Average Error: 1.5 → 0.4
Time: 14.1s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\frac{4}{y} + \left(\frac{x}{y} - z \cdot \frac{x}{y}\right)\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \left(\frac{x}{y} - z \cdot \frac{x}{y}\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -16246640829.1726360321044921875:\\
\;\;\;\;\left|\frac{4}{y} + \left(\frac{x}{y} - z \cdot \frac{x}{y}\right)\right|\\

\mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1015302 = x;
        double r1015303 = 4.0;
        double r1015304 = r1015302 + r1015303;
        double r1015305 = y;
        double r1015306 = r1015304 / r1015305;
        double r1015307 = r1015302 / r1015305;
        double r1015308 = z;
        double r1015309 = r1015307 * r1015308;
        double r1015310 = r1015306 - r1015309;
        double r1015311 = fabs(r1015310);
        return r1015311;
}
↓
double f(double x, double y, double z) {
        double r1015312 = x;
        double r1015313 = -16246640829.172636;
        bool r1015314 = r1015312 <= r1015313;
        double r1015315 = 4.0;
        double r1015316 = y;
        double r1015317 = r1015315 / r1015316;
        double r1015318 = r1015312 / r1015316;
        double r1015319 = z;
        double r1015320 = r1015319 * r1015318;
        double r1015321 = r1015318 - r1015320;
        double r1015322 = r1015317 + r1015321;
        double r1015323 = fabs(r1015322);
        double r1015324 = 1.0296033599100243e-161;
        bool r1015325 = r1015312 <= r1015324;
        double r1015326 = r1015315 + r1015312;
        double r1015327 = r1015319 * r1015312;
        double r1015328 = r1015326 - r1015327;
        double r1015329 = r1015328 / r1015316;
        double r1015330 = fabs(r1015329);
        double r1015331 = r1015325 ? r1015330 : r1015323;
        double r1015332 = r1015314 ? r1015323 : r1015331;
        return r1015332;
}

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 < -16246640829.172636 or 1.0296033599100243e-161 < x

    1. Initial program 0.7

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

    2. Taylor expanded around 0 6.2

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

    3. Simplified0.7

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

    if -16246640829.172636 < x < 1.0296033599100243e-161

    1. Initial program 2.5

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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-div0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\frac{4}{y} + \left(\frac{x}{y} - z \cdot \frac{x}{y}\right)\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + \left(\frac{x}{y} - z \cdot \frac{x}{y}\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))