Average Error: 1.6 → 0.2
Time: 19.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.269126464312056 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 2.30898552077866 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.269126464312056 \cdot 10^{+79}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 2.30898552077866 \cdot 10^{-11}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1155676 = x;
        double r1155677 = 4.0;
        double r1155678 = r1155676 + r1155677;
        double r1155679 = y;
        double r1155680 = r1155678 / r1155679;
        double r1155681 = r1155676 / r1155679;
        double r1155682 = z;
        double r1155683 = r1155681 * r1155682;
        double r1155684 = r1155680 - r1155683;
        double r1155685 = fabs(r1155684);
        return r1155685;
}
↓
double f(double x, double y, double z) {
        double r1155686 = x;
        double r1155687 = -4.269126464312056e+79;
        bool r1155688 = r1155686 <= r1155687;
        double r1155689 = 4.0;
        double r1155690 = r1155689 + r1155686;
        double r1155691 = y;
        double r1155692 = r1155690 / r1155691;
        double r1155693 = z;
        double r1155694 = r1155693 / r1155691;
        double r1155695 = r1155686 * r1155694;
        double r1155696 = r1155692 - r1155695;
        double r1155697 = fabs(r1155696);
        double r1155698 = 2.30898552077866e-11;
        bool r1155699 = r1155686 <= r1155698;
        double r1155700 = r1155693 * r1155686;
        double r1155701 = r1155700 / r1155691;
        double r1155702 = r1155692 - r1155701;
        double r1155703 = fabs(r1155702);
        double r1155704 = r1155699 ? r1155703 : r1155697;
        double r1155705 = r1155688 ? r1155697 : r1155704;
        return r1155705;
}

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 < -4.269126464312056e+79 or 2.30898552077866e-11 < x

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.6

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

    4. Applied *-un-lft-identity0.6

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

    5. Applied times-frac0.6

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

    6. Applied associate-*l*5.8

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

    7. Taylor expanded around inf 9.8

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

    9. Applied *-un-lft-identity9.8

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

    10. Applied times-frac0.1

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

    11. Simplified0.1

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

    if -4.269126464312056e+79 < x < 2.30898552077866e-11

    1. Initial program 2.4

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

    3. Applied add-cube-cbrt2.6

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

    4. Applied *-un-lft-identity2.6

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

    5. Applied times-frac2.6

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

    6. Applied associate-*l*0.5

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

    7. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.269126464312056 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 2.30898552077866 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))