Average Error: 1.5 → 0.4
Time: 22.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -16246640829.1726360321044921875:\\
\;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\

\mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r1853710 = x;
        double r1853711 = 4.0;
        double r1853712 = r1853710 + r1853711;
        double r1853713 = y;
        double r1853714 = r1853712 / r1853713;
        double r1853715 = r1853710 / r1853713;
        double r1853716 = z;
        double r1853717 = r1853715 * r1853716;
        double r1853718 = r1853714 - r1853717;
        double r1853719 = fabs(r1853718);
        return r1853719;
}
↓
double f(double x, double y, double z) {
        double r1853720 = x;
        double r1853721 = -16246640829.172636;
        bool r1853722 = r1853720 <= r1853721;
        double r1853723 = 1.0;
        double r1853724 = z;
        double r1853725 = r1853723 - r1853724;
        double r1853726 = y;
        double r1853727 = r1853720 / r1853726;
        double r1853728 = r1853725 * r1853727;
        double r1853729 = 4.0;
        double r1853730 = r1853729 / r1853726;
        double r1853731 = r1853728 + r1853730;
        double r1853732 = fabs(r1853731);
        double r1853733 = 1.0296033599100243e-161;
        bool r1853734 = r1853720 <= r1853733;
        double r1853735 = r1853729 + r1853720;
        double r1853736 = r1853720 * r1853724;
        double r1853737 = r1853735 - r1853736;
        double r1853738 = r1853737 / r1853726;
        double r1853739 = fabs(r1853738);
        double r1853740 = r1853734 ? r1853739 : r1853732;
        double r1853741 = r1853722 ? r1853732 : r1853740;
        return r1853741;
}

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 < -16246640829.172636 or 1.0296033599100243e-161 < x

    1. Initial program 0.7

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

    2. Taylor expanded around 0 6.2

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

    3. Simplified0.7

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

    if -16246640829.172636 < x < 1.0296033599100243e-161

    1. Initial program 2.5

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))