Average Error: 27.2 → 21.3
Time: 20.2s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.343517699116082085988013028938453324825 \cdot 10^{182}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.375028386241234800074588117756530326546 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.118643715046927542610803400339886881246 \cdot 10^{-149}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.16974325889302484717093484562260440578 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;x \le 1.110981286409301297083753758046103104188 \cdot 10^{-125}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 7.771333049681687694819501234115560782491 \cdot 10^{171}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]
\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}\;x \le -1.343517699116082085988013028938453324825 \cdot 10^{182}:\\
\;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le -1.375028386241234800074588117756530326546 \cdot 10^{-50}:\\
\;\;\;\;\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{elif}\;x \le -1.118643715046927542610803400339886881246 \cdot 10^{-149}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le -1.16974325889302484717093484562260440578 \cdot 10^{-281}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\

\mathbf{elif}\;x \le 1.110981286409301297083753758046103104188 \cdot 10^{-125}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le 7.771333049681687694819501234115560782491 \cdot 10^{171}:\\
\;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{else}:\\
\;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r573836 = x;
        double r573837 = y;
        double r573838 = r573836 + r573837;
        double r573839 = z;
        double r573840 = r573838 * r573839;
        double r573841 = t;
        double r573842 = r573841 + r573837;
        double r573843 = a;
        double r573844 = r573842 * r573843;
        double r573845 = r573840 + r573844;
        double r573846 = b;
        double r573847 = r573837 * r573846;
        double r573848 = r573845 - r573847;
        double r573849 = r573836 + r573841;
        double r573850 = r573849 + r573837;
        double r573851 = r573848 / r573850;
        return r573851;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r573852 = x;
        double r573853 = -1.343517699116082e+182;
        bool r573854 = r573852 <= r573853;
        double r573855 = z;
        double r573856 = y;
        double r573857 = b;
        double r573858 = t;
        double r573859 = r573852 + r573858;
        double r573860 = r573859 + r573856;
        double r573861 = r573857 / r573860;
        double r573862 = r573856 * r573861;
        double r573863 = r573855 - r573862;
        double r573864 = -1.3750283862412348e-50;
        bool r573865 = r573852 <= r573864;
        double r573866 = r573852 + r573856;
        double r573867 = r573866 * r573855;
        double r573868 = r573858 + r573856;
        double r573869 = a;
        double r573870 = r573868 * r573869;
        double r573871 = r573867 + r573870;
        double r573872 = r573871 / r573860;
        double r573873 = r573860 / r573857;
        double r573874 = r573856 / r573873;
        double r573875 = r573872 - r573874;
        double r573876 = -1.1186437150469275e-149;
        bool r573877 = r573852 <= r573876;
        double r573878 = r573869 - r573862;
        double r573879 = -1.1697432588930248e-281;
        bool r573880 = r573852 <= r573879;
        double r573881 = r573856 / r573860;
        double r573882 = 1.0;
        double r573883 = r573882 / r573857;
        double r573884 = r573881 / r573883;
        double r573885 = r573872 - r573884;
        double r573886 = 1.1109812864093013e-125;
        bool r573887 = r573852 <= r573886;
        double r573888 = 7.771333049681688e+171;
        bool r573889 = r573852 <= r573888;
        double r573890 = r573860 / r573871;
        double r573891 = r573882 / r573890;
        double r573892 = r573891 - r573862;
        double r573893 = r573889 ? r573892 : r573863;
        double r573894 = r573887 ? r573878 : r573893;
        double r573895 = r573880 ? r573885 : r573894;
        double r573896 = r573877 ? r573878 : r573895;
        double r573897 = r573865 ? r573875 : r573896;
        double r573898 = r573854 ? r573863 : r573897;
        return r573898;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original27.2
Target11.6
Herbie21.3
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if x < -1.343517699116082e+182 or 7.771333049681688e+171 < x

    1. Initial program 37.3

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

    3. Applied div-sub37.3

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

    5. Applied *-un-lft-identity37.3

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

    6. Applied times-frac34.7

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

    7. Simplified34.7

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

    8. Taylor expanded around inf 20.8

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

    if -1.343517699116082e+182 < x < -1.3750283862412348e-50

    1. Initial program 26.2

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

    3. Applied div-sub26.2

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

    5. Applied associate-/l*23.4

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

    if -1.3750283862412348e-50 < x < -1.1186437150469275e-149 or -1.1697432588930248e-281 < x < 1.1109812864093013e-125

    1. Initial program 24.0

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

    3. Applied div-sub24.0

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

    5. Applied *-un-lft-identity24.0

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

    6. Applied times-frac21.8

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

    7. Simplified21.8

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

    8. Taylor expanded around 0 21.0

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

    if -1.1186437150469275e-149 < x < -1.1697432588930248e-281

    1. Initial program 23.3

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

    3. Applied div-sub23.3

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

    5. Applied associate-/l*22.4

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

    7. Applied div-inv22.5

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

    8. Applied associate-/r*19.7

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

    if 1.1109812864093013e-125 < x < 7.771333049681688e+171

    1. Initial program 23.6

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

    3. Applied div-sub23.6

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

    5. Applied *-un-lft-identity23.6

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

    6. Applied times-frac21.1

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

    7. Simplified21.1

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

    9. Applied clear-num21.2

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}} - y \cdot \frac{b}{\left(x + t\right) + y}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.343517699116082085988013028938453324825 \cdot 10^{182}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.375028386241234800074588117756530326546 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.118643715046927542610803400339886881246 \cdot 10^{-149}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.16974325889302484717093484562260440578 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;x \le 1.110981286409301297083753758046103104188 \cdot 10^{-125}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 7.771333049681687694819501234115560782491 \cdot 10^{171}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))