Average Error: 1.7 → 1.9
Time: 14.6s
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 r40991 = x;
        double r40992 = 4.0;
        double r40993 = r40991 + r40992;
        double r40994 = y;
        double r40995 = r40993 / r40994;
        double r40996 = r40991 / r40994;
        double r40997 = z;
        double r40998 = r40996 * r40997;
        double r40999 = r40995 - r40998;
        double r41000 = fabs(r40999);
        return r41000;
}
↓
double f(double x, double y, double z) {
        double r41001 = x;
        double r41002 = 4.0;
        double r41003 = r41001 + r41002;
        double r41004 = y;
        double r41005 = r41003 / r41004;
        double r41006 = cbrt(r41004);
        double r41007 = r41006 * r41006;
        double r41008 = r41001 / r41007;
        double r41009 = z;
        double r41010 = r41009 / r41006;
        double r41011 = r41008 * r41010;
        double r41012 = r41005 - r41011;
        double r41013 = fabs(r41012);
        return r41013;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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