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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.939662278312858132841061974784859759209 \cdot 10^{305}:\\ \;\;\;\;z\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 9.823560067484486557199205658011723736571 \cdot 10^{276}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

\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}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.939662278312858132841061974784859759209 \cdot 10^{305}:\\
\;\;\;\;z\\

\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} \le 9.823560067484486557199205658011723736571 \cdot 10^{276}:\\
\;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\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 r579114 = x;
        double r579115 = y;
        double r579116 = r579114 + r579115;
        double r579117 = z;
        double r579118 = r579116 * r579117;
        double r579119 = t;
        double r579120 = r579119 + r579115;
        double r579121 = a;
        double r579122 = r579120 * r579121;
        double r579123 = r579118 + r579122;
        double r579124 = b;
        double r579125 = r579115 * r579124;
        double r579126 = r579123 - r579125;
        double r579127 = r579114 + r579119;
        double r579128 = r579127 + r579115;
        double r579129 = r579126 / r579128;
        return r579129;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r579130 = x;
        double r579131 = y;
        double r579132 = r579130 + r579131;
        double r579133 = z;
        double r579134 = r579132 * r579133;
        double r579135 = t;
        double r579136 = r579135 + r579131;
        double r579137 = a;
        double r579138 = r579136 * r579137;
        double r579139 = r579134 + r579138;
        double r579140 = b;
        double r579141 = r579131 * r579140;
        double r579142 = r579139 - r579141;
        double r579143 = r579130 + r579135;
        double r579144 = r579143 + r579131;
        double r579145 = r579142 / r579144;
        double r579146 = -6.939662278312858e+305;
        bool r579147 = r579145 <= r579146;
        double r579148 = 9.823560067484487e+276;
        bool r579149 = r579145 <= r579148;
        double r579150 = r579149 ? r579145 : r579133;
        double r579151 = r579147 ? r579133 : r579150;
        return r579151;
}

Target

Original	26.8
Target	11.1
Herbie	17.1

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

Derivation

Split input into 2 regimes
if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -6.939662278312858e+305 or 9.823560067484487e+276 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))
1. Initial program 63.1
  \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
2. Taylor expanded around inf 40.1
  \[\leadsto \color{blue}{z}\]
if -6.939662278312858e+305 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 9.823560067484487e+276
1. Initial program 0.3
  \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
Recombined 2 regimes into one program.
Final simplification17.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le -6.939662278312858132841061974784859759209 \cdot 10^{305}:\\ \;\;\;\;z\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 9.823560067484486557199205658011723736571 \cdot 10^{276}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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)) -3.5813117084150564e153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e82) (/ 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)))

Error

Try it out

Target

Derivation

`if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -6.939662278312858e+305 or 9.823560067484487e+276 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))`

`if -6.939662278312858e+305 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 9.823560067484487e+276`

Reproduce

In
Out