Average Error: 26.4 → 16.3
Time: 19.1s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.8800874783927961 \cdot 10^{115} \lor \neg \left(y \le 1.29181664689404663 \cdot 10^{42}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \end{array}\]
\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}\;y \le -1.8800874783927961 \cdot 10^{115} \lor \neg \left(y \le 1.29181664689404663 \cdot 10^{42}\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}
double f(double x, double y, double z, double t, double a, double b) {
        double r832828 = x;
        double r832829 = y;
        double r832830 = r832828 + r832829;
        double r832831 = z;
        double r832832 = r832830 * r832831;
        double r832833 = t;
        double r832834 = r832833 + r832829;
        double r832835 = a;
        double r832836 = r832834 * r832835;
        double r832837 = r832832 + r832836;
        double r832838 = b;
        double r832839 = r832829 * r832838;
        double r832840 = r832837 - r832839;
        double r832841 = r832828 + r832833;
        double r832842 = r832841 + r832829;
        double r832843 = r832840 / r832842;
        return r832843;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r832844 = y;
        double r832845 = -1.880087478392796e+115;
        bool r832846 = r832844 <= r832845;
        double r832847 = 1.2918166468940466e+42;
        bool r832848 = r832844 <= r832847;
        double r832849 = !r832848;
        bool r832850 = r832846 || r832849;
        double r832851 = a;
        double r832852 = z;
        double r832853 = r832851 + r832852;
        double r832854 = b;
        double r832855 = r832853 - r832854;
        double r832856 = x;
        double r832857 = r832856 + r832844;
        double r832858 = r832857 * r832852;
        double r832859 = t;
        double r832860 = r832859 + r832844;
        double r832861 = r832860 * r832851;
        double r832862 = r832858 + r832861;
        double r832863 = r832844 * r832854;
        double r832864 = r832862 - r832863;
        double r832865 = 1.0;
        double r832866 = r832856 + r832859;
        double r832867 = r832866 + r832844;
        double r832868 = r832865 / r832867;
        double r832869 = r832864 * r832868;
        double r832870 = r832850 ? r832855 : r832869;
        return r832870;
}

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

Original26.4
Target11.3
Herbie16.3
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.880087478392796e+115 or 1.2918166468940466e+42 < y

    1. Initial program 42.7

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

    3. Applied clear-num42.8

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

    4. Simplified42.8

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

    5. Taylor expanded around 0 15.0

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

    if -1.880087478392796e+115 < y < 1.2918166468940466e+42

    1. Initial program 16.9

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

    3. Applied div-inv17.0

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.8800874783927961 \cdot 10^{115} \lor \neg \left(y \le 1.29181664689404663 \cdot 10^{42}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right) \cdot \frac{1}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))