Average Error: 27.0 → 20.5
Time: 7.1s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.299632096055188359732666430858197676587 \cdot 10^{182}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le -6.58097239450565769445350948474438013136 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le -5.561955879913683304630071073532749837049 \cdot 10^{-33}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 2.188382713534194168391592823268801705419 \cdot 10^{-265}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 3.531204250326267435165734577025015017466 \cdot 10^{-186}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 9.836792356441501745440262935078248892751 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 7.569840457596660380072714567516066108349 \cdot 10^{-75}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 4.992279324067695797541286300663075135612 \cdot 10^{56}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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}\;z \le -3.299632096055188359732666430858197676587 \cdot 10^{182}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le -6.58097239450565769445350948474438013136 \cdot 10^{-7}:\\
\;\;\;\;\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}\;z \le -5.561955879913683304630071073532749837049 \cdot 10^{-33}:\\
\;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le 2.188382713534194168391592823268801705419 \cdot 10^{-265}:\\
\;\;\;\;\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}\;z \le 3.531204250326267435165734577025015017466 \cdot 10^{-186}:\\
\;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le 9.836792356441501745440262935078248892751 \cdot 10^{-126}:\\
\;\;\;\;\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}\;z \le 7.569840457596660380072714567516066108349 \cdot 10^{-75}:\\
\;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le 4.992279324067695797541286300663075135612 \cdot 10^{56}:\\
\;\;\;\;\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{else}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r931825 = x;
        double r931826 = y;
        double r931827 = r931825 + r931826;
        double r931828 = z;
        double r931829 = r931827 * r931828;
        double r931830 = t;
        double r931831 = r931830 + r931826;
        double r931832 = a;
        double r931833 = r931831 * r931832;
        double r931834 = r931829 + r931833;
        double r931835 = b;
        double r931836 = r931826 * r931835;
        double r931837 = r931834 - r931836;
        double r931838 = r931825 + r931830;
        double r931839 = r931838 + r931826;
        double r931840 = r931837 / r931839;
        return r931840;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r931841 = z;
        double r931842 = -3.2996320960551884e+182;
        bool r931843 = r931841 <= r931842;
        double r931844 = y;
        double r931845 = x;
        double r931846 = t;
        double r931847 = r931845 + r931846;
        double r931848 = r931847 + r931844;
        double r931849 = b;
        double r931850 = r931848 / r931849;
        double r931851 = r931844 / r931850;
        double r931852 = r931841 - r931851;
        double r931853 = -6.580972394505658e-07;
        bool r931854 = r931841 <= r931853;
        double r931855 = r931845 + r931844;
        double r931856 = r931855 * r931841;
        double r931857 = r931846 + r931844;
        double r931858 = a;
        double r931859 = r931857 * r931858;
        double r931860 = r931856 + r931859;
        double r931861 = r931860 / r931848;
        double r931862 = r931844 / r931848;
        double r931863 = 1.0;
        double r931864 = r931863 / r931849;
        double r931865 = r931862 / r931864;
        double r931866 = r931861 - r931865;
        double r931867 = -5.561955879913683e-33;
        bool r931868 = r931841 <= r931867;
        double r931869 = r931858 - r931851;
        double r931870 = 2.1883827135341942e-265;
        bool r931871 = r931841 <= r931870;
        double r931872 = 3.5312042503262674e-186;
        bool r931873 = r931841 <= r931872;
        double r931874 = 9.836792356441502e-126;
        bool r931875 = r931841 <= r931874;
        double r931876 = 7.56984045759666e-75;
        bool r931877 = r931841 <= r931876;
        double r931878 = 4.992279324067696e+56;
        bool r931879 = r931841 <= r931878;
        double r931880 = r931879 ? r931866 : r931852;
        double r931881 = r931877 ? r931869 : r931880;
        double r931882 = r931875 ? r931866 : r931881;
        double r931883 = r931873 ? r931869 : r931882;
        double r931884 = r931871 ? r931866 : r931883;
        double r931885 = r931868 ? r931869 : r931884;
        double r931886 = r931854 ? r931866 : r931885;
        double r931887 = r931843 ? r931852 : r931886;
        return r931887;
}

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

Original27.0
Target11.3
Herbie20.5
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.2996320960551884e+182 or 4.992279324067696e+56 < z

    1. Initial program 40.3

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

    3. Applied div-sub40.3

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

    5. Applied associate-/l*40.4

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

    6. Taylor expanded around inf 25.5

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

    if -3.2996320960551884e+182 < z < -6.580972394505658e-07 or -5.561955879913683e-33 < z < 2.1883827135341942e-265 or 3.5312042503262674e-186 < z < 9.836792356441502e-126 or 7.56984045759666e-75 < z < 4.992279324067696e+56

    1. Initial program 21.4

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

    3. Applied div-sub21.4

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

    5. Applied associate-/l*18.2

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

    7. Applied div-inv18.3

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

    8. Applied associate-/r*17.6

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

    if -6.580972394505658e-07 < z < -5.561955879913683e-33 or 2.1883827135341942e-265 < z < 3.5312042503262674e-186 or 9.836792356441502e-126 < z < 7.56984045759666e-75

    1. Initial program 20.2

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

    3. Applied div-sub20.2

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

    5. Applied associate-/l*16.1

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

    6. Taylor expanded around 0 21.0

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

  4. Final simplification20.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.299632096055188359732666430858197676587 \cdot 10^{182}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le -6.58097239450565769445350948474438013136 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le -5.561955879913683304630071073532749837049 \cdot 10^{-33}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 2.188382713534194168391592823268801705419 \cdot 10^{-265}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 3.531204250326267435165734577025015017466 \cdot 10^{-186}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 9.836792356441501745440262935078248892751 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;z \le 7.569840457596660380072714567516066108349 \cdot 10^{-75}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 4.992279324067695797541286300663075135612 \cdot 10^{56}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))