Average Error: 24.5 → 15.2
Time: 20.4s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.2195156445786848 \cdot 10^{+79}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 1752559586117.6345:\\ \;\;\;\;\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right) \cdot \frac{1}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\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 -1.2195156445786848 \cdot 10^{+79}:\\
\;\;\;\;\left(a + z\right) - b\\

\mathbf{elif}\;y \le 1752559586117.6345:\\
\;\;\;\;\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right) \cdot \frac{1}{\left(y + t\right) + x}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r37264129 = x;
        double r37264130 = y;
        double r37264131 = r37264129 + r37264130;
        double r37264132 = z;
        double r37264133 = r37264131 * r37264132;
        double r37264134 = t;
        double r37264135 = r37264134 + r37264130;
        double r37264136 = a;
        double r37264137 = r37264135 * r37264136;
        double r37264138 = r37264133 + r37264137;
        double r37264139 = b;
        double r37264140 = r37264130 * r37264139;
        double r37264141 = r37264138 - r37264140;
        double r37264142 = r37264129 + r37264134;
        double r37264143 = r37264142 + r37264130;
        double r37264144 = r37264141 / r37264143;
        return r37264144;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r37264145 = y;
        double r37264146 = -1.2195156445786848e+79;
        bool r37264147 = r37264145 <= r37264146;
        double r37264148 = a;
        double r37264149 = z;
        double r37264150 = r37264148 + r37264149;
        double r37264151 = b;
        double r37264152 = r37264150 - r37264151;
        double r37264153 = 1752559586117.6345;
        bool r37264154 = r37264145 <= r37264153;
        double r37264155 = x;
        double r37264156 = t;
        double r37264157 = r37264152 * r37264145;
        double r37264158 = fma(r37264148, r37264156, r37264157);
        double r37264159 = fma(r37264149, r37264155, r37264158);
        double r37264160 = 1.0;
        double r37264161 = r37264145 + r37264156;
        double r37264162 = r37264161 + r37264155;
        double r37264163 = r37264160 / r37264162;
        double r37264164 = r37264159 * r37264163;
        double r37264165 = r37264154 ? r37264164 : r37264152;
        double r37264166 = r37264147 ? r37264152 : r37264165;
        return r37264166;
}

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

Original24.5
Target11.6
Herbie15.2
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.2195156445786848e+79 or 1752559586117.6345 < y

    1. Initial program 38.5

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

    2. Simplified38.4

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

    3. Taylor expanded around 0 16.1

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

    if -1.2195156445786848e+79 < y < 1752559586117.6345

    1. Initial program 14.4

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

    2. Simplified14.4

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

    4. Applied div-inv14.5

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.2195156445786848 \cdot 10^{+79}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 1752559586117.6345:\\ \;\;\;\;\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right) \cdot \frac{1}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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 (/ (+ (+ 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)))