Average Error: 26.8 → 21.3
Time: 17.8s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -11807057696238675968:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le -2.201752731199656916658222202068793830749 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 1.425433211750613024926014778141499378436 \cdot 10^{-69}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 2.119164531549616961423837193835311333795 \cdot 10^{105}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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 -11807057696238675968:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le -2.201752731199656916658222202068793830749 \cdot 10^{-249}:\\
\;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le 1.425433211750613024926014778141499378436 \cdot 10^{-69}:\\
\;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;z \le 2.119164531549616961423837193835311333795 \cdot 10^{105}:\\
\;\;\;\;\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}:\\
\;\;\;\;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 r578948 = x;
        double r578949 = y;
        double r578950 = r578948 + r578949;
        double r578951 = z;
        double r578952 = r578950 * r578951;
        double r578953 = t;
        double r578954 = r578953 + r578949;
        double r578955 = a;
        double r578956 = r578954 * r578955;
        double r578957 = r578952 + r578956;
        double r578958 = b;
        double r578959 = r578949 * r578958;
        double r578960 = r578957 - r578959;
        double r578961 = r578948 + r578953;
        double r578962 = r578961 + r578949;
        double r578963 = r578960 / r578962;
        return r578963;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r578964 = z;
        double r578965 = -1.1807057696238676e+19;
        bool r578966 = r578964 <= r578965;
        double r578967 = y;
        double r578968 = x;
        double r578969 = t;
        double r578970 = r578968 + r578969;
        double r578971 = r578970 + r578967;
        double r578972 = b;
        double r578973 = r578971 / r578972;
        double r578974 = r578967 / r578973;
        double r578975 = r578964 - r578974;
        double r578976 = -2.201752731199657e-249;
        bool r578977 = r578964 <= r578976;
        double r578978 = r578968 + r578967;
        double r578979 = r578978 * r578964;
        double r578980 = r578969 + r578967;
        double r578981 = a;
        double r578982 = r578980 * r578981;
        double r578983 = r578979 + r578982;
        double r578984 = 1.0;
        double r578985 = r578984 / r578971;
        double r578986 = r578983 * r578985;
        double r578987 = r578986 - r578974;
        double r578988 = 1.425433211750613e-69;
        bool r578989 = r578964 <= r578988;
        double r578990 = r578981 - r578974;
        double r578991 = 2.119164531549617e+105;
        bool r578992 = r578964 <= r578991;
        double r578993 = r578983 / r578971;
        double r578994 = r578967 / r578971;
        double r578995 = r578994 * r578972;
        double r578996 = r578993 - r578995;
        double r578997 = r578992 ? r578996 : r578975;
        double r578998 = r578989 ? r578990 : r578997;
        double r578999 = r578977 ? r578987 : r578998;
        double r579000 = r578966 ? r578975 : r578999;
        return r579000;
}

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

Original26.8
Target11.2
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 4 regimes

  2. if z < -1.1807057696238676e+19 or 2.119164531549617e+105 < z

    1. Initial program 37.0

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

    3. Applied div-sub37.0

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

    5. Applied associate-/l*36.6

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

    6. Taylor expanded around inf 26.1

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

    if -1.1807057696238676e+19 < z < -2.201752731199657e-249

    1. Initial program 19.4

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

    3. Applied div-sub19.4

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

    5. Applied associate-/l*15.1

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

    7. Applied div-inv15.1

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

    if -2.201752731199657e-249 < z < 1.425433211750613e-69

    1. Initial program 18.6

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

    3. Applied div-sub18.6

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

    5. Applied associate-/l*15.5

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

    6. Taylor expanded around 0 19.6

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

    if 1.425433211750613e-69 < z < 2.119164531549617e+105

    1. Initial program 23.8

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

    3. Applied div-sub23.8

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

    5. Applied associate-/l*21.3

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

    7. Applied associate-/r/20.2

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{\frac{y}{\left(x + t\right) + y} \cdot b}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -11807057696238675968:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le -2.201752731199656916658222202068793830749 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) \cdot \frac{1}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 1.425433211750613024926014778141499378436 \cdot 10^{-69}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;z \le 2.119164531549616961423837193835311333795 \cdot 10^{105}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))