Average Error: 1.7 → 0.3
Time: 14.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.359236426285955 \cdot 10^{-86}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.6596617673403336 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.359236426285955 \cdot 10^{-86}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 2.6596617673403336 \cdot 10^{-59}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1022762 = x;
        double r1022763 = 4.0;
        double r1022764 = r1022762 + r1022763;
        double r1022765 = y;
        double r1022766 = r1022764 / r1022765;
        double r1022767 = r1022762 / r1022765;
        double r1022768 = z;
        double r1022769 = r1022767 * r1022768;
        double r1022770 = r1022766 - r1022769;
        double r1022771 = fabs(r1022770);
        return r1022771;
}
↓
double f(double x, double y, double z) {
        double r1022772 = x;
        double r1022773 = -6.359236426285955e-86;
        bool r1022774 = r1022772 <= r1022773;
        double r1022775 = 4.0;
        double r1022776 = y;
        double r1022777 = r1022775 / r1022776;
        double r1022778 = r1022772 / r1022776;
        double r1022779 = r1022777 + r1022778;
        double r1022780 = z;
        double r1022781 = r1022778 * r1022780;
        double r1022782 = r1022779 - r1022781;
        double r1022783 = fabs(r1022782);
        double r1022784 = 2.6596617673403336e-59;
        bool r1022785 = r1022772 <= r1022784;
        double r1022786 = r1022775 + r1022772;
        double r1022787 = r1022772 * r1022780;
        double r1022788 = r1022786 - r1022787;
        double r1022789 = r1022788 / r1022776;
        double r1022790 = fabs(r1022789);
        double r1022791 = r1022780 / r1022776;
        double r1022792 = r1022772 * r1022791;
        double r1022793 = r1022779 - r1022792;
        double r1022794 = fabs(r1022793);
        double r1022795 = r1022785 ? r1022790 : r1022794;
        double r1022796 = r1022774 ? r1022783 : r1022795;
        return r1022796;
}

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 < -6.359236426285955e-86

    1. Initial program 0.7

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

    2. Taylor expanded around -inf 0.7

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

    3. Simplified0.7

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

    if -6.359236426285955e-86 < x < 2.6596617673403336e-59

    1. Initial program 3.0

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

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

    if 2.6596617673403336e-59 < x

    1. Initial program 0.3

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    5. Applied div-inv0.3

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

    6. Applied associate-*l*0.4

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

    7. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.359236426285955 \cdot 10^{-86}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.6596617673403336 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))