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

Average Error: 26.3 → 20.6

Time: 22.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.329351671379600395696054258455065074018 \cdot 10^{162}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 5.325165846082585887271445202627113987297 \cdot 10^{-212}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{b}{\sqrt[3]{\left(x + t\right) + y}}\\ \mathbf{elif}\;a \le 1.908584443905937974655168865478710611307 \cdot 10^{-45}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 1.978965818958528848241507275261864808485 \cdot 10^{164}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{else}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r716999 = x;
        double r717000 = y;
        double r717001 = r716999 + r717000;
        double r717002 = z;
        double r717003 = r717001 * r717002;
        double r717004 = t;
        double r717005 = r717004 + r717000;
        double r717006 = a;
        double r717007 = r717005 * r717006;
        double r717008 = r717003 + r717007;
        double r717009 = b;
        double r717010 = r717000 * r717009;
        double r717011 = r717008 - r717010;
        double r717012 = r716999 + r717004;
        double r717013 = r717012 + r717000;
        double r717014 = r717011 / r717013;
        return r717014;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r717015 = a;
        double r717016 = -1.3293516713796004e+162;
        bool r717017 = r717015 <= r717016;
        double r717018 = y;
        double r717019 = b;
        double r717020 = x;
        double r717021 = t;
        double r717022 = r717020 + r717021;
        double r717023 = r717022 + r717018;
        double r717024 = r717019 / r717023;
        double r717025 = r717018 * r717024;
        double r717026 = r717015 - r717025;
        double r717027 = 5.325165846082586e-212;
        bool r717028 = r717015 <= r717027;
        double r717029 = r717020 + r717018;
        double r717030 = z;
        double r717031 = r717029 * r717030;
        double r717032 = r717021 + r717018;
        double r717033 = r717032 * r717015;
        double r717034 = r717031 + r717033;
        double r717035 = r717034 / r717023;
        double r717036 = cbrt(r717023);
        double r717037 = r717036 * r717036;
        double r717038 = r717018 / r717037;
        double r717039 = r717019 / r717036;
        double r717040 = r717038 * r717039;
        double r717041 = r717035 - r717040;
        double r717042 = 1.908584443905938e-45;
        bool r717043 = r717015 <= r717042;
        double r717044 = r717030 - r717025;
        double r717045 = 1.9789658189585288e+164;
        bool r717046 = r717015 <= r717045;
        double r717047 = r717023 / r717019;
        double r717048 = r717018 / r717047;
        double r717049 = r717035 - r717048;
        double r717050 = r717046 ? r717049 : r717026;
        double r717051 = r717043 ? r717044 : r717050;
        double r717052 = r717028 ? r717041 : r717051;
        double r717053 = r717017 ? r717026 : r717052;
        return r717053;
}

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	26.3
Target	11.6
Herbie	20.6

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

Derivation

1. Split input into 4 regimes

2. if a < -1.3293516713796004e+162 or 1.9789658189585288e+164 < a

   1. Initial program 42.9

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

   3. Applied div-sub 42.9

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

   5. Applied *-un-lft-identity 42.9

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

   6. Applied times-frac 43.6

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

   7. Simplified 43.6

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

   8. Taylor expanded around 0 25.0

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

   if -1.3293516713796004e+162 < a < 5.325165846082586e-212

   1. Initial program 20.3

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

   3. Applied div-sub 20.3

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

   5. Applied *-un-lft-identity 20.3

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

   6. Applied times-frac 17.5

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

   7. Simplified 17.5

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

   9. Applied add-cube-cbrt 17.7

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

   10. Applied *-un-lft-identity 17.7

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{\color{blue}{1 \cdot b}}{\left(\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}\right) \cdot \sqrt[3]{\left(x + t\right) + y}}\]

   11. Applied times-frac 17.7

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{b}{\sqrt[3]{\left(x + t\right) + y}}\right)}\]

   12. Applied associate-*r* 16.9

       \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{\left(y \cdot \frac{1}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}}\right) \cdot \frac{b}{\sqrt[3]{\left(x + t\right) + y}}}\]

   13. Simplified 16.9

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

   if 5.325165846082586e-212 < a < 1.908584443905938e-45

   1. Initial program 18.8

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

   3. Applied div-sub 18.8

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

   5. Applied *-un-lft-identity 18.8

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

   6. Applied times-frac 15.4

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

   7. Simplified 15.4

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

   8. Taylor expanded around inf 22.0

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

   if 1.908584443905938e-45 < a < 1.9789658189585288e+164

   1. Initial program 26.4

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

   3. Applied div-sub 26.4

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

   5. Applied associate-/l* 23.7

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{\frac{y}{\frac{\left(x + t\right) + y}{b}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 20.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.329351671379600395696054258455065074018 \cdot 10^{162}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 5.325165846082585887271445202627113987297 \cdot 10^{-212}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{b}{\sqrt[3]{\left(x + t\right) + y}}\\ \mathbf{elif}\;a \le 1.908584443905937974655168865478710611307 \cdot 10^{-45}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;a \le 1.978965818958528848241507275261864808485 \cdot 10^{164}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{else}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)) -3.5813117084150564e153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e82) (/ 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)))