Average Error: 1.6 → 0.8
Time: 3.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 249476286.057907402515411376953125:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le 249476286.057907402515411376953125:\\
\;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\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 r27244 = x;
        double r27245 = 4.0;
        double r27246 = r27244 + r27245;
        double r27247 = y;
        double r27248 = r27246 / r27247;
        double r27249 = r27244 / r27247;
        double r27250 = z;
        double r27251 = r27249 * r27250;
        double r27252 = r27248 - r27251;
        double r27253 = fabs(r27252);
        return r27253;
}
↓
double f(double x, double y, double z) {
        double r27254 = y;
        double r27255 = 249476286.0579074;
        bool r27256 = r27254 <= r27255;
        double r27257 = x;
        double r27258 = r27257 / r27254;
        double r27259 = 1.0;
        double r27260 = z;
        double r27261 = r27259 - r27260;
        double r27262 = r27258 * r27261;
        double r27263 = 4.0;
        double r27264 = r27259 / r27254;
        double r27265 = r27263 * r27264;
        double r27266 = r27262 + r27265;
        double r27267 = fabs(r27266);
        double r27268 = r27257 + r27263;
        double r27269 = r27268 / r27254;
        double r27270 = r27260 / r27254;
        double r27271 = r27257 * r27270;
        double r27272 = r27269 - r27271;
        double r27273 = fabs(r27272);
        double r27274 = r27256 ? r27267 : r27273;
        return r27274;
}

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 y < 249476286.0579074

    1. Initial program 1.1

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

    2. Taylor expanded around 0 2.3

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

    3. Simplified1.1

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

    if 249476286.0579074 < y

    1. Initial program 2.6

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

    3. Applied div-inv2.6

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

    4. Applied associate-*l*0.1

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

    5. Simplified0.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 249476286.057907402515411376953125:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))