fabs fraction 1

Average Error: 1.7 → 0.1

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5562.218186525574:\\ \;\;\;\;\left|\frac{x + \left(4 - x \cdot z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) fabs(((double) ((((double) (x + 4.0)) / y) - ((double) ((x / y) * z)))))) <= 5562.218186525574)) {
		VAR = ((double) fabs((((double) (x + ((double) (4.0 - ((double) (x * z)))))) / y)));
	} else {
		VAR = ((double) fabs(((double) ((((double) (x + 4.0)) / y) - ((double) ((x / y) * z))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 2 regimes

2. if (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) < 5562.21818652557431

   1. Initial program 4.0

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

   2. Simplified 0.1

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

   if 5562.21818652557431 < (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))

   1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5562.218186525574:\\ \;\;\;\;\left|\frac{x + \left(4 - x \cdot z\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))