Average Error: 26.7 → 7.8
Time: 15.1s
Precision: binary64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
↓
\[\begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \left(z + a\right) - b\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 1.4278703304685316 \cdot 10^{+264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x + y, \mathsf{fma}\left(a, t, y \cdot \left(a - b\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
\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 := y + \left(x + t\right)\\
t_2 := \left(z + a\right) - b\\
t_3 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\
\mathbf{if}\;t_3 \leq -\infty:\\
\;\;\;\;t_2\\

\mathbf{elif}\;t_3 \leq 1.4278703304685316 \cdot 10^{+264}:\\
\;\;\;\;\frac{\mathsf{fma}\left(z, x + y, \mathsf{fma}\left(a, t, y \cdot \left(a - b\right)\right)\right)}{t_1}\\

\mathbf{else}:\\
\;\;\;\;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 (+ y (+ x t)))
        (t_2 (- (+ z a) b))
        (t_3 (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) t_1)))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 1.4278703304685316e+264)
       (/ (fma z (+ x y) (fma a t (* y (- a b)))) t_1)
       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 = y + (x + t);
	double t_2 = (z + a) - b;
	double t_3 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 1.4278703304685316e+264) {
		tmp = fma(z, (x + y), fma(a, t, (y * (a - b)))) / t_1;
	} else {
		tmp = 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

Target

Original26.7
Target11.9
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

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 1.4278703304685316e264 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

    1. Initial program 62.5

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

    2. Taylor expanded in y around inf 17.9

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

    if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.4278703304685316e264

    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. Taylor expanded in x around 0 0.4

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

    3. Simplified0.3

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 1.4278703304685316 \cdot 10^{+264}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x + y, \mathsf{fma}\left(a, t, y \cdot \left(a - b\right)\right)\right)}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))