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

Average Error: 26.7 → 8.1

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -2.1989661724650308 \cdot 10^{195} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 3.1253587992544404 \cdot 10^{277}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r773061 = x;
        double r773062 = y;
        double r773063 = r773061 + r773062;
        double r773064 = z;
        double r773065 = r773063 * r773064;
        double r773066 = t;
        double r773067 = r773066 + r773062;
        double r773068 = a;
        double r773069 = r773067 * r773068;
        double r773070 = r773065 + r773069;
        double r773071 = b;
        double r773072 = r773062 * r773071;
        double r773073 = r773070 - r773072;
        double r773074 = r773061 + r773066;
        double r773075 = r773074 + r773062;
        double r773076 = r773073 / r773075;
        return r773076;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r773077 = x;
        double r773078 = y;
        double r773079 = r773077 + r773078;
        double r773080 = z;
        double r773081 = r773079 * r773080;
        double r773082 = t;
        double r773083 = r773082 + r773078;
        double r773084 = a;
        double r773085 = r773083 * r773084;
        double r773086 = r773081 + r773085;
        double r773087 = b;
        double r773088 = r773078 * r773087;
        double r773089 = r773086 - r773088;
        double r773090 = r773077 + r773082;
        double r773091 = r773090 + r773078;
        double r773092 = r773089 / r773091;
        double r773093 = -2.1989661724650308e+195;
        bool r773094 = r773092 <= r773093;
        double r773095 = 3.1253587992544404e+277;
        bool r773096 = r773092 <= r773095;
        double r773097 = !r773096;
        bool r773098 = r773094 || r773097;
        double r773099 = r773084 + r773080;
        double r773100 = r773099 - r773087;
        double r773101 = 1.0;
        double r773102 = r773101 / r773091;
        double r773103 = r773089 * r773102;
        double r773104 = r773098 ? r773100 : r773103;
        return r773104;
}

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	26.7
Target	11.5
Herbie	8.1

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

Derivation

1. Split input into 2 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -2.1989661724650308e+195 or 3.1253587992544404e+277 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 58.3

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

   3. Applied clear-num 58.3

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

   4. Taylor expanded around 0 17.3

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

   if -2.1989661724650308e+195 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 3.1253587992544404e+277

   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

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -2.1989661724650308 \cdot 10^{195} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 3.1253587992544404 \cdot 10^{277}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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)))