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

Average Error: 26.2 → 19.5

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.201158076696508984152866869783675785771 \cdot 10^{177} \lor \neg \left(a \le 2.893978670696539514448294922890376187072 \cdot 10^{101}\right):\\ \;\;\;\;a - \left(\sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}} \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\right) \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \left(\sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}} \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\right) \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r909885 = x;
        double r909886 = y;
        double r909887 = r909885 + r909886;
        double r909888 = z;
        double r909889 = r909887 * r909888;
        double r909890 = t;
        double r909891 = r909890 + r909886;
        double r909892 = a;
        double r909893 = r909891 * r909892;
        double r909894 = r909889 + r909893;
        double r909895 = b;
        double r909896 = r909886 * r909895;
        double r909897 = r909894 - r909896;
        double r909898 = r909885 + r909890;
        double r909899 = r909898 + r909886;
        double r909900 = r909897 / r909899;
        return r909900;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r909901 = a;
        double r909902 = -7.201158076696509e+177;
        bool r909903 = r909901 <= r909902;
        double r909904 = 2.8939786706965395e+101;
        bool r909905 = r909901 <= r909904;
        double r909906 = !r909905;
        bool r909907 = r909903 || r909906;
        double r909908 = y;
        double r909909 = b;
        double r909910 = x;
        double r909911 = t;
        double r909912 = r909910 + r909911;
        double r909913 = r909912 + r909908;
        double r909914 = r909909 / r909913;
        double r909915 = r909908 * r909914;
        double r909916 = cbrt(r909915);
        double r909917 = r909916 * r909916;
        double r909918 = r909917 * r909916;
        double r909919 = r909901 - r909918;
        double r909920 = r909910 + r909908;
        double r909921 = z;
        double r909922 = r909920 * r909921;
        double r909923 = r909911 + r909908;
        double r909924 = r909923 * r909901;
        double r909925 = r909922 + r909924;
        double r909926 = r909925 / r909913;
        double r909927 = r909926 - r909918;
        double r909928 = r909907 ? r909919 : r909927;
        return r909928;
}

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	26.2
Target	10.9
Herbie	19.5

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

Derivation

1. Split input into 2 regimes

2. if a < -7.201158076696509e+177 or 2.8939786706965395e+101 < a

   1. Initial program 41.9

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

   3. Applied div-sub 41.9

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

   5. Applied *-un-lft-identity 41.9

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

   6. Applied times-frac 42.3

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

   7. Simplified 42.3

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

   9. Applied add-cube-cbrt 42.3

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

   10. Taylor expanded around 0 23.4

       \[\leadsto \color{blue}{a} - \left(\sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}} \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\right) \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\]

   if -7.201158076696509e+177 < a < 2.8939786706965395e+101

   1. Initial program 20.5

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

   3. Applied div-sub 20.5

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

   5. Applied *-un-lft-identity 20.5

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

   6. Applied times-frac 17.8

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

   7. Simplified 17.8

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

   9. Applied add-cube-cbrt 18.0

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

4. Final simplification 19.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.201158076696508984152866869783675785771 \cdot 10^{177} \lor \neg \left(a \le 2.893978670696539514448294922890376187072 \cdot 10^{101}\right):\\ \;\;\;\;a - \left(\sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}} \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\right) \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \left(\sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}} \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\right) \cdot \sqrt[3]{y \cdot \frac{b}{\left(x + t\right) + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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)))