Average Error: 1.4 → 0.1
Time: 12.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.323311533620367095344248379509361504445 \cdot 10^{50} \lor \neg \left(x \le 4.452832877986661053171246749116107821465\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.323311533620367095344248379509361504445 \cdot 10^{50} \lor \neg \left(x \le 4.452832877986661053171246749116107821465\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 r31061 = x;
        double r31062 = 4.0;
        double r31063 = r31061 + r31062;
        double r31064 = y;
        double r31065 = r31063 / r31064;
        double r31066 = r31061 / r31064;
        double r31067 = z;
        double r31068 = r31066 * r31067;
        double r31069 = r31065 - r31068;
        double r31070 = fabs(r31069);
        return r31070;
}
↓
double f(double x, double y, double z) {
        double r31071 = x;
        double r31072 = -5.323311533620367e+50;
        bool r31073 = r31071 <= r31072;
        double r31074 = 4.452832877986661;
        bool r31075 = r31071 <= r31074;
        double r31076 = !r31075;
        bool r31077 = r31073 || r31076;
        double r31078 = 4.0;
        double r31079 = r31071 + r31078;
        double r31080 = y;
        double r31081 = r31079 / r31080;
        double r31082 = z;
        double r31083 = r31082 / r31080;
        double r31084 = r31071 * r31083;
        double r31085 = r31081 - r31084;
        double r31086 = fabs(r31085);
        double r31087 = r31071 * r31082;
        double r31088 = r31079 - r31087;
        double r31089 = r31088 / r31080;
        double r31090 = fabs(r31089);
        double r31091 = r31077 ? r31086 : r31090;
        return r31091;
}

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 < -5.323311533620367e+50 or 4.452832877986661 < x

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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*0.2

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

    5. Simplified0.1

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

    if -5.323311533620367e+50 < x < 4.452832877986661

    1. Initial program 2.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.323311533620367095344248379509361504445 \cdot 10^{50} \lor \neg \left(x \le 4.452832877986661053171246749116107821465\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))