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

Average Error: 26.4 → 7.8

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.93403233532442538 \cdot 10^{221} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 1.135513662883736 \cdot 10^{248}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r776032 = x;
        double r776033 = y;
        double r776034 = r776032 + r776033;
        double r776035 = z;
        double r776036 = r776034 * r776035;
        double r776037 = t;
        double r776038 = r776037 + r776033;
        double r776039 = a;
        double r776040 = r776038 * r776039;
        double r776041 = r776036 + r776040;
        double r776042 = b;
        double r776043 = r776033 * r776042;
        double r776044 = r776041 - r776043;
        double r776045 = r776032 + r776037;
        double r776046 = r776045 + r776033;
        double r776047 = r776044 / r776046;
        return r776047;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r776048 = x;
        double r776049 = y;
        double r776050 = r776048 + r776049;
        double r776051 = z;
        double r776052 = r776050 * r776051;
        double r776053 = t;
        double r776054 = r776053 + r776049;
        double r776055 = a;
        double r776056 = r776054 * r776055;
        double r776057 = r776052 + r776056;
        double r776058 = b;
        double r776059 = r776049 * r776058;
        double r776060 = r776057 - r776059;
        double r776061 = r776048 + r776053;
        double r776062 = r776061 + r776049;
        double r776063 = r776060 / r776062;
        double r776064 = -6.934032335324425e+221;
        bool r776065 = r776063 <= r776064;
        double r776066 = 1.135513662883736e+248;
        bool r776067 = r776063 <= r776066;
        double r776068 = !r776067;
        bool r776069 = r776065 || r776068;
        double r776070 = r776055 + r776051;
        double r776071 = r776070 - r776058;
        double r776072 = 1.0;
        double r776073 = r776062 / r776060;
        double r776074 = r776072 / r776073;
        double r776075 = r776069 ? r776071 : r776074;
        return r776075;
}

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	26.4
Target	10.9
Herbie	7.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -6.934032335324425e+221 or 1.135513662883736e+248 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 58.4

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

   3. Applied clear-num 58.4

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

   4. Taylor expanded around 0 16.8

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

   if -6.934032335324425e+221 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 1.135513662883736e+248

   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 0.5

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

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.93403233532442538 \cdot 10^{221} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 1.135513662883736 \cdot 10^{248}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))