Average Error: 1.6 → 1.2
Time: 18.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.720264343127334 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{x}{\sqrt[3]{y}}\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.720264343127334 \cdot 10^{-40}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{y} \cdot x\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{x}{\sqrt[3]{y}}\right)\right|\\

\end{array}
double f(double x, double y, double z) {
        double r890717 = x;
        double r890718 = 4.0;
        double r890719 = r890717 + r890718;
        double r890720 = y;
        double r890721 = r890719 / r890720;
        double r890722 = r890717 / r890720;
        double r890723 = z;
        double r890724 = r890722 * r890723;
        double r890725 = r890721 - r890724;
        double r890726 = fabs(r890725);
        return r890726;
}
↓
double f(double x, double y, double z) {
        double r890727 = y;
        double r890728 = -2.720264343127334e-40;
        bool r890729 = r890727 <= r890728;
        double r890730 = x;
        double r890731 = 4.0;
        double r890732 = r890730 + r890731;
        double r890733 = r890732 / r890727;
        double r890734 = z;
        double r890735 = r890734 / r890727;
        double r890736 = r890735 * r890730;
        double r890737 = r890733 - r890736;
        double r890738 = fabs(r890737);
        double r890739 = 1.0;
        double r890740 = cbrt(r890727);
        double r890741 = r890740 * r890740;
        double r890742 = r890739 / r890741;
        double r890743 = r890730 / r890740;
        double r890744 = r890734 * r890743;
        double r890745 = r890742 * r890744;
        double r890746 = r890733 - r890745;
        double r890747 = fabs(r890746);
        double r890748 = r890729 ? r890738 : r890747;
        return r890748;
}

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 y < -2.720264343127334e-40

    1. Initial program 2.3

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

    3. Applied div-inv2.3

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

    4. Applied associate-*l*0.2

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

    5. Simplified0.2

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

    if -2.720264343127334e-40 < y

    1. Initial program 1.3

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

    3. Applied add-cube-cbrt1.5

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

    4. Applied *-un-lft-identity1.5

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

    5. Applied times-frac1.5

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

    6. Applied associate-*l*1.7

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{x}{\sqrt[3]{y}} \cdot z\right)}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.720264343127334 \cdot 10^{-40}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{x}{\sqrt[3]{y}}\right)\right|\\ \end{array}\]

Reproduce

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