Average Error: 26.5 → 7.4
Time: 6.9s
Precision: binary64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_1 \leq -\infty \lor \neg \left(t_1 \leq 2.9085101000265782 \cdot 10^{+261}\right) \end{array}:\\ \;\;\;\;a + \left(z - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, a - b, \mathsf{fma}\left(x + y, z, t \cdot a\right)\right)}{x + \left(y + t\right)}\\ \end{array} \]
\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}\;\begin{array}{l}
t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\
t_1 \leq -\infty \lor \neg \left(t_1 \leq 2.9085101000265782 \cdot 10^{+261}\right)
\end{array}:\\
\;\;\;\;a + \left(z - b\right)\\

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


\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
 (if (let* ((t_1
             (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) (+ y (+ x t)))))
       (or (<= t_1 (- INFINITY)) (not (<= t_1 2.9085101000265782e+261))))
   (+ a (- z b))
   (/ (fma y (- a b) (fma (+ x y) z (* t a))) (+ x (+ y t)))))
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)) - (y * b)) / (y + (x + t));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2.9085101000265782e+261)) {
		tmp = a + (z - b);
	} else {
		tmp = fma(y, (a - b), fma((x + y), z, (t * a))) / (x + (y + t));
	}
	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

Target

Original26.5
Target11.2
Herbie7.4
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

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 2.90851010002657823e261 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

    1. Initial program 62.1

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

    2. Simplified62.1

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

    3. Taylor expanded in y around inf 17.0

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

    4. Applied associate--l+_binary6417.0

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

    if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.90851010002657823e261

    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. Simplified0.3

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2.9085101000265782 \cdot 10^{+261}\right):\\ \;\;\;\;a + \left(z - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, a - b, \mathsf{fma}\left(x + y, z, t \cdot a\right)\right)}{x + \left(y + t\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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)))