Average Error: 26.8 → 7.7
Time: 13.1s
Precision: binary64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
↓
\[\begin{array}{l} t_1 := y + \left(x + t\right)\\ \mathbf{if}\;\begin{array}{l} t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\ t_2 \leq -\infty \lor \neg \left(t_2 \leq 6.409190456738072 \cdot 10^{+247}\right) \end{array}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + y, z, \mathsf{fma}\left(a, t, y \cdot \left(a - b\right)\right)\right)}{t_1}\\ \end{array} \]
\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)\\
\mathbf{if}\;\begin{array}{l}
t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{t_1}\\
t_2 \leq -\infty \lor \neg \left(t_2 \leq 6.409190456738072 \cdot 10^{+247}\right)
\end{array}:\\
\;\;\;\;\left(z + a\right) - b\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x + y, z, \mathsf{fma}\left(a, t, y \cdot \left(a - b\right)\right)\right)}{t_1}\\


\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))))
   (if (let* ((t_2 (/ (- (+ (* (+ x y) z) (* (+ y t) a)) (* y b)) t_1)))
         (or (<= t_2 (- INFINITY)) (not (<= t_2 6.409190456738072e+247))))
     (- (+ z a) b)
     (/ (fma (+ x y) z (fma a t (* y (- a b)))) t_1))))
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 = ((((x + y) * z) + ((y + t) * a)) - (y * b)) / t_1;
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 6.409190456738072e+247)) {
		tmp = (z + a) - b;
	} else {
		tmp = fma((x + y), z, fma(a, t, (y * (a - b)))) / t_1;
	}
	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

Original26.8
Target11.3
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

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 6.4091904567380718e247 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

    1. Initial program 61.8

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

    2. Taylor expanded in y around inf 17.5

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

    if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 6.4091904567380718e247

    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. Applied associate--l+_binary640.3

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

    3. Simplified0.3

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

    4. Applied fma-def_binary640.3

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 6.409190456738072 \cdot 10^{+247}\right):\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + y, z, \mathsf{fma}\left(a, t, y \cdot \left(a - b\right)\right)\right)}{y + \left(x + t\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(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)))