Average Error: 1.7 → 0.3
Time: 2.7s
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 r27320 = x;
        double r27321 = 4.0;
        double r27322 = r27320 + r27321;
        double r27323 = y;
        double r27324 = r27322 / r27323;
        double r27325 = r27320 / r27323;
        double r27326 = z;
        double r27327 = r27325 * r27326;
        double r27328 = r27324 - r27327;
        double r27329 = fabs(r27328);
        return r27329;
}
↓
double f(double x, double y, double z) {
        double r27330 = x;
        double r27331 = 4.0;
        double r27332 = r27330 + r27331;
        double r27333 = y;
        double r27334 = r27332 / r27333;
        double r27335 = r27330 / r27333;
        double r27336 = z;
        double r27337 = r27335 * r27336;
        double r27338 = r27334 - r27337;
        double r27339 = -5.608942803594382e+23;
        bool r27340 = r27338 <= r27339;
        double r27341 = 3.5059797311862954e-146;
        bool r27342 = r27338 <= r27341;
        double r27343 = !r27342;
        bool r27344 = r27340 || r27343;
        double r27345 = fabs(r27338);
        double r27346 = r27336 / r27333;
        double r27347 = r27330 * r27346;
        double r27348 = r27334 - r27347;
        double r27349 = fabs(r27348);
        double r27350 = r27344 ? r27345 : r27349;
        return r27350;
}

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 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))