Average Error: 26.4 → 23.9
Time: 25.0s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.061610428241891645480708199708681011513 \cdot 10^{213}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \le -79984826952145731778421884972492363137020:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, \mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)\right)}{\left(x + t\right) + y}\\ \mathbf{elif}\;z \le -1.299216067398793496033270482516750923125 \cdot 10^{-13}:\\ \;\;\;\;a\\ \mathbf{elif}\;z \le 1.515482064664791338217368179176752602296 \cdot 10^{167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, \mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]
\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}\;z \le -1.061610428241891645480708199708681011513 \cdot 10^{213}:\\
\;\;\;\;z\\

\mathbf{elif}\;z \le -79984826952145731778421884972492363137020:\\
\;\;\;\;\frac{\mathsf{fma}\left(a, t + y, \mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)\right)}{\left(x + t\right) + y}\\

\mathbf{elif}\;z \le -1.299216067398793496033270482516750923125 \cdot 10^{-13}:\\
\;\;\;\;a\\

\mathbf{elif}\;z \le 1.515482064664791338217368179176752602296 \cdot 10^{167}:\\
\;\;\;\;\frac{\mathsf{fma}\left(a, t + y, \mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)\right)}{\left(x + t\right) + y}\\

\mathbf{else}:\\
\;\;\;\;z\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r624791 = x;
        double r624792 = y;
        double r624793 = r624791 + r624792;
        double r624794 = z;
        double r624795 = r624793 * r624794;
        double r624796 = t;
        double r624797 = r624796 + r624792;
        double r624798 = a;
        double r624799 = r624797 * r624798;
        double r624800 = r624795 + r624799;
        double r624801 = b;
        double r624802 = r624792 * r624801;
        double r624803 = r624800 - r624802;
        double r624804 = r624791 + r624796;
        double r624805 = r624804 + r624792;
        double r624806 = r624803 / r624805;
        return r624806;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r624807 = z;
        double r624808 = -1.0616104282418916e+213;
        bool r624809 = r624807 <= r624808;
        double r624810 = -7.998482695214573e+40;
        bool r624811 = r624807 <= r624810;
        double r624812 = a;
        double r624813 = t;
        double r624814 = y;
        double r624815 = r624813 + r624814;
        double r624816 = x;
        double r624817 = b;
        double r624818 = r624807 - r624817;
        double r624819 = r624814 * r624818;
        double r624820 = fma(r624816, r624807, r624819);
        double r624821 = fma(r624812, r624815, r624820);
        double r624822 = r624816 + r624813;
        double r624823 = r624822 + r624814;
        double r624824 = r624821 / r624823;
        double r624825 = -1.2992160673987935e-13;
        bool r624826 = r624807 <= r624825;
        double r624827 = 1.5154820646647913e+167;
        bool r624828 = r624807 <= r624827;
        double r624829 = r624828 ? r624824 : r624807;
        double r624830 = r624826 ? r624812 : r624829;
        double r624831 = r624811 ? r624824 : r624830;
        double r624832 = r624809 ? r624807 : r624831;
        return r624832;
}

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

Original26.4
Target11.2
Herbie23.9
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.581311708415056427521064305370896655752 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.228596430831560895857110658734089400289 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.0616104282418916e+213 or 1.5154820646647913e+167 < z

    1. Initial program 43.8

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

    2. Simplified43.8

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

    3. Taylor expanded around 0 25.6

      \[\leadsto \color{blue}{z}\]

    if -1.0616104282418916e+213 < z < -7.998482695214573e+40 or -1.2992160673987935e-13 < z < 1.5154820646647913e+167

    1. Initial program 22.3

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

    2. Simplified22.3

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

    if -7.998482695214573e+40 < z < -1.2992160673987935e-13

    1. Initial program 22.3

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

    2. Simplified22.3

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

    3. Taylor expanded around inf 42.4

      \[\leadsto \color{blue}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.061610428241891645480708199708681011513 \cdot 10^{213}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \le -79984826952145731778421884972492363137020:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, \mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)\right)}{\left(x + t\right) + y}\\ \mathbf{elif}\;z \le -1.299216067398793496033270482516750923125 \cdot 10^{-13}:\\ \;\;\;\;a\\ \mathbf{elif}\;z \le 1.515482064664791338217368179176752602296 \cdot 10^{167}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, \mathsf{fma}\left(x, z, y \cdot \left(z - b\right)\right)\right)}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))