Average Error: 1.7 → 0.3
Time: 4.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -5.608942803594382 \cdot 10^{23} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 3.5059797311862954 \cdot 10^{-146}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -5.608942803594382 \cdot 10^{23} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 3.5059797311862954 \cdot 10^{-146}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\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 r34425 = x;
        double r34426 = 4.0;
        double r34427 = r34425 + r34426;
        double r34428 = y;
        double r34429 = r34427 / r34428;
        double r34430 = r34425 / r34428;
        double r34431 = z;
        double r34432 = r34430 * r34431;
        double r34433 = r34429 - r34432;
        double r34434 = fabs(r34433);
        return r34434;
}
↓
double f(double x, double y, double z) {
        double r34435 = x;
        double r34436 = 4.0;
        double r34437 = r34435 + r34436;
        double r34438 = y;
        double r34439 = r34437 / r34438;
        double r34440 = r34435 / r34438;
        double r34441 = z;
        double r34442 = r34440 * r34441;
        double r34443 = r34439 - r34442;
        double r34444 = -5.608942803594382e+23;
        bool r34445 = r34443 <= r34444;
        double r34446 = 3.5059797311862954e-146;
        bool r34447 = r34443 <= r34446;
        double r34448 = !r34447;
        bool r34449 = r34445 || r34448;
        double r34450 = fabs(r34443);
        double r34451 = r34441 / r34438;
        double r34452 = r34435 * r34451;
        double r34453 = r34439 - r34452;
        double r34454 = fabs(r34453);
        double r34455 = r34449 ? r34450 : r34454;
        return r34455;
}

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 4.0) y) (* (/ x y) z)) < -5.608942803594382e+23 or 3.5059797311862954e-146 < (- (/ (+ x 4.0) y) (* (/ x y) z))

    1. Initial program 0.4

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

    3. Applied associate-*l/5.2

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

    5. Applied associate-/l*4.4

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

    7. Applied associate-/r/0.4

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

    if -5.608942803594382e+23 < (- (/ (+ x 4.0) y) (* (/ x y) z)) < 3.5059797311862954e-146

    1. Initial program 4.6

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

    3. Applied div-inv4.6

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

    4. Applied associate-*l*0.1

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

    5. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -5.608942803594382 \cdot 10^{23} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 3.5059797311862954 \cdot 10^{-146}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))