Average Error: 24.5 → 14.8
Time: 16.1s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.2841481611231983 \cdot 10^{+83}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 10735475308532.344:\\ \;\;\;\;\mathsf{fma}\left(y, a - b, \mathsf{fma}\left(a, t, z \cdot \left(x + y\right)\right)\right) \cdot \frac{1}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\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 -1.2841481611231983 \cdot 10^{+83}:\\
\;\;\;\;\left(a + z\right) - b\\

\mathbf{elif}\;y \le 10735475308532.344:\\
\;\;\;\;\mathsf{fma}\left(y, a - b, \mathsf{fma}\left(a, t, z \cdot \left(x + y\right)\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 r16518104 = x;
        double r16518105 = y;
        double r16518106 = r16518104 + r16518105;
        double r16518107 = z;
        double r16518108 = r16518106 * r16518107;
        double r16518109 = t;
        double r16518110 = r16518109 + r16518105;
        double r16518111 = a;
        double r16518112 = r16518110 * r16518111;
        double r16518113 = r16518108 + r16518112;
        double r16518114 = b;
        double r16518115 = r16518105 * r16518114;
        double r16518116 = r16518113 - r16518115;
        double r16518117 = r16518104 + r16518109;
        double r16518118 = r16518117 + r16518105;
        double r16518119 = r16518116 / r16518118;
        return r16518119;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r16518120 = y;
        double r16518121 = -1.2841481611231983e+83;
        bool r16518122 = r16518120 <= r16518121;
        double r16518123 = a;
        double r16518124 = z;
        double r16518125 = r16518123 + r16518124;
        double r16518126 = b;
        double r16518127 = r16518125 - r16518126;
        double r16518128 = 10735475308532.344;
        bool r16518129 = r16518120 <= r16518128;
        double r16518130 = r16518123 - r16518126;
        double r16518131 = t;
        double r16518132 = x;
        double r16518133 = r16518132 + r16518120;
        double r16518134 = r16518124 * r16518133;
        double r16518135 = fma(r16518123, r16518131, r16518134);
        double r16518136 = fma(r16518120, r16518130, r16518135);
        double r16518137 = 1.0;
        double r16518138 = r16518120 + r16518131;
        double r16518139 = r16518138 + r16518132;
        double r16518140 = r16518137 / r16518139;
        double r16518141 = r16518136 * r16518140;
        double r16518142 = r16518129 ? r16518141 : r16518127;
        double r16518143 = r16518122 ? r16518127 : r16518142;
        return r16518143;
}

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
Target10.7
Herbie14.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.2841481611231983e+83 or 10735475308532.344 < y

    1. Initial program 38.1

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

    2. Simplified38.1

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

    3. Taylor expanded around inf 15.2

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

    if -1.2841481611231983e+83 < y < 10735475308532.344

    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

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

    4. Applied div-inv14.5

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.2841481611231983 \cdot 10^{+83}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 10735475308532.344:\\ \;\;\;\;\mathsf{fma}\left(y, a - b, \mathsf{fma}\left(a, t, z \cdot \left(x + y\right)\right)\right) \cdot \frac{1}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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)))