Average Error: 27.1 → 8.4
Time: 7.1s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -3.371023939266602 \cdot 10^{224} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 6.4510149534974711 \cdot 10^{259}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\mathsf{fma}\left(z, x + y, \left(t + y\right) \cdot a - y \cdot b\right)}}\\ \end{array}\]
\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}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -3.371023939266602 \cdot 10^{224} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 6.4510149534974711 \cdot 10^{259}\right):\\
\;\;\;\;\left(a + z\right) - b\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r1049917 = x;
        double r1049918 = y;
        double r1049919 = r1049917 + r1049918;
        double r1049920 = z;
        double r1049921 = r1049919 * r1049920;
        double r1049922 = t;
        double r1049923 = r1049922 + r1049918;
        double r1049924 = a;
        double r1049925 = r1049923 * r1049924;
        double r1049926 = r1049921 + r1049925;
        double r1049927 = b;
        double r1049928 = r1049918 * r1049927;
        double r1049929 = r1049926 - r1049928;
        double r1049930 = r1049917 + r1049922;
        double r1049931 = r1049930 + r1049918;
        double r1049932 = r1049929 / r1049931;
        return r1049932;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1049933 = x;
        double r1049934 = y;
        double r1049935 = r1049933 + r1049934;
        double r1049936 = z;
        double r1049937 = r1049935 * r1049936;
        double r1049938 = t;
        double r1049939 = r1049938 + r1049934;
        double r1049940 = a;
        double r1049941 = r1049939 * r1049940;
        double r1049942 = r1049937 + r1049941;
        double r1049943 = b;
        double r1049944 = r1049934 * r1049943;
        double r1049945 = r1049942 - r1049944;
        double r1049946 = r1049933 + r1049938;
        double r1049947 = r1049946 + r1049934;
        double r1049948 = r1049945 / r1049947;
        double r1049949 = -3.3710239392666025e+224;
        bool r1049950 = r1049948 <= r1049949;
        double r1049951 = 6.451014953497471e+259;
        bool r1049952 = r1049948 <= r1049951;
        double r1049953 = !r1049952;
        bool r1049954 = r1049950 || r1049953;
        double r1049955 = r1049940 + r1049936;
        double r1049956 = r1049955 - r1049943;
        double r1049957 = 1.0;
        double r1049958 = r1049941 - r1049944;
        double r1049959 = fma(r1049936, r1049935, r1049958);
        double r1049960 = r1049947 / r1049959;
        double r1049961 = r1049957 / r1049960;
        double r1049962 = r1049954 ? r1049956 : r1049961;
        return r1049962;
}

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

Original27.1
Target11.5
Herbie8.4
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -3.3710239392666025e+224 or 6.451014953497471e+259 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

    1. Initial program 59.3

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

    3. Applied clear-num59.3

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

    4. Simplified59.3

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

    5. Taylor expanded around 0 18.0

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

    if -3.3710239392666025e+224 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 6.451014953497471e+259

    1. Initial program 0.3

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

    3. Applied clear-num0.4

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

    4. Simplified0.4

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -3.371023939266602 \cdot 10^{224} \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 6.4510149534974711 \cdot 10^{259}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\mathsf{fma}\left(z, x + y, \left(t + y\right) \cdot a - y \cdot b\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(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)))