Average Error: 1.6 → 0.2
Time: 9.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -393523888002957312000 \lor \neg \left(x \le 4.65838789560541676126034813512897433168 \cdot 10^{-85}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -393523888002957312000 \lor \neg \left(x \le 4.65838789560541676126034813512897433168 \cdot 10^{-85}\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 r26797 = x;
        double r26798 = 4.0;
        double r26799 = r26797 + r26798;
        double r26800 = y;
        double r26801 = r26799 / r26800;
        double r26802 = r26797 / r26800;
        double r26803 = z;
        double r26804 = r26802 * r26803;
        double r26805 = r26801 - r26804;
        double r26806 = fabs(r26805);
        return r26806;
}
↓
double f(double x, double y, double z) {
        double r26807 = x;
        double r26808 = -3.935238880029573e+20;
        bool r26809 = r26807 <= r26808;
        double r26810 = 4.658387895605417e-85;
        bool r26811 = r26807 <= r26810;
        double r26812 = !r26811;
        bool r26813 = r26809 || r26812;
        double r26814 = 4.0;
        double r26815 = r26807 + r26814;
        double r26816 = y;
        double r26817 = r26815 / r26816;
        double r26818 = z;
        double r26819 = r26818 / r26816;
        double r26820 = r26807 * r26819;
        double r26821 = r26817 - r26820;
        double r26822 = fabs(r26821);
        double r26823 = r26807 * r26818;
        double r26824 = r26815 - r26823;
        double r26825 = r26824 / r26816;
        double r26826 = fabs(r26825);
        double r26827 = r26813 ? r26822 : r26826;
        return r26827;
}

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 < -3.935238880029573e+20 or 4.658387895605417e-85 < x

    1. Initial program 0.4

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

    3. Applied div-inv0.4

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

    4. Applied associate-*l*0.4

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

    5. Simplified0.4

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

    if -3.935238880029573e+20 < x < 4.658387895605417e-85

    1. Initial program 2.6

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -393523888002957312000 \lor \neg \left(x \le 4.65838789560541676126034813512897433168 \cdot 10^{-85}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))