Average Error: 27.1 → 22.4
Time: 7.0s
Precision: binary64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.942880152641725 \cdot 10^{228}:\\ \;\;\;\;z - \frac{y}{x + \left(y + t\right)} \cdot b\\ \mathbf{elif}\;z \le -2.1287153099689439 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\ \mathbf{elif}\;z \le 7.1336840955988063 \cdot 10^{-70}:\\ \;\;\;\;a - \frac{y}{x + \left(y + t\right)} \cdot b\\ \mathbf{elif}\;z \le 1.6625525016520592 \cdot 10^{121}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{x + \left(y + t\right)} \cdot b\\ \end{array}\]
\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}\;z \le -1.942880152641725 \cdot 10^{228}:\\
\;\;\;\;z - \frac{y}{x + \left(y + t\right)} \cdot b\\

\mathbf{elif}\;z \le -2.1287153099689439 \cdot 10^{-228}:\\
\;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\

\mathbf{elif}\;z \le 7.1336840955988063 \cdot 10^{-70}:\\
\;\;\;\;a - \frac{y}{x + \left(y + t\right)} \cdot b\\

\mathbf{elif}\;z \le 1.6625525016520592 \cdot 10^{121}:\\
\;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\

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

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (((double) (((double) (((double) (x + y)) * z)) + ((double) (((double) (t + y)) * a)))) - ((double) (y * b)))) / ((double) (((double) (x + t)) + y))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((z <= -1.942880152641725e+228)) {
		VAR = ((double) (z - ((double) (((double) (y / ((double) (x + ((double) (y + t)))))) * b))));
	} else {
		double VAR_1;
		if ((z <= -2.128715309968944e-228)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (x + ((double) (y + t)))) / ((double) (((double) (z * ((double) (y + x)))) + ((double) (((double) (t * a)) + ((double) (y * ((double) (a - b))))))))))));
		} else {
			double VAR_2;
			if ((z <= 7.133684095598806e-70)) {
				VAR_2 = ((double) (a - ((double) (((double) (y / ((double) (x + ((double) (y + t)))))) * b))));
			} else {
				double VAR_3;
				if ((z <= 1.6625525016520592e+121)) {
					VAR_3 = ((double) (1.0 / ((double) (((double) (x + ((double) (y + t)))) / ((double) (((double) (z * ((double) (y + x)))) + ((double) (((double) (t * a)) + ((double) (y * ((double) (a - b))))))))))));
				} else {
					VAR_3 = ((double) (z - ((double) (((double) (y / ((double) (x + ((double) (y + t)))))) * b))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original27.1
Target12.1
Herbie22.4
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.942880152641725e228 or 1.6625525016520592e121 < z

    1. Initial program 42.4

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

    3. Applied div-sub42.4

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

    4. Simplified42.4

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

    5. Simplified42.0

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

    6. Taylor expanded around inf 24.0

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

    if -1.942880152641725e228 < z < -2.1287153099689439e-228 or 7.1336840955988063e-70 < z < 1.6625525016520592e121

    1. Initial program 23.3

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

    3. Applied clear-num23.4

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

    4. Simplified23.4

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

    if -2.1287153099689439e-228 < z < 7.1336840955988063e-70

    1. Initial program 21.3

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

    3. Applied div-sub21.3

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

    4. Simplified21.3

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

    5. Simplified16.7

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

    6. Taylor expanded around 0 19.1

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.942880152641725 \cdot 10^{228}:\\ \;\;\;\;z - \frac{y}{x + \left(y + t\right)} \cdot b\\ \mathbf{elif}\;z \le -2.1287153099689439 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\ \mathbf{elif}\;z \le 7.1336840955988063 \cdot 10^{-70}:\\ \;\;\;\;a - \frac{y}{x + \left(y + t\right)} \cdot b\\ \mathbf{elif}\;z \le 1.6625525016520592 \cdot 10^{121}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + t\right)}{z \cdot \left(y + x\right) + \left(t \cdot a + y \cdot \left(a - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;z - \frac{y}{x + \left(y + t\right)} \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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) (/ 1.0 (/ (+ (+ 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)))