Average Error: 1.5 → 0.8
Time: 3.7s
Precision: binary64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -1.7752144640494725 \cdot 10^{+134} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 5.586231394031493 \cdot 10^{+200}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -1.7752144640494725 \cdot 10^{+134} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 5.586231394031493 \cdot 10^{+200}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\end{array}
(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= (- (/ (+ x 4.0) y) (* (/ x y) z)) -1.7752144640494725e+134)
         (not (<= (- (/ (+ x 4.0) y) (* (/ x y) z)) 5.586231394031493e+200)))
   (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))
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 + 4.0) / y) - ((x / y) * z)) <= -1.7752144640494725e+134) || !((((x + 4.0) / y) - ((x / y) * z)) <= 5.586231394031493e+200)) {
		tmp = fabs(((x + 4.0) / y) - ((x / y) * z));
	} else {
		tmp = fabs(((x + 4.0) / y) - (x * (z / y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < -1.7752144640494725e134 or 5.58623139403149307e200 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

    1. Initial program 0.1

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

    if -1.7752144640494725e134 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 5.58623139403149307e200

    1. Initial program 2.1

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

    3. Applied div-inv_binary64_752.1

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

    4. Applied associate-*l*_binary64_191.1

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

    5. Simplified1.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -1.7752144640494725 \cdot 10^{+134} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 5.586231394031493 \cdot 10^{+200}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))