Average Error: 27.1 → 20.9
Time: 21.2s
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 r648351 = x;
        double r648352 = y;
        double r648353 = r648351 + r648352;
        double r648354 = z;
        double r648355 = r648353 * r648354;
        double r648356 = t;
        double r648357 = r648356 + r648352;
        double r648358 = a;
        double r648359 = r648357 * r648358;
        double r648360 = r648355 + r648359;
        double r648361 = b;
        double r648362 = r648352 * r648361;
        double r648363 = r648360 - r648362;
        double r648364 = r648351 + r648356;
        double r648365 = r648364 + r648352;
        double r648366 = r648363 / r648365;
        return r648366;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r648367 = a;
        double r648368 = -6.147203584124048e+75;
        bool r648369 = r648367 <= r648368;
        double r648370 = y;
        double r648371 = x;
        double r648372 = t;
        double r648373 = r648371 + r648372;
        double r648374 = r648373 + r648370;
        double r648375 = b;
        double r648376 = r648374 / r648375;
        double r648377 = r648370 / r648376;
        double r648378 = r648367 - r648377;
        double r648379 = -6.155686702486673e-170;
        bool r648380 = r648367 <= r648379;
        double r648381 = r648371 + r648370;
        double r648382 = z;
        double r648383 = r648381 * r648382;
        double r648384 = r648372 + r648370;
        double r648385 = r648384 * r648367;
        double r648386 = r648383 + r648385;
        double r648387 = cbrt(r648386);
        double r648388 = r648387 * r648387;
        double r648389 = cbrt(r648374);
        double r648390 = r648389 * r648389;
        double r648391 = r648388 / r648390;
        double r648392 = r648387 / r648389;
        double r648393 = r648391 * r648392;
        double r648394 = r648370 / r648374;
        double r648395 = r648394 * r648375;
        double r648396 = r648393 - r648395;
        double r648397 = 6.340261835080083e-175;
        bool r648398 = r648367 <= r648397;
        double r648399 = r648382 - r648377;
        double r648400 = 7.121581141661449e+174;
        bool r648401 = r648367 <= r648400;
        double r648402 = r648386 / r648374;
        double r648403 = r648375 / r648374;
        double r648404 = r648370 * r648403;
        double r648405 = r648402 - r648404;
        double r648406 = r648401 ? r648405 : r648378;
        double r648407 = r648398 ? r648399 : r648406;
        double r648408 = r648380 ? r648396 : r648407;
        double r648409 = r648369 ? r648378 : r648408;
        return r648409;
}

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)))