fabs fraction 1

Average Error: 1.7 → 0.1

Time: 2.9s

Precision: binary64

Math

\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.80615219986552692 \cdot 10^{-4}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 3.91743995991523077 \cdot 10^{31}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (((double) (x / y)) * z))))));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if ((x <= -0.0002806152199865527)) {
		VAR = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (x * ((double) (z / y))))))));
	} else {
		double VAR_1;
		if ((x <= 3.9174399599152308e+31)) {
			VAR_1 = ((double) fabs(((double) (((double) (((double) (x + 4.0)) - ((double) (x * z)))) / y))));
		} else {
			VAR_1 = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (((double) (x / y)) * z))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 3 regimes

2. if x < -2.80615219986552692e-4

   1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   2. Using strategy rm

   3. Applied div-inv 0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]

   4. Applied associate-*l* 0.2

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]

   5. Simplified 0.1

      \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]

   if -2.80615219986552692e-4 < x < 3.91743995991523077e31

   1. Initial program 2.6

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   2. Using strategy rm

   3. Applied associate-*l/ 0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]

   4. Applied sub-div 0.1

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]

   if 3.91743995991523077e31 < x

   1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.80615219986552692 \cdot 10^{-4}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 3.91743995991523077 \cdot 10^{31}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))