Average Error: 1.5 → 2.4
Time: 3.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 8.04631218254802011 \cdot 10^{144}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le 8.04631218254802011 \cdot 10^{144}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\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 r32114 = x;
        double r32115 = 4.0;
        double r32116 = r32114 + r32115;
        double r32117 = y;
        double r32118 = r32116 / r32117;
        double r32119 = r32114 / r32117;
        double r32120 = z;
        double r32121 = r32119 * r32120;
        double r32122 = r32118 - r32121;
        double r32123 = fabs(r32122);
        return r32123;
}
↓
double f(double x, double y, double z) {
        double r32124 = y;
        double r32125 = 8.04631218254802e+144;
        bool r32126 = r32124 <= r32125;
        double r32127 = x;
        double r32128 = 4.0;
        double r32129 = r32127 + r32128;
        double r32130 = r32129 / r32124;
        double r32131 = z;
        double r32132 = r32127 * r32131;
        double r32133 = r32132 / r32124;
        double r32134 = r32130 - r32133;
        double r32135 = fabs(r32134);
        double r32136 = r32131 / r32124;
        double r32137 = r32127 * r32136;
        double r32138 = r32130 - r32137;
        double r32139 = fabs(r32138);
        double r32140 = r32126 ? r32135 : r32139;
        return r32140;
}

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 y < 8.04631218254802e+144

    1. Initial program 0.9

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

    3. Applied associate-*l/2.8

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

    if 8.04631218254802e+144 < y

    1. Initial program 4.4

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

    3. Applied div-inv4.4

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

    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 simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 8.04631218254802011 \cdot 10^{144}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))