Average Error: 26.8 → 23.1
Time: 6.7s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.51042017274667401 \cdot 10^{171}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.50708683770980189 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -485115.40011078131:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -6.6876587271108054 \cdot 10^{-170}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -2.860753366813772 \cdot 10^{-271}:\\ \;\;\;\;\frac{1}{\left(x + t\right) + y} \cdot \left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right)\\ \mathbf{elif}\;x \le 6.7642958276510964 \cdot 10^{-203}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 9.7109844173215245 \cdot 10^{230}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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}\;x \le -1.51042017274667401 \cdot 10^{171}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;x \le -1.50708683770980189 \cdot 10^{134}:\\
\;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;x \le -485115.40011078131:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;x \le -6.6876587271108054 \cdot 10^{-170}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le -2.860753366813772 \cdot 10^{-271}:\\
\;\;\;\;\frac{1}{\left(x + t\right) + y} \cdot \left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right)\\

\mathbf{elif}\;x \le 6.7642958276510964 \cdot 10^{-203}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le 9.7109844173215245 \cdot 10^{230}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\

\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 r886061 = x;
        double r886062 = y;
        double r886063 = r886061 + r886062;
        double r886064 = z;
        double r886065 = r886063 * r886064;
        double r886066 = t;
        double r886067 = r886066 + r886062;
        double r886068 = a;
        double r886069 = r886067 * r886068;
        double r886070 = r886065 + r886069;
        double r886071 = b;
        double r886072 = r886062 * r886071;
        double r886073 = r886070 - r886072;
        double r886074 = r886061 + r886066;
        double r886075 = r886074 + r886062;
        double r886076 = r886073 / r886075;
        return r886076;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r886077 = x;
        double r886078 = -1.510420172746674e+171;
        bool r886079 = r886077 <= r886078;
        double r886080 = z;
        double r886081 = y;
        double r886082 = t;
        double r886083 = r886077 + r886082;
        double r886084 = r886083 + r886081;
        double r886085 = b;
        double r886086 = r886084 / r886085;
        double r886087 = r886081 / r886086;
        double r886088 = r886080 - r886087;
        double r886089 = -1.5070868377098019e+134;
        bool r886090 = r886077 <= r886089;
        double r886091 = 1.0;
        double r886092 = r886077 + r886081;
        double r886093 = r886092 * r886080;
        double r886094 = r886082 + r886081;
        double r886095 = a;
        double r886096 = r886094 * r886095;
        double r886097 = r886093 + r886096;
        double r886098 = r886084 / r886097;
        double r886099 = r886091 / r886098;
        double r886100 = r886099 - r886087;
        double r886101 = -485115.4001107813;
        bool r886102 = r886077 <= r886101;
        double r886103 = -6.687658727110805e-170;
        bool r886104 = r886077 <= r886103;
        double r886105 = r886085 / r886084;
        double r886106 = r886081 * r886105;
        double r886107 = r886095 - r886106;
        double r886108 = -2.8607533668137723e-271;
        bool r886109 = r886077 <= r886108;
        double r886110 = r886091 / r886084;
        double r886111 = r886081 * r886085;
        double r886112 = r886097 - r886111;
        double r886113 = r886110 * r886112;
        double r886114 = 6.764295827651096e-203;
        bool r886115 = r886077 <= r886114;
        double r886116 = 9.710984417321525e+230;
        bool r886117 = r886077 <= r886116;
        double r886118 = r886097 / r886084;
        double r886119 = r886118 - r886106;
        double r886120 = r886117 ? r886119 : r886088;
        double r886121 = r886115 ? r886107 : r886120;
        double r886122 = r886109 ? r886113 : r886121;
        double r886123 = r886104 ? r886107 : r886122;
        double r886124 = r886102 ? r886088 : r886123;
        double r886125 = r886090 ? r886100 : r886124;
        double r886126 = r886079 ? r886088 : r886125;
        return r886126;
}

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
Target10.8
Herbie23.1
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if x < -1.510420172746674e+171 or -1.5070868377098019e+134 < x < -485115.4001107813 or 9.710984417321525e+230 < x

    1. Initial program 33.1

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

    3. Applied div-sub33.1

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

    5. Applied associate-/l*30.2

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

    6. Taylor expanded around inf 23.8

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

    if -1.510420172746674e+171 < x < -1.5070868377098019e+134

    1. Initial program 31.7

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

    3. Applied div-sub31.7

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

    5. Applied associate-/l*29.2

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

    7. Applied clear-num29.2

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

    if -485115.4001107813 < x < -6.687658727110805e-170 or -2.8607533668137723e-271 < x < 6.764295827651096e-203

    1. Initial program 23.6

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

    3. Applied div-sub23.6

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

    5. Applied *-un-lft-identity23.6

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

    6. Applied times-frac22.2

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

    7. Simplified22.2

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

    8. Taylor expanded around 0 22.6

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

    if -6.687658727110805e-170 < x < -2.8607533668137723e-271

    1. Initial program 21.0

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

    3. Applied div-sub21.0

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

    5. Applied div-inv21.0

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

    6. Applied div-inv21.1

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

    7. Applied distribute-rgt-out--21.1

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

    if 6.764295827651096e-203 < x < 9.710984417321525e+230

    1. Initial program 25.0

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

    3. Applied div-sub25.0

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

    5. Applied *-un-lft-identity25.0

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

    6. Applied times-frac22.9

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

    7. Simplified22.9

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{y} \cdot \frac{b}{\left(x + t\right) + y}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification23.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.51042017274667401 \cdot 10^{171}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.50708683770980189 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -485115.40011078131:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -6.6876587271108054 \cdot 10^{-170}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -2.860753366813772 \cdot 10^{-271}:\\ \;\;\;\;\frac{1}{\left(x + t\right) + y} \cdot \left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right)\\ \mathbf{elif}\;x \le 6.7642958276510964 \cdot 10^{-203}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 9.7109844173215245 \cdot 10^{230}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))