Average Error: 1.5 → 1.9
Time: 26.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 2.7430283332273266 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(z \cdot \sqrt[3]{\frac{1}{y}}\right) \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le 2.7430283332273266 \cdot 10^{-83}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(z \cdot \sqrt[3]{\frac{1}{y}}\right) \cdot x\right)\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1344556 = x;
        double r1344557 = 4.0;
        double r1344558 = r1344556 + r1344557;
        double r1344559 = y;
        double r1344560 = r1344558 / r1344559;
        double r1344561 = r1344556 / r1344559;
        double r1344562 = z;
        double r1344563 = r1344561 * r1344562;
        double r1344564 = r1344560 - r1344563;
        double r1344565 = fabs(r1344564);
        return r1344565;
}
↓
double f(double x, double y, double z) {
        double r1344566 = y;
        double r1344567 = 2.7430283332273266e-83;
        bool r1344568 = r1344566 <= r1344567;
        double r1344569 = x;
        double r1344570 = 4.0;
        double r1344571 = r1344569 + r1344570;
        double r1344572 = r1344571 / r1344566;
        double r1344573 = 1.0;
        double r1344574 = cbrt(r1344566);
        double r1344575 = r1344574 * r1344574;
        double r1344576 = r1344573 / r1344575;
        double r1344577 = z;
        double r1344578 = r1344573 / r1344566;
        double r1344579 = cbrt(r1344578);
        double r1344580 = r1344577 * r1344579;
        double r1344581 = r1344580 * r1344569;
        double r1344582 = r1344576 * r1344581;
        double r1344583 = r1344572 - r1344582;
        double r1344584 = fabs(r1344583);
        double r1344585 = r1344577 / r1344566;
        double r1344586 = r1344569 * r1344585;
        double r1344587 = r1344572 - r1344586;
        double r1344588 = fabs(r1344587);
        double r1344589 = r1344568 ? r1344584 : r1344588;
        return r1344589;
}

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 < 2.7430283332273266e-83

    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-frac1.6

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

    6. Applied associate-*l*1.8

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

    7. Taylor expanded around inf 48.9

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

    8. Simplified2.7

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

    if 2.7430283332273266e-83 < y

    1. Initial program 1.9

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

    3. Applied div-inv2.0

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

    4. Applied associate-*l*0.6

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

    5. Simplified0.6

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 2.7430283332273266 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(z \cdot \sqrt[3]{\frac{1}{y}}\right) \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))