Average Error: 1.6 → 0.2
Time: 7.0m
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.433505417011087 \cdot 10^{-11}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 8.367091182980687 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.433505417011087 \cdot 10^{-11}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 8.367091182980687 \cdot 10^{-37}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r17568981 = x;
        double r17568982 = 4.0;
        double r17568983 = r17568981 + r17568982;
        double r17568984 = y;
        double r17568985 = r17568983 / r17568984;
        double r17568986 = r17568981 / r17568984;
        double r17568987 = z;
        double r17568988 = r17568986 * r17568987;
        double r17568989 = r17568985 - r17568988;
        double r17568990 = fabs(r17568989);
        return r17568990;
}
↓
double f(double x, double y, double z) {
        double r17568991 = x;
        double r17568992 = -4.433505417011087e-11;
        bool r17568993 = r17568991 <= r17568992;
        double r17568994 = 4.0;
        double r17568995 = y;
        double r17568996 = r17568994 / r17568995;
        double r17568997 = r17568991 / r17568995;
        double r17568998 = r17568996 + r17568997;
        double r17568999 = z;
        double r17569000 = r17568997 * r17568999;
        double r17569001 = r17568998 - r17569000;
        double r17569002 = fabs(r17569001);
        double r17569003 = 8.367091182980687e-37;
        bool r17569004 = r17568991 <= r17569003;
        double r17569005 = r17568994 + r17568991;
        double r17569006 = r17569005 / r17568995;
        double r17569007 = r17568991 * r17568999;
        double r17569008 = r17569007 / r17568995;
        double r17569009 = r17569006 - r17569008;
        double r17569010 = fabs(r17569009);
        double r17569011 = r17568995 / r17568991;
        double r17569012 = r17568999 / r17569011;
        double r17569013 = r17569006 - r17569012;
        double r17569014 = fabs(r17569013);
        double r17569015 = r17569004 ? r17569010 : r17569014;
        double r17569016 = r17568993 ? r17569002 : r17569015;
        return r17569016;
}

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 < -4.433505417011087e-11

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-cube-cbrt0.9

      \[\leadsto \left|\frac{\color{blue}{\left(\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}\right) \cdot \sqrt[3]{x + 4}}}{1 \cdot y} - \frac{x}{y} \cdot z\right|\]

    5. Applied times-frac0.9

      \[\leadsto \left|\color{blue}{\frac{\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}}{1} \cdot \frac{\sqrt[3]{x + 4}}{y}} - \frac{x}{y} \cdot z\right|\]

    6. Applied prod-diff0.9

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\left(\frac{\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}}{1}\right), \left(\frac{\sqrt[3]{x + 4}}{y}\right), \left(-z \cdot \frac{x}{y}\right)\right) + \mathsf{fma}\left(\left(-z\right), \left(\frac{x}{y}\right), \left(z \cdot \frac{x}{y}\right)\right)}\right|\]

    7. Simplified0.1

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

    8. Simplified0.1

      \[\leadsto \left|\left(\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right) + \color{blue}{0}\right|\]

    9. Taylor expanded around -inf 8.5

      \[\leadsto \left|\color{blue}{\left(\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}\right)} + 0\right|\]

    10. Simplified0.1

      \[\leadsto \left|\color{blue}{\left(\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{y} \cdot z\right)} + 0\right|\]

    if -4.433505417011087e-11 < x < 8.367091182980687e-37

    1. Initial program 2.6

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

    2. Taylor expanded around 0 0.1

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

    if 8.367091182980687e-37 < x

    1. Initial program 0.4

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.4

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

    4. Applied add-cube-cbrt1.1

      \[\leadsto \left|\frac{\color{blue}{\left(\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}\right) \cdot \sqrt[3]{x + 4}}}{1 \cdot y} - \frac{x}{y} \cdot z\right|\]

    5. Applied times-frac1.1

      \[\leadsto \left|\color{blue}{\frac{\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}}{1} \cdot \frac{\sqrt[3]{x + 4}}{y}} - \frac{x}{y} \cdot z\right|\]

    6. Applied prod-diff1.1

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\left(\frac{\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}}{1}\right), \left(\frac{\sqrt[3]{x + 4}}{y}\right), \left(-z \cdot \frac{x}{y}\right)\right) + \mathsf{fma}\left(\left(-z\right), \left(\frac{x}{y}\right), \left(z \cdot \frac{x}{y}\right)\right)}\right|\]

    7. Simplified0.5

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

    8. Simplified0.5

      \[\leadsto \left|\left(\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right) + \color{blue}{0}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.433505417011087 \cdot 10^{-11}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 8.367091182980687 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))