Average Error: 1.6 → 0.4
Time: 27.4s
Precision: 64
\[\left|\frac{x + 4.0}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4.0 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.018719735392733 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\left(4.0 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \left(\frac{4.0}{y} - \frac{x}{\frac{y}{z}}\right)\right|\\ \end{array}\]
\left|\frac{x + 4.0}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\
\;\;\;\;\left|\frac{4.0 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 8.018719735392733 \cdot 10^{-129}:\\
\;\;\;\;\left|\frac{\left(4.0 + x\right) - z \cdot x}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x}{y} + \left(\frac{4.0}{y} - \frac{x}{\frac{y}{z}}\right)\right|\\

\end{array}
double f(double x, double y, double z) {
        double r1171712 = x;
        double r1171713 = 4.0;
        double r1171714 = r1171712 + r1171713;
        double r1171715 = y;
        double r1171716 = r1171714 / r1171715;
        double r1171717 = r1171712 / r1171715;
        double r1171718 = z;
        double r1171719 = r1171717 * r1171718;
        double r1171720 = r1171716 - r1171719;
        double r1171721 = fabs(r1171720);
        return r1171721;
}
↓
double f(double x, double y, double z) {
        double r1171722 = x;
        double r1171723 = -4.5186598216349587e+49;
        bool r1171724 = r1171722 <= r1171723;
        double r1171725 = 4.0;
        double r1171726 = r1171725 + r1171722;
        double r1171727 = y;
        double r1171728 = r1171726 / r1171727;
        double r1171729 = z;
        double r1171730 = r1171729 / r1171727;
        double r1171731 = r1171722 * r1171730;
        double r1171732 = r1171728 - r1171731;
        double r1171733 = fabs(r1171732);
        double r1171734 = 8.018719735392733e-129;
        bool r1171735 = r1171722 <= r1171734;
        double r1171736 = r1171729 * r1171722;
        double r1171737 = r1171726 - r1171736;
        double r1171738 = r1171737 / r1171727;
        double r1171739 = fabs(r1171738);
        double r1171740 = r1171722 / r1171727;
        double r1171741 = r1171725 / r1171727;
        double r1171742 = r1171727 / r1171729;
        double r1171743 = r1171722 / r1171742;
        double r1171744 = r1171741 - r1171743;
        double r1171745 = r1171740 + r1171744;
        double r1171746 = fabs(r1171745);
        double r1171747 = r1171735 ? r1171739 : r1171746;
        double r1171748 = r1171724 ? r1171733 : r1171747;
        return r1171748;
}

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 < -4.5186598216349587e+49

    1. Initial program 0.1

      \[\left|\frac{x + 4.0}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv0.2

      \[\leadsto \left|\frac{x + 4.0}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]

    4. Applied associate-*l*0.2

      \[\leadsto \left|\frac{x + 4.0}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]

    5. Simplified0.1

      \[\leadsto \left|\frac{x + 4.0}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]

    if -4.5186598216349587e+49 < x < 8.018719735392733e-129

    1. Initial program 2.4

      \[\left|\frac{x + 4.0}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied associate-*l/0.2

      \[\leadsto \left|\frac{x + 4.0}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]

    4. Applied sub-div0.2

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4.0\right) - x \cdot z}{y}}\right|\]

    if 8.018719735392733e-129 < x

    1. Initial program 0.9

      \[\left|\frac{x + 4.0}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv1.0

      \[\leadsto \left|\frac{x + 4.0}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]

    4. Applied associate-*l*1.1

      \[\leadsto \left|\frac{x + 4.0}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]

    5. Simplified1.0

      \[\leadsto \left|\frac{x + 4.0}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]

    6. Taylor expanded around 0 6.1

      \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4.0 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}}\right|\]

    7. Simplified0.9

      \[\leadsto \left|\color{blue}{\left(\frac{4.0}{y} - \frac{x}{\frac{y}{z}}\right) + \frac{x}{y}}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4.0 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.018719735392733 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\left(4.0 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \left(\frac{4.0}{y} - \frac{x}{\frac{y}{z}}\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))