Average Error: 26.3 → 17.4
Time: 18.4s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.243078244809433359419741519989651106527 \cdot 10^{-52}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 4617794.309636129997670650482177734375:\\ \;\;\;\;\frac{1}{\frac{\left(t + y\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(a + z\right) - b\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]
\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 -1.243078244809433359419741519989651106527 \cdot 10^{-52}:\\
\;\;\;\;\left(a + z\right) - b\\

\mathbf{elif}\;y \le 4617794.309636129997670650482177734375:\\
\;\;\;\;\frac{1}{\frac{\left(t + y\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(a + z\right) - b\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 r31940890 = x;
        double r31940891 = y;
        double r31940892 = r31940890 + r31940891;
        double r31940893 = z;
        double r31940894 = r31940892 * r31940893;
        double r31940895 = t;
        double r31940896 = r31940895 + r31940891;
        double r31940897 = a;
        double r31940898 = r31940896 * r31940897;
        double r31940899 = r31940894 + r31940898;
        double r31940900 = b;
        double r31940901 = r31940891 * r31940900;
        double r31940902 = r31940899 - r31940901;
        double r31940903 = r31940890 + r31940895;
        double r31940904 = r31940903 + r31940891;
        double r31940905 = r31940902 / r31940904;
        return r31940905;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r31940906 = y;
        double r31940907 = -1.2430782448094334e-52;
        bool r31940908 = r31940906 <= r31940907;
        double r31940909 = a;
        double r31940910 = z;
        double r31940911 = r31940909 + r31940910;
        double r31940912 = b;
        double r31940913 = r31940911 - r31940912;
        double r31940914 = 4617794.30963613;
        bool r31940915 = r31940906 <= r31940914;
        double r31940916 = 1.0;
        double r31940917 = t;
        double r31940918 = r31940917 + r31940906;
        double r31940919 = x;
        double r31940920 = r31940918 + r31940919;
        double r31940921 = r31940906 * r31940913;
        double r31940922 = fma(r31940909, r31940917, r31940921);
        double r31940923 = fma(r31940910, r31940919, r31940922);
        double r31940924 = r31940920 / r31940923;
        double r31940925 = r31940916 / r31940924;
        double r31940926 = r31940915 ? r31940925 : r31940913;
        double r31940927 = r31940908 ? r31940913 : r31940926;
        return r31940927;
}

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

Original26.3
Target11.2
Herbie17.4
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.2430782448094334e-52 or 4617794.30963613 < y

    1. Initial program 36.0

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

    2. Simplified36.0

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

    3. Taylor expanded around 0 19.3

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

    if -1.2430782448094334e-52 < y < 4617794.30963613

    1. Initial program 15.1

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

    2. Simplified15.1

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

    4. Applied clear-num15.2

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.243078244809433359419741519989651106527 \cdot 10^{-52}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \le 4617794.309636129997670650482177734375:\\ \;\;\;\;\frac{1}{\frac{\left(t + y\right) + x}{\mathsf{fma}\left(z, x, \mathsf{fma}\left(a, t, y \cdot \left(\left(a + z\right) - b\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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)))