Average Error: 1.5 → 0.2
Time: 3.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -1.41012956374722444 \cdot 10^{118} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 7.6391539351210826 \cdot 10^{58}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\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 -1.41012956374722444 \cdot 10^{118} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 7.6391539351210826 \cdot 10^{58}\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 r31196 = x;
        double r31197 = 4.0;
        double r31198 = r31196 + r31197;
        double r31199 = y;
        double r31200 = r31198 / r31199;
        double r31201 = r31196 / r31199;
        double r31202 = z;
        double r31203 = r31201 * r31202;
        double r31204 = r31200 - r31203;
        double r31205 = fabs(r31204);
        return r31205;
}
↓
double f(double x, double y, double z) {
        double r31206 = x;
        double r31207 = 4.0;
        double r31208 = r31206 + r31207;
        double r31209 = y;
        double r31210 = r31208 / r31209;
        double r31211 = r31206 / r31209;
        double r31212 = z;
        double r31213 = r31211 * r31212;
        double r31214 = r31210 - r31213;
        double r31215 = -1.4101295637472244e+118;
        bool r31216 = r31214 <= r31215;
        double r31217 = 7.639153935121083e+58;
        bool r31218 = r31214 <= r31217;
        double r31219 = !r31218;
        bool r31220 = r31216 || r31219;
        double r31221 = fabs(r31214);
        double r31222 = r31212 / r31209;
        double r31223 = r31206 * r31222;
        double r31224 = r31210 - r31223;
        double r31225 = fabs(r31224);
        double r31226 = r31220 ? r31221 : r31225;
        return r31226;
}

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)) < -1.4101295637472244e+118 or 7.639153935121083e+58 < (- (/ (+ x 4.0) y) (* (/ x y) z))

    1. Initial program 0.1

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

    if -1.4101295637472244e+118 < (- (/ (+ x 4.0) y) (* (/ x y) z)) < 7.639153935121083e+58

    1. Initial program 2.5

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

    3. Applied div-inv2.6

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

    4. Applied associate-*l*0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -1.41012956374722444 \cdot 10^{118} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 7.6391539351210826 \cdot 10^{58}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))