Average Error: 26.8 → 23.1
Time: 6.9s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.51042017274667401 \cdot 10^{171}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.50708683770980189 \cdot 10^{134}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -485115.40011078131:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -6.6876587271108054 \cdot 10^{-170}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -2.860753366813772 \cdot 10^{-271}:\\ \;\;\;\;\frac{1}{\left(x + t\right) + y} \cdot \left(\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\right)\\ \mathbf{elif}\;x \le 6.7642958276510964 \cdot 10^{-203}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 9.7109844173215245 \cdot 10^{230}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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 r979622 = x;
        double r979623 = y;
        double r979624 = r979622 + r979623;
        double r979625 = z;
        double r979626 = r979624 * r979625;
        double r979627 = t;
        double r979628 = r979627 + r979623;
        double r979629 = a;
        double r979630 = r979628 * r979629;
        double r979631 = r979626 + r979630;
        double r979632 = b;
        double r979633 = r979623 * r979632;
        double r979634 = r979631 - r979633;
        double r979635 = r979622 + r979627;
        double r979636 = r979635 + r979623;
        double r979637 = r979634 / r979636;
        return r979637;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r979638 = x;
        double r979639 = -1.510420172746674e+171;
        bool r979640 = r979638 <= r979639;
        double r979641 = z;
        double r979642 = y;
        double r979643 = t;
        double r979644 = r979638 + r979643;
        double r979645 = r979644 + r979642;
        double r979646 = b;
        double r979647 = r979645 / r979646;
        double r979648 = r979642 / r979647;
        double r979649 = r979641 - r979648;
        double r979650 = -1.5070868377098019e+134;
        bool r979651 = r979638 <= r979650;
        double r979652 = 1.0;
        double r979653 = r979638 + r979642;
        double r979654 = r979653 * r979641;
        double r979655 = r979643 + r979642;
        double r979656 = a;
        double r979657 = r979655 * r979656;
        double r979658 = r979654 + r979657;
        double r979659 = r979645 / r979658;
        double r979660 = r979652 / r979659;
        double r979661 = r979660 - r979648;
        double r979662 = -485115.4001107813;
        bool r979663 = r979638 <= r979662;
        double r979664 = -6.687658727110805e-170;
        bool r979665 = r979638 <= r979664;
        double r979666 = r979646 / r979645;
        double r979667 = r979642 * r979666;
        double r979668 = r979656 - r979667;
        double r979669 = -2.8607533668137723e-271;
        bool r979670 = r979638 <= r979669;
        double r979671 = r979652 / r979645;
        double r979672 = r979642 * r979646;
        double r979673 = r979658 - r979672;
        double r979674 = r979671 * r979673;
        double r979675 = 6.764295827651096e-203;
        bool r979676 = r979638 <= r979675;
        double r979677 = 9.710984417321525e+230;
        bool r979678 = r979638 <= r979677;
        double r979679 = r979658 / r979645;
        double r979680 = r979679 - r979667;
        double r979681 = r979678 ? r979680 : r979649;
        double r979682 = r979676 ? r979668 : r979681;
        double r979683 = r979670 ? r979674 : r979682;
        double r979684 = r979665 ? r979668 : r979683;
        double r979685 = r979663 ? r979649 : r979684;
        double r979686 = r979651 ? r979661 : r979685;
        double r979687 = r979640 ? r979649 : r979686;
        return r979687;
}

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)))