Average Error: 26.8 → 16.0
Time: 7.2s
Precision: 64
\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.7900535216185051 \cdot 10^{35} \lor \neg \left(y \le 3.0691536757249856 \cdot 10^{32}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \mathsf{fma}\left(y, a - b, a \cdot t\right)}{\left(x + t\right) + y}\\ \end{array}\]
\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}\;y \le -3.7900535216185051 \cdot 10^{35} \lor \neg \left(y \le 3.0691536757249856 \cdot 10^{32}\right):\\
\;\;\;\;\left(a + z\right) - b\\

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

\end{array}
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 VAR;
	if (((y <= -3.790053521618505e+35) || !(y <= 3.0691536757249856e+32))) {
		VAR = ((a + z) - b);
	} else {
		VAR = ((((x + y) * z) + fma(y, (a - b), (a * t))) / ((x + t) + y));
	}
	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

Original26.8
Target11.6
Herbie16.0
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.790053521618505e+35 or 3.0691536757249856e+32 < y

    1. Initial program 40.7

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

    3. Applied clear-num40.7

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

    4. Simplified40.7

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

    5. Taylor expanded around 0 16.3

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

    if -3.790053521618505e+35 < y < 3.0691536757249856e+32

    1. Initial program 15.7

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

    3. Applied associate--l+15.7

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

    4. Simplified15.7

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.7900535216185051 \cdot 10^{35} \lor \neg \left(y \le 3.0691536757249856 \cdot 10^{32}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot z + \mathsf{fma}\left(y, a - b, a \cdot t\right)}{\left(x + t\right) + y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(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 (/ (+ (+ 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)))