = y;
        double r596590 = r596588 + r596589;
        double r596591 = z;
        double r596592 = r596590 * r596591;
        double r596593 = t;
        double r596594 = r596593 + r596589;
        double r596595 = a;
        double r596596 = r596594 * r596595;
        double r596597 = r596592 + r596596;
        double r596598 = b;
        double r596599 = r596589 * r596598;
        double r596600 = r596597 - r596599;
        double r596601 = r596588 + r596593;
        double r596602 = r596601 + r596589;
        double r596603 = r596600 / r596602;
        return r596603;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r596604 = y;
        double r596605 = -1.3348760602382656e+125;
        bool r596606 = r596604 <= r596605;
        double r596607 = 4.7164385787340076e+80;
        bool r596608 = r596604 <= r596607;
        double r596609 = !r596608;
        bool r596610 = r596606 || r596609;
        double r596611 = z;
        double r596612 = a;
        double r596613 = r596611 + r596612;
        double r596614 = 1.0;
        double r596615 = t;
        double r596616 = r596604 + r596615;
        double r596617 = x;
        double r596618 = r596616 + r596617;
        double r596619 = r596618 / r596604;
        double r596620 = b;
        double r596621 = r596619 / r596620;
        double r596622 = r596614 / r596621;
        double r596623 = r596613 - r596622;
        double r596624 = r596612 * r596616;
        double r596625 = r596604 + r596617;
        double r596626 = r596611 * r596625;
        double r596627 = r596624 + r596626;
        double r596628 = r596618 / r596627;
        double r596629 = r596614 / r596628;
        double r596630 = r596615 + r596625;
        double r596631 = r596630 / r596620;
        double r596632 = r596604 / r596631;
        double r596633 = r596629 - r596632;
        double r596634 = r596610 ? r596623 : r596633;
        return r596634;
}

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	11.3
Herbie	14.5

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

Derivation

1. Split input into 2 regimes

2. if y < -1.3348760602382656e+125 or 4.7164385787340076e+80 < y

   1. Initial program 45.6

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

   2. Simplified 45.6

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

   4. Applied div-sub 45.6

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

   5. Simplified 45.6

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

   6. Simplified 38.7

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

   8. Applied clear-num 38.7

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

   9. Simplified 37.7

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

   10. Taylor expanded around inf 7.9

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

   11. Simplified 7.9

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

   if -1.3348760602382656e+125 < y < 4.7164385787340076e+80

   1. Initial program 17.6

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

   2. Simplified 17.6

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

   4. Applied div-sub 17.6

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

   5. Simplified 17.6

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

   6. Simplified 17.7

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

   8. Applied clear-num 17.8

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

   9. Simplified 17.8

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

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.334876060238265610113965261028670807354 \cdot 10^{125} \lor \neg \left(y \le 4.716438578734007639407707901307795091714 \cdot 10^{80}\right):\\ \;\;\;\;\left(z + a\right) - \frac{1}{\frac{\frac{\left(y + t\right) + x}{y}}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(y + t\right) + x}{a \cdot \left(y + t\right) + z \cdot \left(y + x\right)}} - \frac{y}{\frac{t + \left(y + x\right)}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"

  :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)))