Average Error: 26.7 → 7.9
Time: 18.2s
Precision: binary64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
↓
\[\begin{array}{l} t_1 := \left(x + y\right) \cdot z + \left(y + t\right) \cdot a\\ t_2 := y + \left(x + t\right)\\ \mathbf{if}\;\begin{array}{l} t_3 := \frac{t_1 - y \cdot b}{t_2}\\ t_3 \leq -\infty \lor \neg \left(t_3 \leq 2.4876467548670116 \cdot 10^{+258}\right) \end{array}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_2} - \frac{y \cdot b}{t_2}\\ \end{array} \]
\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}
t_1 := \left(x + y\right) \cdot z + \left(y + t\right) \cdot a\\
t_2 := y + \left(x + t\right)\\
\mathbf{if}\;\begin{array}{l}
t_3 := \frac{t_1 - y \cdot b}{t_2}\\
t_3 \leq -\infty \lor \neg \left(t_3 \leq 2.4876467548670116 \cdot 10^{+258}\right)
\end{array}:\\
\;\;\;\;\left(z + a\right) - b\\

\mathbf{else}:\\
\;\;\;\;\frac{t_1}{t_2} - \frac{y \cdot b}{t_2}\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (+ x y) z) (* (+ y t) a))) (t_2 (+ y (+ x t))))
   (if (let* ((t_3 (/ (- t_1 (* y b)) t_2)))
         (or (<= t_3 (- INFINITY)) (not (<= t_3 2.4876467548670116e+258))))
     (- (+ z a) b)
     (- (/ t_1 t_2) (/ (* y b) t_2)))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x + y) * z) + ((y + t) * a);
	double t_2 = y + (x + t);
	double t_3 = (t_1 - (y * b)) / t_2;
	double tmp;
	if ((t_3 <= -((double) INFINITY)) || !(t_3 <= 2.4876467548670116e+258)) {
		tmp = (z + a) - b;
	} else {
		tmp = (t_1 / t_2) - ((y * b) / t_2);
	}
	return tmp;
}

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

Original26.7
Target11.6
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 2.4876467548670116e258 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

    1. Initial program 62.2

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

    3. Applied clear-num_binary6462.2

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

    4. Simplified62.2

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

    5. Taylor expanded around inf 18.1

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

    6. Simplified18.1

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

    if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.4876467548670116e258

    1. Initial program 0.3

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

    3. Applied div-sub_binary640.3

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

    4. Simplified0.3

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

    5. Simplified0.3

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2.4876467548670116 \cdot 10^{+258}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \left(y + t\right) \cdot a}{y + \left(x + t\right)} - \frac{y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021190 
(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)))