fabs fraction 1

Average Error: 1.4 → 2.3

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 4.2510798617965315 \cdot 10^{-152}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

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 ((y <= 4.2510798617965315e-152)) {
		VAR = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (((double) (x * z)) * ((double) (1.0 / y))))))));
	} else {
		VAR = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (x / ((double) (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 y < 4.2510798617965315e-152

   1. Initial program 1.2

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

   3. Applied associate-*l/ 3.0

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

   5. Applied div-inv 3.0

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

   if 4.2510798617965315e-152 < y

   1. Initial program 1.7

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

   3. Applied associate-*l/ 4.1

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

   5. Applied associate-/l* 1.2

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 4.2510798617965315 \cdot 10^{-152}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))