Average Error: 1.7 → 1.8
Time: 13.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.331683792302665078276548190002856814797 \cdot 10^{-52}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.331683792302665078276548190002856814797 \cdot 10^{-52}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1076756 = x;
        double r1076757 = 4.0;
        double r1076758 = r1076756 + r1076757;
        double r1076759 = y;
        double r1076760 = r1076758 / r1076759;
        double r1076761 = r1076756 / r1076759;
        double r1076762 = z;
        double r1076763 = r1076761 * r1076762;
        double r1076764 = r1076760 - r1076763;
        double r1076765 = fabs(r1076764);
        return r1076765;
}
↓
double f(double x, double y, double z) {
        double r1076766 = x;
        double r1076767 = 2.331683792302665e-52;
        bool r1076768 = r1076766 <= r1076767;
        double r1076769 = 4.0;
        double r1076770 = y;
        double r1076771 = r1076769 / r1076770;
        double r1076772 = r1076766 / r1076770;
        double r1076773 = r1076771 + r1076772;
        double r1076774 = z;
        double r1076775 = r1076766 * r1076774;
        double r1076776 = r1076775 / r1076770;
        double r1076777 = r1076773 - r1076776;
        double r1076778 = fabs(r1076777);
        double r1076779 = r1076770 / r1076774;
        double r1076780 = r1076766 / r1076779;
        double r1076781 = r1076773 - r1076780;
        double r1076782 = fabs(r1076781);
        double r1076783 = r1076768 ? r1076778 : r1076782;
        return r1076783;
}

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 < 2.331683792302665e-52

    1. Initial program 2.1

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

    2. Taylor expanded around 0 2.2

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

    3. Simplified2.2

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

    5. Applied associate-*l/2.3

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

    if 2.331683792302665e-52 < x

    1. Initial program 0.3

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

    2. Taylor expanded around 0 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

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

    5. Applied associate-*l/6.9

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

    7. Applied associate-/l*0.3

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{\frac{x}{\frac{y}{z}}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.331683792302665078276548190002856814797 \cdot 10^{-52}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))