Average Error: 1.7 → 1.9
Time: 15.1s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{x + 4}{y} - \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{x + 4}{y} - \frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}\right|
double f(double x, double y, double z) {
        double r42724 = x;
        double r42725 = 4.0;
        double r42726 = r42724 + r42725;
        double r42727 = y;
        double r42728 = r42726 / r42727;
        double r42729 = r42724 / r42727;
        double r42730 = z;
        double r42731 = r42729 * r42730;
        double r42732 = r42728 - r42731;
        double r42733 = fabs(r42732);
        return r42733;
}
↓
double f(double x, double y, double z) {
        double r42734 = x;
        double r42735 = 4.0;
        double r42736 = r42734 + r42735;
        double r42737 = y;
        double r42738 = r42736 / r42737;
        double r42739 = cbrt(r42737);
        double r42740 = r42739 * r42739;
        double r42741 = r42734 / r42740;
        double r42742 = z;
        double r42743 = r42742 / r42739;
        double r42744 = r42741 * r42743;
        double r42745 = r42738 - r42744;
        double r42746 = fabs(r42745);
        return r42746;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.7

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

  3. Applied div-inv1.7

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

  4. Applied associate-*l*3.5

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

  5. Simplified3.5

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

  7. Applied add-cube-cbrt3.8

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

  8. Applied *-un-lft-identity3.8

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

  9. Applied times-frac3.8

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

  10. Applied associate-*r*1.9

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

  11. Simplified1.9

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

  12. Final simplification1.9

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))