Average Error: 1.7 → 0.4
Time: 3.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.4733731582962843 \cdot 10^{-74}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 6.7358812145801762 \cdot 10^{-103}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-\left(x + 4\right)}{-y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.4733731582962843 \cdot 10^{-74}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 6.7358812145801762 \cdot 10^{-103}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

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

\end{array}
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 temp;
	if ((x <= -1.4733731582962843e-74)) {
		temp = fabs((((x + 4.0) / y) - (x * (z / y))));
	} else {
		double temp_1;
		if ((x <= 6.735881214580176e-103)) {
			temp_1 = fabs((((x + 4.0) - (x * z)) / y));
		} else {
			temp_1 = fabs(((-(x + 4.0) / -y) - ((x / y) * z)));
		}
		temp = temp_1;
	}
	return temp;
}

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 3 regimes

  2. if x < -1.4733731582962843e-74

    1. Initial program 0.5

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

    3. Applied div-inv0.6

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

    4. Applied associate-*l*0.7

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

    5. Simplified0.7

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

    if -1.4733731582962843e-74 < x < 6.735881214580176e-103

    1. Initial program 3.1

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.0

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

    if 6.735881214580176e-103 < x

    1. Initial program 0.6

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

    3. Applied frac-2neg0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4733731582962843 \cdot 10^{-74}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 6.7358812145801762 \cdot 10^{-103}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-\left(x + 4\right)}{-y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))