Average Error: 26.3 → 18.8
Time: 19.9s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.936654150029262825244469501044403686678 \cdot 10^{95}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;a \le 1.208693876705612329500717088392973051565 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;a \le 1.14480033436725852809181048882882069905 \cdot 10^{-209}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;a \le 2.680679968880847721751866050236761271388 \cdot 10^{110}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \end{array}\]
\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}\;a \le -9.936654150029262825244469501044403686678 \cdot 10^{95}:\\
\;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\

\mathbf{elif}\;a \le 1.208693876705612329500717088392973051565 \cdot 10^{-302}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\

\mathbf{elif}\;a \le 1.14480033436725852809181048882882069905 \cdot 10^{-209}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;a \le 2.680679968880847721751866050236761271388 \cdot 10^{110}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r688938 = x;
        double r688939 = y;
        double r688940 = r688938 + r688939;
        double r688941 = z;
        double r688942 = r688940 * r688941;
        double r688943 = t;
        double r688944 = r688943 + r688939;
        double r688945 = a;
        double r688946 = r688944 * r688945;
        double r688947 = r688942 + r688946;
        double r688948 = b;
        double r688949 = r688939 * r688948;
        double r688950 = r688947 - r688949;
        double r688951 = r688938 + r688943;
        double r688952 = r688951 + r688939;
        double r688953 = r688950 / r688952;
        return r688953;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r688954 = a;
        double r688955 = -9.936654150029263e+95;
        bool r688956 = r688954 <= r688955;
        double r688957 = y;
        double r688958 = x;
        double r688959 = t;
        double r688960 = r688958 + r688959;
        double r688961 = r688960 + r688957;
        double r688962 = r688957 / r688961;
        double r688963 = b;
        double r688964 = r688962 * r688963;
        double r688965 = r688954 - r688964;
        double r688966 = 1.2086938767056123e-302;
        bool r688967 = r688954 <= r688966;
        double r688968 = r688958 + r688957;
        double r688969 = z;
        double r688970 = r688968 * r688969;
        double r688971 = r688959 + r688957;
        double r688972 = r688971 * r688954;
        double r688973 = r688970 + r688972;
        double r688974 = r688973 / r688961;
        double r688975 = r688974 - r688964;
        double r688976 = 1.1448003343672585e-209;
        bool r688977 = r688954 <= r688976;
        double r688978 = r688961 / r688963;
        double r688979 = r688957 / r688978;
        double r688980 = r688969 - r688979;
        double r688981 = 2.6806799688808477e+110;
        bool r688982 = r688954 <= r688981;
        double r688983 = r688982 ? r688975 : r688965;
        double r688984 = r688977 ? r688980 : r688983;
        double r688985 = r688967 ? r688975 : r688984;
        double r688986 = r688956 ? r688965 : r688985;
        return r688986;
}

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

Original26.3
Target11.2
Herbie18.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -9.936654150029263e+95 or 2.6806799688808477e+110 < a

    1. Initial program 39.9

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

    3. Applied div-sub39.9

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

    5. Applied associate-/l*40.3

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

    7. Applied associate-/r/39.4

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

    8. Taylor expanded around 0 24.9

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

    if -9.936654150029263e+95 < a < 1.2086938767056123e-302 or 1.1448003343672585e-209 < a < 2.6806799688808477e+110

    1. Initial program 19.5

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

    3. Applied div-sub19.5

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

    5. Applied associate-/l*16.3

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

    7. Applied associate-/r/15.6

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

    if 1.2086938767056123e-302 < a < 1.1448003343672585e-209

    1. Initial program 20.1

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

    3. Applied div-sub20.1

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

    5. Applied associate-/l*14.9

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

    6. Taylor expanded around inf 17.2

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.936654150029262825244469501044403686678 \cdot 10^{95}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;a \le 1.208693876705612329500717088392973051565 \cdot 10^{-302}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;a \le 1.14480033436725852809181048882882069905 \cdot 10^{-209}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;a \le 2.680679968880847721751866050236761271388 \cdot 10^{110}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\left(x + t\right) + y} \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))