Average Error: 1.7 → 0.2
Time: 3.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -6.23403900926287088 \cdot 10^{81} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 2.88731983193916299 \cdot 10^{143}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\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 \le -6.23403900926287088 \cdot 10^{81} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 2.88731983193916299 \cdot 10^{143}\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}
double f(double x, double y, double z) {
        double r29242 = x;
        double r29243 = 4.0;
        double r29244 = r29242 + r29243;
        double r29245 = y;
        double r29246 = r29244 / r29245;
        double r29247 = r29242 / r29245;
        double r29248 = z;
        double r29249 = r29247 * r29248;
        double r29250 = r29246 - r29249;
        double r29251 = fabs(r29250);
        return r29251;
}
↓
double f(double x, double y, double z) {
        double r29252 = x;
        double r29253 = 4.0;
        double r29254 = r29252 + r29253;
        double r29255 = y;
        double r29256 = r29254 / r29255;
        double r29257 = r29252 / r29255;
        double r29258 = z;
        double r29259 = r29257 * r29258;
        double r29260 = r29256 - r29259;
        double r29261 = -6.234039009262871e+81;
        bool r29262 = r29260 <= r29261;
        double r29263 = 2.887319831939163e+143;
        bool r29264 = r29260 <= r29263;
        double r29265 = !r29264;
        bool r29266 = r29262 || r29265;
        double r29267 = fabs(r29260);
        double r29268 = r29258 / r29255;
        double r29269 = r29252 * r29268;
        double r29270 = r29256 - r29269;
        double r29271 = fabs(r29270);
        double r29272 = r29266 ? r29267 : r29271;
        return r29272;
}

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 4.0) y) (* (/ x y) z)) < -6.234039009262871e+81 or 2.887319831939163e+143 < (- (/ (+ x 4.0) y) (* (/ x y) z))

    1. Initial program 0.1

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

    if -6.234039009262871e+81 < (- (/ (+ x 4.0) y) (* (/ x y) z)) < 2.887319831939163e+143

    1. Initial program 2.7

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

    3. Applied div-inv2.7

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

    4. Applied associate-*l*0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -6.23403900926287088 \cdot 10^{81} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 2.88731983193916299 \cdot 10^{143}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))