Average Error: 27.0 → 22.0
Time: 13.2s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6986254320624221 \cdot 10^{168}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le -4.77185841419090264 \cdot 10^{97}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le -6.2691502506957849 \cdot 10^{93}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \le -2.3344990110164508 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;t \le -1.87134248020096529 \cdot 10^{-164}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le -3.67353273866948948 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \sqrt[3]{{\left(\frac{y}{\left(x + t\right) + y}\right)}^{3}} \cdot b\\ \mathbf{elif}\;t \le 4.38041407240006756 \cdot 10^{-97}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le 2.197975804352339 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \left(y \cdot \frac{1}{\left(x + t\right) + y}\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]
\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}\;t \le -1.6986254320624221 \cdot 10^{168}:\\
\;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;t \le -4.77185841419090264 \cdot 10^{97}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;t \le -6.2691502506957849 \cdot 10^{93}:\\
\;\;\;\;a\\

\mathbf{elif}\;t \le -2.3344990110164508 \cdot 10^{-75}:\\
\;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\left(x + t\right) + y} \cdot b\\

\mathbf{elif}\;t \le -1.87134248020096529 \cdot 10^{-164}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;t \le -3.67353273866948948 \cdot 10^{-268}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \sqrt[3]{{\left(\frac{y}{\left(x + t\right) + y}\right)}^{3}} \cdot b\\

\mathbf{elif}\;t \le 4.38041407240006756 \cdot 10^{-97}:\\
\;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\

\mathbf{elif}\;t \le 2.197975804352339 \cdot 10^{135}:\\
\;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \left(y \cdot \frac{1}{\left(x + t\right) + y}\right) \cdot b\\

\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 r2430 = x;
        double r2431 = y;
        double r2432 = r2430 + r2431;
        double r2433 = z;
        double r2434 = r2432 * r2433;
        double r2435 = t;
        double r2436 = r2435 + r2431;
        double r2437 = a;
        double r2438 = r2436 * r2437;
        double r2439 = r2434 + r2438;
        double r2440 = b;
        double r2441 = r2431 * r2440;
        double r2442 = r2439 - r2441;
        double r2443 = r2430 + r2435;
        double r2444 = r2443 + r2431;
        double r2445 = r2442 / r2444;
        return r2445;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2446 = t;
        double r2447 = -1.6986254320624221e+168;
        bool r2448 = r2446 <= r2447;
        double r2449 = a;
        double r2450 = y;
        double r2451 = x;
        double r2452 = r2451 + r2446;
        double r2453 = r2452 + r2450;
        double r2454 = b;
        double r2455 = r2453 / r2454;
        double r2456 = r2450 / r2455;
        double r2457 = r2449 - r2456;
        double r2458 = -4.771858414190903e+97;
        bool r2459 = r2446 <= r2458;
        double r2460 = z;
        double r2461 = r2460 - r2456;
        double r2462 = -6.269150250695785e+93;
        bool r2463 = r2446 <= r2462;
        double r2464 = -2.334499011016451e-75;
        bool r2465 = r2446 <= r2464;
        double r2466 = 1.0;
        double r2467 = r2451 + r2450;
        double r2468 = r2467 * r2460;
        double r2469 = r2446 + r2450;
        double r2470 = r2469 * r2449;
        double r2471 = r2468 + r2470;
        double r2472 = r2453 / r2471;
        double r2473 = r2466 / r2472;
        double r2474 = r2450 / r2453;
        double r2475 = r2474 * r2454;
        double r2476 = r2473 - r2475;
        double r2477 = -1.8713424802009653e-164;
        bool r2478 = r2446 <= r2477;
        double r2479 = -3.6735327386694895e-268;
        bool r2480 = r2446 <= r2479;
        double r2481 = r2471 / r2453;
        double r2482 = 3.0;
        double r2483 = pow(r2474, r2482);
        double r2484 = cbrt(r2483);
        double r2485 = r2484 * r2454;
        double r2486 = r2481 - r2485;
        double r2487 = 4.3804140724000676e-97;
        bool r2488 = r2446 <= r2487;
        double r2489 = 2.197975804352339e+135;
        bool r2490 = r2446 <= r2489;
        double r2491 = r2466 / r2453;
        double r2492 = r2450 * r2491;
        double r2493 = r2492 * r2454;
        double r2494 = r2481 - r2493;
        double r2495 = r2490 ? r2494 : r2457;
        double r2496 = r2488 ? r2461 : r2495;
        double r2497 = r2480 ? r2486 : r2496;
        double r2498 = r2478 ? r2461 : r2497;
        double r2499 = r2465 ? r2476 : r2498;
        double r2500 = r2463 ? r2449 : r2499;
        double r2501 = r2459 ? r2461 : r2500;
        double r2502 = r2448 ? r2457 : r2501;
        return r2502;
}

