Average Error: 1.7 → 0.2
Time: 3.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.141065750945845758760211222436365665667 \cdot 10^{-15} \lor \neg \left(x \le 4.40041470183000606091280929473849033149 \cdot 10^{-62}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.141065750945845758760211222436365665667 \cdot 10^{-15} \lor \neg \left(x \le 4.40041470183000606091280929473849033149 \cdot 10^{-62}\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 r29032 = x;
        double r29033 = 4.0;
        double r29034 = r29032 + r29033;
        double r29035 = y;
        double r29036 = r29034 / r29035;
        double r29037 = r29032 / r29035;
        double r29038 = z;
        double r29039 = r29037 * r29038;
        double r29040 = r29036 - r29039;
        double r29041 = fabs(r29040);
        return r29041;
}
↓
double f(double x, double y, double z) {
        double r29042 = x;
        double r29043 = -3.1410657509458458e-15;
        bool r29044 = r29042 <= r29043;
        double r29045 = 4.400414701830006e-62;
        bool r29046 = r29042 <= r29045;
        double r29047 = !r29046;
        bool r29048 = r29044 || r29047;
        double r29049 = 4.0;
        double r29050 = r29042 + r29049;
        double r29051 = y;
        double r29052 = r29050 / r29051;
        double r29053 = z;
        double r29054 = r29053 / r29051;
        double r29055 = r29042 * r29054;
        double r29056 = r29052 - r29055;
        double r29057 = fabs(r29056);
        double r29058 = r29042 * r29053;
        double r29059 = r29050 - r29058;
        double r29060 = r29059 / r29051;
        double r29061 = fabs(r29060);
        double r29062 = r29048 ? r29057 : r29061;
        return r29062;
}

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 < -3.1410657509458458e-15 or 4.400414701830006e-62 < x

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    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 -3.1410657509458458e-15 < x < 4.400414701830006e-62

    1. Initial program 2.9

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.141065750945845758760211222436365665667 \cdot 10^{-15} \lor \neg \left(x \le 4.40041470183000606091280929473849033149 \cdot 10^{-62}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))