Average Error: 1.5 → 0.2
Time: 3.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -23494568.222248383 \lor \neg \left(x \le 7.80513442966517965 \cdot 10^{-43}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -23494568.222248383 \lor \neg \left(x \le 7.80513442966517965 \cdot 10^{-43}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\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 r31129 = x;
        double r31130 = 4.0;
        double r31131 = r31129 + r31130;
        double r31132 = y;
        double r31133 = r31131 / r31132;
        double r31134 = r31129 / r31132;
        double r31135 = z;
        double r31136 = r31134 * r31135;
        double r31137 = r31133 - r31136;
        double r31138 = fabs(r31137);
        return r31138;
}
↓
double f(double x, double y, double z) {
        double r31139 = x;
        double r31140 = -23494568.222248383;
        bool r31141 = r31139 <= r31140;
        double r31142 = 7.80513442966518e-43;
        bool r31143 = r31139 <= r31142;
        double r31144 = !r31143;
        bool r31145 = r31141 || r31144;
        double r31146 = 4.0;
        double r31147 = r31139 + r31146;
        double r31148 = y;
        double r31149 = r31147 / r31148;
        double r31150 = z;
        double r31151 = r31150 / r31148;
        double r31152 = r31139 * r31151;
        double r31153 = r31149 - r31152;
        double r31154 = fabs(r31153);
        double r31155 = r31139 * r31150;
        double r31156 = r31147 - r31155;
        double r31157 = r31156 / r31148;
        double r31158 = fabs(r31157);
        double r31159 = r31145 ? r31154 : r31158;
        return r31159;
}

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 < -23494568.222248383 or 7.80513442966518e-43 < x

    1. Initial program 0.2

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

    3. Applied div-inv0.2

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

    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

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

    if -23494568.222248383 < x < 7.80513442966518e-43

    1. Initial program 2.5

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -23494568.222248383 \lor \neg \left(x \le 7.80513442966517965 \cdot 10^{-43}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))