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

Original27.0
Target11.5
Herbie22.0
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 6 regimes

  2. if t < -1.6986254320624221e+168 or 2.197975804352339e+135 < t

    1. Initial program 35.1

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

    3. Applied div-sub35.1

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

    5. Applied associate-/l*32.9

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

    6. Taylor expanded around 0 21.6

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

    if -1.6986254320624221e+168 < t < -4.771858414190903e+97 or -2.334499011016451e-75 < t < -1.8713424802009653e-164 or -3.6735327386694895e-268 < t < 4.3804140724000676e-97

    1. Initial program 24.8

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

    3. Applied div-sub24.8

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

    5. Applied associate-/l*23.7

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

    6. Taylor expanded around inf 24.1

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

    if -4.771858414190903e+97 < t < -6.269150250695785e+93

    1. Initial program 32.0

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

    2. Taylor expanded around 0 27.7

      \[\leadsto \color{blue}{a}\]

    if -6.269150250695785e+93 < t < -2.334499011016451e-75

    1. Initial program 23.8

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

    3. Applied div-sub23.8

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

    5. Applied associate-/l*19.9

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

    7. Applied associate-/r/19.8

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

    9. Applied clear-num19.9

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

    if -1.8713424802009653e-164 < t < -3.6735327386694895e-268

    1. Initial program 24.3

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

    3. Applied div-sub24.3

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

    5. Applied associate-/l*21.5

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

    7. Applied associate-/r/19.9

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

    9. Applied add-cbrt-cube32.5

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

    10. Applied add-cbrt-cube38.2

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

    11. Applied cbrt-undiv38.5

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

    12. Simplified22.6

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

    if 4.3804140724000676e-97 < t < 2.197975804352339e+135

    1. Initial program 23.4

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

    3. Applied div-sub23.4

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

    5. Applied associate-/l*20.2

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

    7. Applied associate-/r/20.1

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

    9. Applied div-inv20.2

      \[\leadsto \frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \color{blue}{\left(y \cdot \frac{1}{\left(x + t\right) + y}\right)} \cdot b\]
  3. Recombined 6 regimes into one program.

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6986254320624221 \cdot 10^{168}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le -4.77185841419090264 \cdot 10^{97}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le -6.2691502506957849 \cdot 10^{93}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \le -2.3344990110164508 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}} - \frac{y}{\left(x + t\right) + y} \cdot b\\ \mathbf{elif}\;t \le -1.87134248020096529 \cdot 10^{-164}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le -3.67353273866948948 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \sqrt[3]{{\left(\frac{y}{\left(x + t\right) + y}\right)}^{3}} \cdot b\\ \mathbf{elif}\;t \le 4.38041407240006756 \cdot 10^{-97}:\\ \;\;\;\;z - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \mathbf{elif}\;t \le 2.197975804352339 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(t + y\right) \cdot a}{\left(x + t\right) + y} - \left(y \cdot \frac{1}{\left(x + t\right) + y}\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y}{\frac{\left(x + t\right) + y}{b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))