fabs fraction 1

Average Error: 1.7 → 0.6

Time: 3.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.5667262830596557 \cdot 10^{-128} \lor \neg \left(x \le 4.10438320495437101 \cdot 10^{-107}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(\left(x + 4\right) - z \cdot x\right)\right|\\ \end{array}\]

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 <= -1.5667262830596557e-128) || !(x <= 4.104383204954371e-107))) {
		VAR = ((double) fabs(((double) (((double) (((double) (x + 4.0)) / y)) - ((double) (z / ((double) (y / x))))))));
	} else {
		VAR = ((double) fabs(((double) (((double) (1.0 / y)) * ((double) (((double) (x + 4.0)) - ((double) (z * x))))))));
	}
	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 < -1.5667262830596557e-128 or 4.104383204954371e-107 < x

   1. Initial program 0.8

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

   3. Applied clear-num 1.0

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

   4. Applied associate-*l/ 0.9

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

   5. Simplified 0.9

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

   if -1.5667262830596557e-128 < x < 4.104383204954371e-107

   1. Initial program 3.0

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

   3. Applied clear-num 3.2

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

   4. Applied associate-*l/ 3.2

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

   5. Simplified 3.2

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

   7. Applied clear-num 3.2

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

   8. Applied associate-/r/ 3.0

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

   9. Simplified 3.0

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

   11. Applied div-inv 3.1

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

   12. Applied associate-*r* 0.1

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

   13. Applied div-inv 0.1

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

   14. Applied distribute-rgt-out-- 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5667262830596557 \cdot 10^{-128} \lor \neg \left(x \le 4.10438320495437101 \cdot 10^{-107}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(\left(x + 4\right) - z \cdot x\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))