Average Error: 1.8 → 0.3
Time: 10.5s
Precision: binary64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0327798710615128 \cdot 10^{+36} \lor \neg \left(x \leq 3.292944409666188 \cdot 10^{-105}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0327798710615128 \cdot 10^{+36} \lor \neg \left(x \leq 3.292944409666188 \cdot 10^{-105}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot 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 -1.0327798710615128e+36) (not (<= x 3.292944409666188e-105)))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))
   (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 <= -1.0327798710615128e+36) || !(x <= 3.292944409666188e-105)) {
		tmp = fabs(((x + 4.0) / y) - (x * (z / y)));
	} 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 x < -1.0327798710615128e36 or 3.29294440966618795e-105 < x

    1. Initial program 0.6

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

    3. Applied div-inv_binary64_750.6

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

    4. Applied associate-*l*_binary64_190.6

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

    5. Simplified0.6

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

    if -1.0327798710615128e36 < x < 3.29294440966618795e-105

    1. Initial program 2.8

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

    2. Taylor expanded around 0 0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0327798710615128 \cdot 10^{+36} \lor \neg \left(x \leq 3.292944409666188 \cdot 10^{-105}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))