fabs fraction 1

Average Error: 1.6 → 0.1

Time: 6.4s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -548550455324647.94 \lor \neg \left(x \leq 1.6983471831252163 \cdot 10^{+25}\right):\\ \;\;\;\;\left|\frac{x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -548550455324647.94) (not (<= x 1.6983471831252163e+25)))
   (fabs (- (/ x y) (* (/ x y) z)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -548550455324647.94) || !(x <= 1.6983471831252163e+25)) {
		tmp = fabs((x / y) - ((x / y) * z));
	} else {
		tmp = fabs(((x + 4.0) - (x * z)) / y);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Alternatives

Alternative 1
Error	2.3
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -13091502411.24689 \lor \neg \left(z \leq 2.5687676999185847\right):\\ \;\;\;\;\left|\frac{4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Alternative 2
Error	9.2
Cost	7304

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934540579464493 \cdot 10^{-34} \lor \neg \left(x \leq 7.633301677712175 \cdot 10^{-32}\right):\\ \;\;\;\;\left|\frac{x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Alternative 3
Error	9.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3934540579464493 \cdot 10^{-34} \lor \neg \left(x \leq 7.633301677712175 \cdot 10^{-32}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Alternative 4
Error	11.4
Cost	7818

\[\begin{array}{l} \mathbf{if}\;z \leq -4.590732991688508 \cdot 10^{+151}:\\ \;\;\;\;\left|-\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq -6.706000536155162 \cdot 10^{+142}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq -1.2839354414049617 \cdot 10^{+48} \lor \neg \left(z \leq 4.8117962760082794 \cdot 10^{+42}\right):\\ \;\;\;\;\left|-\frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Alternative 5
Error	23.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -7.687075550799982 \cdot 10^{-36} \lor \neg \left(x \leq 3.1747198910273813 \cdot 10^{-18}\right):\\ \;\;\;\;\left|-\frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array}\]

Alternative 6
Error	33.0
Cost	6592

\[\left|\frac{4}{y}\right|\]

Alternative 7
Error	60.5
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if x < -548550455324647.938 or 1.6983471831252163e25 < x

   1. Initial program 0.1

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

   if -548550455324647.938 < x < 1.6983471831252163e25

   1. Initial program 2.5

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

   2. Simplified 0.1

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

   3. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -548550455324647.94 \lor \neg \left(x \leq 1.6983471831252163 \cdot 10^{+25}\right):\\ \;\;\;\;\left|\frac{x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))