Average Error: 1.7 → 0.2
Time: 4.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -274859649089261516357632 \lor \neg \left(x \le 1.086562483217625862476281364036535503525 \cdot 10^{-64}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot \frac{z}{y}}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -274859649089261516357632 \lor \neg \left(x \le 1.086562483217625862476281364036535503525 \cdot 10^{-64}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot \frac{z}{y}}{1}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r38329 = x;
        double r38330 = 4.0;
        double r38331 = r38329 + r38330;
        double r38332 = y;
        double r38333 = r38331 / r38332;
        double r38334 = r38329 / r38332;
        double r38335 = z;
        double r38336 = r38334 * r38335;
        double r38337 = r38333 - r38336;
        double r38338 = fabs(r38337);
        return r38338;
}
↓
double f(double x, double y, double z) {
        double r38339 = x;
        double r38340 = -2.748596490892615e+23;
        bool r38341 = r38339 <= r38340;
        double r38342 = 1.0865624832176259e-64;
        bool r38343 = r38339 <= r38342;
        double r38344 = !r38343;
        bool r38345 = r38341 || r38344;
        double r38346 = 4.0;
        double r38347 = r38339 + r38346;
        double r38348 = y;
        double r38349 = r38347 / r38348;
        double r38350 = z;
        double r38351 = r38350 / r38348;
        double r38352 = r38339 * r38351;
        double r38353 = 1.0;
        double r38354 = r38352 / r38353;
        double r38355 = r38349 - r38354;
        double r38356 = fabs(r38355);
        double r38357 = r38339 * r38350;
        double r38358 = r38347 - r38357;
        double r38359 = r38358 / r38348;
        double r38360 = fabs(r38359);
        double r38361 = r38345 ? r38356 : r38360;
        return r38361;
}

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 < -2.748596490892615e+23 or 1.0865624832176259e-64 < x

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied add-cube-cbrt0.7

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

    5. Applied times-frac0.7

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

    6. Applied associate-*l*0.7

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

    8. Applied associate-*l/0.7

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

    9. Simplified0.3

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

    if -2.748596490892615e+23 < x < 1.0865624832176259e-64

    1. Initial program 2.8

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -274859649089261516357632 \lor \neg \left(x \le 1.086562483217625862476281364036535503525 \cdot 10^{-64}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot \frac{z}{y}}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))