Average Error: 27.0 → 16.0
Time: 24.6s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -13199050022600241102316371968:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 2.111340660358068790178853732464209512348 \cdot 10^{96}:\\ \;\;\;\;\frac{1}{\frac{\left(y + t\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(z - b\right) + a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\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}\;y \le -13199050022600241102316371968:\\
\;\;\;\;\left(a + z\right) - b\\

\mathbf{elif}\;y \le 2.111340660358068790178853732464209512348 \cdot 10^{96}:\\
\;\;\;\;\frac{1}{\frac{\left(y + t\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(z - b\right) + a\right)\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\left(a + z\right) - b\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r35916483 = x;
        double r35916484 = y;
        double r35916485 = r35916483 + r35916484;
        double r35916486 = z;
        double r35916487 = r35916485 * r35916486;
        double r35916488 = t;
        double r35916489 = r35916488 + r35916484;
        double r35916490 = a;
        double r35916491 = r35916489 * r35916490;
        double r35916492 = r35916487 + r35916491;
        double r35916493 = b;
        double r35916494 = r35916484 * r35916493;
        double r35916495 = r35916492 - r35916494;
        double r35916496 = r35916483 + r35916488;
        double r35916497 = r35916496 + r35916484;
        double r35916498 = r35916495 / r35916497;
        return r35916498;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r35916499 = y;
        double r35916500 = -1.3199050022600241e+28;
        bool r35916501 = r35916499 <= r35916500;
        double r35916502 = a;
        double r35916503 = z;
        double r35916504 = r35916502 + r35916503;
        double r35916505 = b;
        double r35916506 = r35916504 - r35916505;
        double r35916507 = 2.1113406603580688e+96;
        bool r35916508 = r35916499 <= r35916507;
        double r35916509 = 1.0;
        double r35916510 = t;
        double r35916511 = r35916499 + r35916510;
        double r35916512 = x;
        double r35916513 = r35916511 + r35916512;
        double r35916514 = r35916503 - r35916505;
        double r35916515 = r35916514 + r35916502;
        double r35916516 = r35916499 * r35916515;
        double r35916517 = fma(r35916502, r35916510, r35916516);
        double r35916518 = fma(r35916503, r35916512, r35916517);
        double r35916519 = r35916513 / r35916518;
        double r35916520 = r35916509 / r35916519;
        double r35916521 = r35916508 ? r35916520 : r35916506;
        double r35916522 = r35916501 ? r35916506 : r35916521;
        return r35916522;
}

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

Target

Original27.0
Target11.5
Herbie16.0
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.3199050022600241e+28 or 2.1113406603580688e+96 < y

    1. Initial program 42.1

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

    2. Simplified42.1

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

    3. Taylor expanded around 0 15.6

      \[\leadsto \color{blue}{\left(a + z\right) - b}\]

    if -1.3199050022600241e+28 < y < 2.1113406603580688e+96

    1. Initial program 16.1

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

    2. Simplified16.1

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

    4. Applied fma-udef16.1

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

    6. Applied clear-num16.2

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

    7. Simplified16.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(t + y\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(z - b\right) + a\right)\right)\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -13199050022600241102316371968:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 2.111340660358068790178853732464209512348 \cdot 10^{96}:\\ \;\;\;\;\frac{1}{\frac{\left(y + t\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(z - b\right) + a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"

  :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) (/ 1.0 (/ (+ (+ 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)))