Average Error: 1.7 → 1.9
Time: 3.9s
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 r36395 = x;
        double r36396 = 4.0;
        double r36397 = r36395 + r36396;
        double r36398 = y;
        double r36399 = r36397 / r36398;
        double r36400 = r36395 / r36398;
        double r36401 = z;
        double r36402 = r36400 * r36401;
        double r36403 = r36399 - r36402;
        double r36404 = fabs(r36403);
        return r36404;
}
↓
double f(double x, double y, double z) {
        double r36405 = x;
        double r36406 = 4.0;
        double r36407 = r36405 + r36406;
        double r36408 = y;
        double r36409 = r36407 / r36408;
        double r36410 = cbrt(r36408);
        double r36411 = r36410 * r36410;
        double r36412 = r36405 / r36411;
        double r36413 = z;
        double r36414 = r36413 / r36410;
        double r36415 = r36412 * r36414;
        double r36416 = r36409 - r36415;
        double r36417 = fabs(r36416);
        return r36417;
}

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