Average Error: 27.2 → 21.3
Time: 7.7s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.343517699116082085988013028938453324825 \cdot 10^{182}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.375028386241234800074588117756530326546 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.118643715046927542610803400339886881246 \cdot 10^{-149}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.16974325889302484717093484562260440578 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;x \le 1.110981286409301297083753758046103104188 \cdot 10^{-125}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 7.771333049681687694819501234115560782491 \cdot 10^{171}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]
\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 -1.343517699116082085988013028938453324825 \cdot 10^{182}:\\
\;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le -1.375028386241234800074588117756530326546 \cdot 10^{-50}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;x \le -1.118643715046927542610803400339886881246 \cdot 10^{-149}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le -1.16974325889302484717093484562260440578 \cdot 10^{-281}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\

\mathbf{elif}\;x \le 1.110981286409301297083753758046103104188 \cdot 10^{-125}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le 7.771333049681687694819501234115560782491 \cdot 10^{171}:\\
\;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - y \cdot \frac{b}{\left(x + t\right) + y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r815526 = x;
        double r815527 = y;
        double r815528 = r815526 + r815527;
        double r815529 = z;
        double r815530 = r815528 * r815529;
        double r815531 = t;
        double r815532 = r815531 + r815527;
        double r815533 = a;
        double r815534 = r815532 * r815533;
        double r815535 = r815530 + r815534;
        double r815536 = b;
        double r815537 = r815527 * r815536;
        double r815538 = r815535 - r815537;
        double r815539 = r815526 + r815531;
        double r815540 = r815539 + r815527;
        double r815541 = r815538 / r815540;
        return r815541;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r815542 = x;
        double r815543 = -1.343517699116082e+182;
        bool r815544 = r815542 <= r815543;
        double r815545 = z;
        double r815546 = y;
        double r815547 = b;
        double r815548 = t;
        double r815549 = r815542 + r815548;
        double r815550 = r815549 + r815546;
        double r815551 = r815547 / r815550;
        double r815552 = r815546 * r815551;
        double r815553 = r815545 - r815552;
        double r815554 = -1.3750283862412348e-50;
        bool r815555 = r815542 <= r815554;
        double r815556 = r815542 + r815546;
        double r815557 = r815556 * r815545;
        double r815558 = r815548 + r815546;
        double r815559 = a;
        double r815560 = r815558 * r815559;
        double r815561 = r815557 + r815560;
        double r815562 = r815561 / r815550;
        double r815563 = r815550 / r815547;
        double r815564 = r815546 / r815563;
        double r815565 = r815562 - r815564;
        double r815566 = -1.1186437150469275e-149;
        bool r815567 = r815542 <= r815566;
        double r815568 = r815559 - r815552;
        double r815569 = -1.1697432588930248e-281;
        bool r815570 = r815542 <= r815569;
        double r815571 = r815546 / r815550;
        double r815572 = 1.0;
        double r815573 = r815572 / r815547;
        double r815574 = r815571 / r815573;
        double r815575 = r815562 - r815574;
        double r815576 = 1.1109812864093013e-125;
        bool r815577 = r815542 <= r815576;
        double r815578 = 7.771333049681688e+171;
        bool r815579 = r815542 <= r815578;
        double r815580 = r815550 / r815561;
        double r815581 = r815572 / r815580;
        double r815582 = r815581 - r815552;
        double r815583 = r815579 ? r815582 : r815553;
        double r815584 = r815577 ? r815568 : r815583;
        double r815585 = r815570 ? r815575 : r815584;
        double r815586 = r815567 ? r815568 : r815585;
        double r815587 = r815555 ? r815565 : r815586;
        double r815588 = r815544 ? r815553 : r815587;
        return r815588;
}

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

Original27.2
Target11.6
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 5 regimes

  2. if x < -1.343517699116082e+182 or 7.771333049681688e+171 < x

    1. Initial program 37.3

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

    3. Applied div-sub37.3

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

    5. Applied *-un-lft-identity37.3

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

    6. Applied times-frac34.7

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

    7. Simplified34.7

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

    8. Taylor expanded around inf 20.8

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

    if -1.343517699116082e+182 < x < -1.3750283862412348e-50

    1. Initial program 26.2

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

    3. Applied div-sub26.2

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

    5. Applied associate-/l*23.4

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

    if -1.3750283862412348e-50 < x < -1.1186437150469275e-149 or -1.1697432588930248e-281 < x < 1.1109812864093013e-125

    1. Initial program 24.0

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

    3. Applied div-sub24.0

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

    5. Applied *-un-lft-identity24.0

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

    6. Applied times-frac21.8

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

    7. Simplified21.8

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

    8. Taylor expanded around 0 21.0

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

    if -1.1186437150469275e-149 < x < -1.1697432588930248e-281

    1. Initial program 23.3

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

    3. Applied div-sub23.3

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

    5. Applied associate-/l*22.4

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

    7. Applied div-inv22.5

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

    8. Applied associate-/r*19.7

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

    if 1.1109812864093013e-125 < x < 7.771333049681688e+171

    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-frac21.1

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

    7. Simplified21.1

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

    9. Applied clear-num21.2

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}}} - y \cdot \frac{b}{\left(x + t\right) + y}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification21.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.343517699116082085988013028938453324825 \cdot 10^{182}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.375028386241234800074588117756530326546 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -1.118643715046927542610803400339886881246 \cdot 10^{-149}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le -1.16974325889302484717093484562260440578 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{\frac{y}{\left(x + t\right) + y}}{\frac{1}{b}}\\ \mathbf{elif}\;x \le 1.110981286409301297083753758046103104188 \cdot 10^{-125}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 7.771333049681687694819501234115560782491 \cdot 10^{171}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z - y \cdot \frac{b}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))