Average Error: 27.0 → 16.0
Time: 22.0s
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 r43077287 = x;
        double r43077288 = y;
        double r43077289 = r43077287 + r43077288;
        double r43077290 = z;
        double r43077291 = r43077289 * r43077290;
        double r43077292 = t;
        double r43077293 = r43077292 + r43077288;
        double r43077294 = a;
        double r43077295 = r43077293 * r43077294;
        double r43077296 = r43077291 + r43077295;
        double r43077297 = b;
        double r43077298 = r43077288 * r43077297;
        double r43077299 = r43077296 - r43077298;
        double r43077300 = r43077287 + r43077292;
        double r43077301 = r43077300 + r43077288;
        double r43077302 = r43077299 / r43077301;
        return r43077302;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r43077303 = y;
        double r43077304 = -1.3199050022600241e+28;
        bool r43077305 = r43077303 <= r43077304;
        double r43077306 = a;
        double r43077307 = z;
        double r43077308 = r43077306 + r43077307;
        double r43077309 = b;
        double r43077310 = r43077308 - r43077309;
        double r43077311 = 2.1113406603580688e+96;
        bool r43077312 = r43077303 <= r43077311;
        double r43077313 = 1.0;
        double r43077314 = t;
        double r43077315 = r43077303 + r43077314;
        double r43077316 = x;
        double r43077317 = r43077315 + r43077316;
        double r43077318 = r43077307 - r43077309;
        double r43077319 = r43077318 + r43077306;
        double r43077320 = r43077303 * r43077319;
        double r43077321 = fma(r43077306, r43077314, r43077320);
        double r43077322 = fma(r43077307, r43077316, r43077321);
        double r43077323 = r43077317 / r43077322;
        double r43077324 = r43077313 / r43077323;
        double r43077325 = r43077312 ? r43077324 : r43077310;
        double r43077326 = r43077305 ? r43077310 : r43077325;
        return r43077326;
}

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)))