Average Error: 1.4 → 0.2
Time: 34.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.8868127395774704 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \left(\frac{4}{y} - \frac{z}{\frac{y}{x}}\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 1.8868127395774704 \cdot 10^{-28}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x}{y} + \left(\frac{4}{y} - \frac{z}{\frac{y}{x}}\right)\right|\\

\end{array}
double f(double x, double y, double z) {
        double r787734 = x;
        double r787735 = 4.0;
        double r787736 = r787734 + r787735;
        double r787737 = y;
        double r787738 = r787736 / r787737;
        double r787739 = r787734 / r787737;
        double r787740 = z;
        double r787741 = r787739 * r787740;
        double r787742 = r787738 - r787741;
        double r787743 = fabs(r787742);
        return r787743;
}
↓
double f(double x, double y, double z) {
        double r787744 = x;
        double r787745 = -2.1618774464985056e+63;
        bool r787746 = r787744 <= r787745;
        double r787747 = 4.0;
        double r787748 = r787747 + r787744;
        double r787749 = y;
        double r787750 = r787748 / r787749;
        double r787751 = r787744 / r787749;
        double r787752 = z;
        double r787753 = r787751 * r787752;
        double r787754 = r787750 - r787753;
        double r787755 = fabs(r787754);
        double r787756 = 1.8868127395774704e-28;
        bool r787757 = r787744 <= r787756;
        double r787758 = r787744 * r787752;
        double r787759 = r787748 - r787758;
        double r787760 = r787759 / r787749;
        double r787761 = fabs(r787760);
        double r787762 = r787747 / r787749;
        double r787763 = r787749 / r787744;
        double r787764 = r787752 / r787763;
        double r787765 = r787762 - r787764;
        double r787766 = r787751 + r787765;
        double r787767 = fabs(r787766);
        double r787768 = r787757 ? r787761 : r787767;
        double r787769 = r787746 ? r787755 : r787768;
        return r787769;
}

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 3 regimes

  2. if x < -2.1618774464985056e+63

    1. Initial program 0.1

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

    if -2.1618774464985056e+63 < x < 1.8868127395774704e-28

    1. Initial program 2.2

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

    3. Applied associate-*l/0.3

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

    4. Applied sub-div0.3

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

    if 1.8868127395774704e-28 < x

    1. Initial program 0.2

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

    2. Taylor expanded around inf 6.6

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

    3. Simplified0.2

      \[\leadsto \left|\color{blue}{\frac{x}{y} + \left(\frac{4}{y} - \frac{z}{\frac{y}{x}}\right)}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.8868127395774704 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \left(\frac{4}{y} - \frac{z}{\frac{y}{x}}\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))