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

Average Error: 26.1 → 17.8

Time: 28.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.1822739620428702 \cdot 10^{282}:\\ \;\;\;\;a\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 1.01559546125992643 \cdot 10^{272}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r759120 = x;
        double r759121 = y;
        double r759122 = r759120 + r759121;
        double r759123 = z;
        double r759124 = r759122 * r759123;
        double r759125 = t;
        double r759126 = r759125 + r759121;
        double r759127 = a;
        double r759128 = r759126 * r759127;
        double r759129 = r759124 + r759128;
        double r759130 = b;
        double r759131 = r759121 * r759130;
        double r759132 = r759129 - r759131;
        double r759133 = r759120 + r759125;
        double r759134 = r759133 + r759121;
        double r759135 = r759132 / r759134;
        return r759135;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r759136 = x;
        double r759137 = y;
        double r759138 = r759136 + r759137;
        double r759139 = z;
        double r759140 = r759138 * r759139;
        double r759141 = t;
        double r759142 = r759141 + r759137;
        double r759143 = a;
        double r759144 = r759142 * r759143;
        double r759145 = r759140 + r759144;
        double r759146 = b;
        double r759147 = r759137 * r759146;
        double r759148 = r759145 - r759147;
        double r759149 = r759136 + r759141;
        double r759150 = r759149 + r759137;
        double r759151 = r759148 / r759150;
        double r759152 = -6.18227396204287e+282;
        bool r759153 = r759151 <= r759152;
        double r759154 = 1.0155954612599264e+272;
        bool r759155 = r759151 <= r759154;
        double r759156 = 1.0;
        double r759157 = r759156 / r759150;
        double r759158 = r759148 * r759157;
        double r759159 = r759155 ? r759158 : r759139;
        double r759160 = r759153 ? r759143 : r759159;
        return r759160;
}

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	26.1
Target	11.4
Herbie	17.8

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

Derivation

1. Split input into 3 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -6.18227396204287e+282

   1. Initial program 62.2

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

   2. Taylor expanded around 0 41.3

      \[\leadsto \color{blue}{a}\]

   if -6.18227396204287e+282 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 1.0155954612599264e+272

   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 div-inv 0.4

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

   if 1.0155954612599264e+272 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 61.9

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

   2. Taylor expanded around inf 42.5

      \[\leadsto \color{blue}{z}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.1822739620428702 \cdot 10^{282}:\\ \;\;\;\;a\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 1.01559546125992643 \cdot 10^{272}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(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)))