Average Error: 27.0 → 20.5
Time: 7.3s
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 r938097 = x;
        double r938098 = y;
        double r938099 = r938097 + r938098;
        double r938100 = z;
        double r938101 = r938099 * r938100;
        double r938102 = t;
        double r938103 = r938102 + r938098;
        double r938104 = a;
        double r938105 = r938103 * r938104;
        double r938106 = r938101 + r938105;
        double r938107 = b;
        double r938108 = r938098 * r938107;
        double r938109 = r938106 - r938108;
        double r938110 = r938097 + r938102;
        double r938111 = r938110 + r938098;
        double r938112 = r938109 / r938111;
        return r938112;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r938113 = z;
        double r938114 = -3.2996320960551884e+182;
        bool r938115 = r938113 <= r938114;
        double r938116 = y;
        double r938117 = x;
        double r938118 = t;
        double r938119 = r938117 + r938118;
        double r938120 = r938119 + r938116;
        double r938121 = b;
        double r938122 = r938120 / r938121;
        double r938123 = r938116 / r938122;
        double r938124 = r938113 - r938123;
        double r938125 = -6.580972394505658e-07;
        bool r938126 = r938113 <= r938125;
        double r938127 = r938117 + r938116;
        double r938128 = r938127 * r938113;
        double r938129 = r938118 + r938116;
        double r938130 = a;
        double r938131 = r938129 * r938130;
        double r938132 = r938128 + r938131;
        double r938133 = r938132 / r938120;
        double r938134 = r938116 / r938120;
        double r938135 = 1.0;
        double r938136 = r938135 / r938121;
        double r938137 = r938134 / r938136;
        double r938138 = r938133 - r938137;
        double r938139 = -5.561955879913683e-33;
        bool r938140 = r938113 <= r938139;
        double r938141 = r938130 - r938123;
        double r938142 = 2.1883827135341942e-265;
        bool r938143 = r938113 <= r938142;
        double r938144 = 3.5312042503262674e-186;
        bool r938145 = r938113 <= r938144;
        double r938146 = 9.836792356441502e-126;
        bool r938147 = r938113 <= r938146;
        double r938148 = 7.56984045759666e-75;
        bool r938149 = r938113 <= r938148;
        double r938150 = 4.992279324067696e+56;
        bool r938151 = r938113 <= r938150;
        double r938152 = r938151 ? r938138 : r938124;
        double r938153 = r938149 ? r938141 : r938152;
        double r938154 = r938147 ? r938138 : r938153;
        double r938155 = r938145 ? r938141 : r938154;
        double r938156 = r938143 ? r938138 : r938155;
        double r938157 = r938140 ? r938141 : r938156;
        double r938158 = r938126 ? r938138 : r938157;
        double r938159 = r938115 ? r938124 : r938158;
        return r938159;
}

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)))