Average Error: 1.6 → 0.1
Time: 4.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 988.771363812410641:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot \frac{z}{y}}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 988.771363812410641:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot \frac{z}{y}}{1}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r36582 = x;
        double r36583 = 4.0;
        double r36584 = r36582 + r36583;
        double r36585 = y;
        double r36586 = r36584 / r36585;
        double r36587 = r36582 / r36585;
        double r36588 = z;
        double r36589 = r36587 * r36588;
        double r36590 = r36586 - r36589;
        double r36591 = fabs(r36590);
        return r36591;
}
↓
double f(double x, double y, double z) {
        double r36592 = x;
        double r36593 = 4.0;
        double r36594 = r36592 + r36593;
        double r36595 = y;
        double r36596 = r36594 / r36595;
        double r36597 = r36592 / r36595;
        double r36598 = z;
        double r36599 = r36597 * r36598;
        double r36600 = r36596 - r36599;
        double r36601 = fabs(r36600);
        double r36602 = 988.7713638124106;
        bool r36603 = r36601 <= r36602;
        double r36604 = r36598 / r36595;
        double r36605 = r36592 * r36604;
        double r36606 = 1.0;
        double r36607 = r36605 / r36606;
        double r36608 = r36596 - r36607;
        double r36609 = fabs(r36608);
        double r36610 = r36603 ? r36609 : r36601;
        return r36610;
}

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 4.0) y) (* (/ x y) z))) < 988.7713638124106

    1. Initial program 3.8

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

    3. Applied *-un-lft-identity3.8

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

    4. Applied add-cube-cbrt4.0

      \[\leadsto \left|\frac{x + 4}{y} - \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot y} \cdot z\right|\]

    5. Applied times-frac4.0

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{y}\right)} \cdot z\right|\]

    6. Applied associate-*l*1.2

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \left(\frac{\sqrt[3]{x}}{y} \cdot z\right)}\right|\]
    7. Using strategy rm

    8. Applied associate-*l/1.2

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{y} \cdot z\right)}{1}}\right|\]

    9. Simplified0.1

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

    if 988.7713638124106 < (fabs (- (/ (+ x 4.0) 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 988.771363812410641:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot \frac{z}{y}}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))