Average Error: 26.4 → 22.5
Time: 14.6s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.167970801017747333585571881926417802986 \cdot 10^{80}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -2.234989953541200263385581111442281856948 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;x \le -4.899515873749284249389445834868375982167 \cdot 10^{-236}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 3.46626155810342549714430232432862157048 \cdot 10^{-290}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;x \le 8.319255981684834624526027052421217572153 \cdot 10^{-47}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 90417508901533.34375:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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}\;x \le -3.167970801017747333585571881926417802986 \cdot 10^{80}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;x \le -2.234989953541200263385581111442281856948 \cdot 10^{-94}:\\
\;\;\;\;\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{elif}\;x \le -4.899515873749284249389445834868375982167 \cdot 10^{-236}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le 3.46626155810342549714430232432862157048 \cdot 10^{-290}:\\
\;\;\;\;\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{elif}\;x \le 8.319255981684834624526027052421217572153 \cdot 10^{-47}:\\
\;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\

\mathbf{elif}\;x \le 90417508901533.34375:\\
\;\;\;\;\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 r956412 = x;
        double r956413 = y;
        double r956414 = r956412 + r956413;
        double r956415 = z;
        double r956416 = r956414 * r956415;
        double r956417 = t;
        double r956418 = r956417 + r956413;
        double r956419 = a;
        double r956420 = r956418 * r956419;
        double r956421 = r956416 + r956420;
        double r956422 = b;
        double r956423 = r956413 * r956422;
        double r956424 = r956421 - r956423;
        double r956425 = r956412 + r956417;
        double r956426 = r956425 + r956413;
        double r956427 = r956424 / r956426;
        return r956427;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r956428 = x;
        double r956429 = -3.1679708010177473e+80;
        bool r956430 = r956428 <= r956429;
        double r956431 = z;
        double r956432 = y;
        double r956433 = t;
        double r956434 = r956428 + r956433;
        double r956435 = r956434 + r956432;
        double r956436 = b;
        double r956437 = r956435 / r956436;
        double r956438 = r956432 / r956437;
        double r956439 = r956431 - r956438;
        double r956440 = -2.2349899535412003e-94;
        bool r956441 = r956428 <= r956440;
        double r956442 = r956428 + r956432;
        double r956443 = r956442 * r956431;
        double r956444 = r956433 + r956432;
        double r956445 = a;
        double r956446 = r956444 * r956445;
        double r956447 = r956443 + r956446;
        double r956448 = r956447 / r956435;
        double r956449 = r956432 / r956435;
        double r956450 = r956449 * r956436;
        double r956451 = r956448 - r956450;
        double r956452 = -4.899515873749284e-236;
        bool r956453 = r956428 <= r956452;
        double r956454 = r956436 / r956435;
        double r956455 = r956432 * r956454;
        double r956456 = r956445 - r956455;
        double r956457 = 3.4662615581034255e-290;
        bool r956458 = r956428 <= r956457;
        double r956459 = 8.319255981684835e-47;
        bool r956460 = r956428 <= r956459;
        double r956461 = 90417508901533.34;
        bool r956462 = r956428 <= r956461;
        double r956463 = r956462 ? r956451 : r956439;
        double r956464 = r956460 ? r956456 : r956463;
        double r956465 = r956458 ? r956451 : r956464;
        double r956466 = r956453 ? r956456 : r956465;
        double r956467 = r956441 ? r956451 : r956466;
        double r956468 = r956430 ? r956439 : r956467;
        return r956468;
}

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

Original26.4
Target11.6
Herbie22.5
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.1679708010177473e+80 or 90417508901533.34 < x

    1. Initial program 31.6

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

    3. Applied div-sub31.6

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

    5. Applied associate-/l*29.0

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

    6. Taylor expanded around inf 24.9

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

    if -3.1679708010177473e+80 < x < -2.2349899535412003e-94 or -4.899515873749284e-236 < x < 3.4662615581034255e-290 or 8.319255981684835e-47 < x < 90417508901533.34

    1. Initial program 23.1

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

    3. Applied div-sub23.1

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

    5. Applied associate-/l*21.4

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

    7. Applied associate-/r/20.1

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

    if -2.2349899535412003e-94 < x < -4.899515873749284e-236 or 3.4662615581034255e-290 < x < 8.319255981684835e-47

    1. Initial program 22.1

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

    3. Applied div-sub22.1

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

    5. Applied *-un-lft-identity22.1

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

    6. Applied times-frac21.5

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

    7. Simplified21.5

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

    8. Taylor expanded around 0 21.3

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

  4. Final simplification22.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.167970801017747333585571881926417802986 \cdot 10^{80}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;x \le -2.234989953541200263385581111442281856948 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;x \le -4.899515873749284249389445834868375982167 \cdot 10^{-236}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 3.46626155810342549714430232432862157048 \cdot 10^{-290}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;x \le 8.319255981684834624526027052421217572153 \cdot 10^{-47}:\\ \;\;\;\;a - y \cdot \frac{b}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \le 90417508901533.34375:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))