Average Error: 1.5 → 0.3
Time: 10.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.0025647343369246 \cdot 10^{88}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.38362149527649136 \cdot 10^{44}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.0025647343369246 \cdot 10^{88}:\\
\;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 2.38362149527649136 \cdot 10^{44}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r26816 = x;
        double r26817 = 4.0;
        double r26818 = r26816 + r26817;
        double r26819 = y;
        double r26820 = r26818 / r26819;
        double r26821 = r26816 / r26819;
        double r26822 = z;
        double r26823 = r26821 * r26822;
        double r26824 = r26820 - r26823;
        double r26825 = fabs(r26824);
        return r26825;
}
↓
double f(double x, double y, double z) {
        double r26826 = x;
        double r26827 = -4.0025647343369246e+88;
        bool r26828 = r26826 <= r26827;
        double r26829 = y;
        double r26830 = r26826 / r26829;
        double r26831 = 4.0;
        double r26832 = r26831 / r26829;
        double r26833 = r26830 + r26832;
        double r26834 = z;
        double r26835 = r26830 * r26834;
        double r26836 = r26833 - r26835;
        double r26837 = fabs(r26836);
        double r26838 = 2.3836214952764914e+44;
        bool r26839 = r26826 <= r26838;
        double r26840 = r26826 + r26831;
        double r26841 = r26826 * r26834;
        double r26842 = r26840 - r26841;
        double r26843 = r26842 / r26829;
        double r26844 = fabs(r26843);
        double r26845 = r26834 / r26829;
        double r26846 = r26826 * r26845;
        double r26847 = r26833 - r26846;
        double r26848 = fabs(r26847);
        double r26849 = r26839 ? r26844 : r26848;
        double r26850 = r26828 ? r26837 : r26849;
        return r26850;
}

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 < -4.0025647343369246e+88

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if -4.0025647343369246e+88 < x < 2.3836214952764914e+44

    1. Initial program 2.0

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 2.3836214952764914e+44 < x

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied div-inv0.2

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

    6. Applied associate-*l*0.2

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

    7. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.0025647343369246 \cdot 10^{88}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.38362149527649136 \cdot 10^{44}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))