Average Error: 27.1 → 20.9
Time: 21.0s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -6.147203584124048199955159448919944745691 \cdot 10^{75}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;a \le -6.155686702486672781294624345907714289504 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y}} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;a \le 6.340261835080082788347680532822021978218 \cdot 10^{-175}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;a \le 7.121581141661448680701955268111578403189 \cdot 10^{174}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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}\;a \le -6.147203584124048199955159448919944745691 \cdot 10^{75}:\\
\;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;a \le -6.155686702486672781294624345907714289504 \cdot 10^{-170}:\\
\;\;\;\;\frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y}} - \frac{y}{\left(x + t\right) + y} \cdot b\\

\mathbf{elif}\;a \le 6.340261835080082788347680532822021978218 \cdot 10^{-175}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;a \le 7.121581141661448680701955268111578403189 \cdot 10^{174}:\\
\;\;\;\;\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}:\\
\;\;\;\;a - \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 r552640 = x;
        double r552641 = y;
        double r552642 = r552640 + r552641;
        double r552643 = z;
        double r552644 = r552642 * r552643;
        double r552645 = t;
        double r552646 = r552645 + r552641;
        double r552647 = a;
        double r552648 = r552646 * r552647;
        double r552649 = r552644 + r552648;
        double r552650 = b;
        double r552651 = r552641 * r552650;
        double r552652 = r552649 - r552651;
        double r552653 = r552640 + r552645;
        double r552654 = r552653 + r552641;
        double r552655 = r552652 / r552654;
        return r552655;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r552656 = a;
        double r552657 = -6.147203584124048e+75;
        bool r552658 = r552656 <= r552657;
        double r552659 = y;
        double r552660 = x;
        double r552661 = t;
        double r552662 = r552660 + r552661;
        double r552663 = r552662 + r552659;
        double r552664 = b;
        double r552665 = r552663 / r552664;
        double r552666 = r552659 / r552665;
        double r552667 = r552656 - r552666;
        double r552668 = -6.155686702486673e-170;
        bool r552669 = r552656 <= r552668;
        double r552670 = r552660 + r552659;
        double r552671 = z;
        double r552672 = r552670 * r552671;
        double r552673 = r552661 + r552659;
        double r552674 = r552673 * r552656;
        double r552675 = r552672 + r552674;
        double r552676 = cbrt(r552675);
        double r552677 = r552676 * r552676;
        double r552678 = cbrt(r552663);
        double r552679 = r552678 * r552678;
        double r552680 = r552677 / r552679;
        double r552681 = r552676 / r552678;
        double r552682 = r552680 * r552681;
        double r552683 = r552659 / r552663;
        double r552684 = r552683 * r552664;
        double r552685 = r552682 - r552684;
        double r552686 = 6.340261835080083e-175;
        bool r552687 = r552656 <= r552686;
        double r552688 = r552671 - r552666;
        double r552689 = 7.121581141661449e+174;
        bool r552690 = r552656 <= r552689;
        double r552691 = r552675 / r552663;
        double r552692 = r552664 / r552663;
        double r552693 = r552659 * r552692;
        double r552694 = r552691 - r552693;
        double r552695 = r552690 ? r552694 : r552667;
        double r552696 = r552687 ? r552688 : r552695;
        double r552697 = r552669 ? r552685 : r552696;
        double r552698 = r552658 ? r552667 : r552697;
        return r552698;
}

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

Original27.1
Target11.4
Herbie20.9
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if a < -6.147203584124048e+75 or 7.121581141661449e+174 < a

    1. Initial program 40.5

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

    3. Applied div-sub40.5

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

    5. Applied associate-/l*40.8

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

    6. Taylor expanded around 0 25.2

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

    if -6.147203584124048e+75 < a < -6.155686702486673e-170

    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*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. Using strategy rm

    7. Applied associate-/r/15.5

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

    9. Applied add-cube-cbrt16.1

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\color{blue}{\left(\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}\right) \cdot \sqrt[3]{\left(x + t\right) + y}}} - \frac{y}{\left(x + t\right) + y} \cdot b\]

    10. Applied add-cube-cbrt16.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}\right) \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}}{\left(\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}\right) \cdot \sqrt[3]{\left(x + t\right) + y}} - \frac{y}{\left(x + t\right) + y} \cdot b\]

    11. Applied times-frac16.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y}}} - \frac{y}{\left(x + t\right) + y} \cdot b\]

    if -6.155686702486673e-170 < a < 6.340261835080083e-175

    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*16.1

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

    6. Taylor expanded around inf 19.0

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

    if 6.340261835080083e-175 < a < 7.121581141661449e+174

    1. Initial program 23.9

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

    3. Applied div-sub23.9

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

    5. Applied *-un-lft-identity23.9

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

    6. Applied times-frac21.2

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

    7. Simplified21.2

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

  4. Final simplification20.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.147203584124048199955159448919944745691 \cdot 10^{75}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;a \le -6.155686702486672781294624345907714289504 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a} \cdot \sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y} \cdot \sqrt[3]{\left(x + t\right) + y}} \cdot \frac{\sqrt[3]{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}{\sqrt[3]{\left(x + t\right) + y}} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;a \le 6.340261835080082788347680532822021978218 \cdot 10^{-175}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;a \le 7.121581141661448680701955268111578403189 \cdot 10^{174}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))