fabs fraction 1

Average Error: 1.8 → 0.6

Time: 2.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3589800915338266 \cdot 10^{122} \lor \neg \left(x \le 3.28172621284575739 \cdot 10^{-81}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \end{array}\]

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 <= -1.3589800915338266e+122) || !(x <= 3.2817262128457574e-81))) {
		VAR = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (x * ((double) (z / y))))))));
	} else {
		VAR = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (((double) (x * z)) * ((double) (1.0 / 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 < -1.3589800915338266e+122 or 3.2817262128457574e-81 < x

   1. Initial program 0.3

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

   3. Applied div-inv 0.4

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

   4. Applied associate-*l* 0.4

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

   5. Simplified 0.3

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

   if -1.3589800915338266e+122 < x < 3.2817262128457574e-81

   1. Initial program 2.7

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

   3. Applied div-inv 2.7

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

   4. Applied associate-*l* 5.0

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

   5. Simplified 5.0

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

   7. Applied div-inv 5.0

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

   8. Applied associate-*r* 0.8

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot z\right) \cdot \frac{1}{y}}\right|\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3589800915338266 \cdot 10^{122} \lor \neg \left(x \le 3.28172621284575739 \cdot 10^{-81}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))