(- (+ (* (* x y) z) (* x y)) (/ (* z y) x))

Average Error: 3.2 → 0.3

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(x \cdot y\right) \cdot z + x \cdot y\right) - \frac{z \cdot y}{x} \le -1.134502737044711 \cdot 10^{-103} \lor \neg \left(\left(\left(x \cdot y\right) \cdot z + x \cdot y\right) - \frac{z \cdot y}{x} \le 1.2758900792983024 \cdot 10^{41}\right):\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot z + x \cdot y\right) - \frac{z \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z + \left(x - \frac{z}{x}\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) (((double) (((double) (((double) (x * y)) * z)) + ((double) (x * y)))) - ((double) (((double) (z * y)) / x))));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (((double) (((double) (((double) (x * y)) * z)) + ((double) (x * y)))) - ((double) (((double) (z * y)) / x)))) <= -1.1345027370447107e-103) || !(((double) (((double) (((double) (((double) (x * y)) * z)) + ((double) (x * y)))) - ((double) (((double) (z * y)) / x)))) <= 1.2758900792983024e+41))) {
		VAR = ((double) (((double) (((double) (((double) (x * y)) * z)) + ((double) (x * y)))) - ((double) (((double) (z * y)) / x))));
	} else {
		VAR = ((double) (y * ((double) (((double) (x * z)) + ((double) (x - ((double) (z / x))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 2 regimes

2. if (- (+ (* (* x y) z) (* x y)) (/ (* z y) x)) < -1.134502737044711e-103 or 1.2758900792983024e41 < (- (+ (* (* x y) z) (* x y)) (/ (* z y) x))

   1. Initial program 0.2

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

   if -1.134502737044711e-103 < (- (+ (* (* x y) z) (* x y)) (/ (* z y) x)) < 1.2758900792983024e41

   1. Initial program 10.5

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

   2. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x \cdot y\right) \cdot z + x \cdot y\right) - \frac{z \cdot y}{x} \le -1.134502737044711 \cdot 10^{-103} \lor \neg \left(\left(\left(x \cdot y\right) \cdot z + x \cdot y\right) - \frac{z \cdot y}{x} \le 1.2758900792983024 \cdot 10^{41}\right):\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot z + x \cdot y\right) - \frac{z \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z + \left(x - \frac{z}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x y z)
  :name "(- (+ (* (* x y) z) (* x y)) (/ (* z y) x))"
  :precision binary64
  (- (+ (* (* x y) z) (* x y)) (/ (* z y) x)))