fabs fraction 1

Average Error: 1.6 → 0.1

Time: 7.1min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.58962761951495524 \cdot 10^{-18} \lor \neg \left(x \le 1.04021454216689316 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]

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 (((x <= -2.5896276195149552e-18) || !(x <= 1.0402145421668932e-21))) {
		VAR = ((double) fabs(((double) ((((double) (x + 4.0)) / y) - ((double) (x * (z / y)))))));
	} else {
		VAR = ((double) fabs(((double) ((((double) (x + 4.0)) / y) - (((double) (x * z)) / y)))));
	}
	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 < -2.58962761951495524e-18 or 1.04021454216689316e-21 < x

   1. Initial program 0.2

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

   3. Applied div-inv 0.2

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

   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 0.2

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

   if -2.58962761951495524e-18 < x < 1.04021454216689316e-21

   1. Initial program 2.7

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

   3. Applied associate-*l/ 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.58962761951495524 \cdot 10^{-18} \lor \neg \left(x \le 1.04021454216689316 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))