Average Error: 1.7 → 0.1
Time: 11.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 1.9019857839603548 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 1.9019857839603548 \cdot 10^{-49}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{y} \cdot x\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r943402 = x;
        double r943403 = 4.0;
        double r943404 = r943402 + r943403;
        double r943405 = y;
        double r943406 = r943404 / r943405;
        double r943407 = r943402 / r943405;
        double r943408 = z;
        double r943409 = r943407 * r943408;
        double r943410 = r943406 - r943409;
        double r943411 = fabs(r943410);
        return r943411;
}
↓
double f(double x, double y, double z) {
        double r943412 = 4.0;
        double r943413 = x;
        double r943414 = r943412 + r943413;
        double r943415 = y;
        double r943416 = r943414 / r943415;
        double r943417 = r943413 / r943415;
        double r943418 = z;
        double r943419 = r943417 * r943418;
        double r943420 = r943416 - r943419;
        double r943421 = fabs(r943420);
        double r943422 = 1.9019857839603548e-49;
        bool r943423 = r943421 <= r943422;
        double r943424 = r943418 / r943415;
        double r943425 = r943424 * r943413;
        double r943426 = r943416 - r943425;
        double r943427 = fabs(r943426);
        double r943428 = r943423 ? r943427 : r943421;
        return r943428;
}

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 (fabs (- (/ (+ x 4) y) (* (/ x y) z))) < 1.9019857839603548e-49

    1. Initial program 5.0

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

    3. Applied div-inv5.1

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

    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

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

    if 1.9019857839603548e-49 < (fabs (- (/ (+ x 4) y) (* (/ x y) z)))

    1. Initial program 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 1.9019857839603548 \cdot 10^{-49}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))