Average Error: 27.0 → 16.1
Time: 58.1s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.672509138302709691714274895411511055373 \cdot 10^{73}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 2.567985143919534345254434311651824859187 \cdot 10^{75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right)}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\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 -2.672509138302709691714274895411511055373 \cdot 10^{73}:\\
\;\;\;\;\left(a + z\right) - b\\

\mathbf{elif}\;y \le 2.567985143919534345254434311651824859187 \cdot 10^{75}:\\
\;\;\;\;\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right)}{\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 r39291910 = x;
        double r39291911 = y;
        double r39291912 = r39291910 + r39291911;
        double r39291913 = z;
        double r39291914 = r39291912 * r39291913;
        double r39291915 = t;
        double r39291916 = r39291915 + r39291911;
        double r39291917 = a;
        double r39291918 = r39291916 * r39291917;
        double r39291919 = r39291914 + r39291918;
        double r39291920 = b;
        double r39291921 = r39291911 * r39291920;
        double r39291922 = r39291919 - r39291921;
        double r39291923 = r39291910 + r39291915;
        double r39291924 = r39291923 + r39291911;
        double r39291925 = r39291922 / r39291924;
        return r39291925;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r39291926 = y;
        double r39291927 = -2.6725091383027097e+73;
        bool r39291928 = r39291926 <= r39291927;
        double r39291929 = a;
        double r39291930 = z;
        double r39291931 = r39291929 + r39291930;
        double r39291932 = b;
        double r39291933 = r39291931 - r39291932;
        double r39291934 = 2.5679851439195343e+75;
        bool r39291935 = r39291926 <= r39291934;
        double r39291936 = x;
        double r39291937 = t;
        double r39291938 = r39291933 * r39291926;
        double r39291939 = fma(r39291929, r39291937, r39291938);
        double r39291940 = fma(r39291930, r39291936, r39291939);
        double r39291941 = r39291926 + r39291937;
        double r39291942 = r39291941 + r39291936;
        double r39291943 = r39291940 / r39291942;
        double r39291944 = r39291935 ? r39291943 : r39291933;
        double r39291945 = r39291928 ? r39291933 : r39291944;
        return r39291945;
}

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
Target11.2
Herbie16.1
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.6725091383027097e+73 or 2.5679851439195343e+75 < y

    1. Initial program 43.4

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

    2. Simplified43.3

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

    3. Taylor expanded around 0 14.9

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

    if -2.6725091383027097e+73 < y < 2.5679851439195343e+75

    1. Initial program 16.8

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

    2. Simplified16.8

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.672509138302709691714274895411511055373 \cdot 10^{73}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 2.567985143919534345254434311651824859187 \cdot 10^{75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, \left(\left(a + z\right) - b\right) \cdot y\right)\right)}{\left(y + t\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))