Average Error: 1.6 → 0.1
Time: 4.4s
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 r35749 = x;
        double r35750 = 4.0;
        double r35751 = r35749 + r35750;
        double r35752 = y;
        double r35753 = r35751 / r35752;
        double r35754 = r35749 / r35752;
        double r35755 = z;
        double r35756 = r35754 * r35755;
        double r35757 = r35753 - r35756;
        double r35758 = fabs(r35757);
        return r35758;
}
↓
double f(double x, double y, double z) {
        double r35759 = x;
        double r35760 = 4.0;
        double r35761 = r35759 + r35760;
        double r35762 = y;
        double r35763 = r35761 / r35762;
        double r35764 = r35759 / r35762;
        double r35765 = z;
        double r35766 = r35764 * r35765;
        double r35767 = r35763 - r35766;
        double r35768 = fabs(r35767);
        double r35769 = 988.7713638124106;
        bool r35770 = r35768 <= r35769;
        double r35771 = r35765 / r35762;
        double r35772 = r35759 * r35771;
        double r35773 = 1.0;
        double r35774 = r35772 / r35773;
        double r35775 = r35763 - r35774;
        double r35776 = fabs(r35775);
        double r35777 = r35770 ? r35776 : r35768;
        return r35777;
}

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