Average Error: 1.5 → 0.9
Time: 14.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{x + 4}{y} - \left(x \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{x + 4}{y} - \left(x \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right|
double f(double x, double y, double z) {
        double r26476 = x;
        double r26477 = 4.0;
        double r26478 = r26476 + r26477;
        double r26479 = y;
        double r26480 = r26478 / r26479;
        double r26481 = r26476 / r26479;
        double r26482 = z;
        double r26483 = r26481 * r26482;
        double r26484 = r26480 - r26483;
        double r26485 = fabs(r26484);
        return r26485;
}
↓
double f(double x, double y, double z) {
        double r26486 = x;
        double r26487 = 4.0;
        double r26488 = r26486 + r26487;
        double r26489 = y;
        double r26490 = r26488 / r26489;
        double r26491 = z;
        double r26492 = cbrt(r26491);
        double r26493 = r26492 * r26492;
        double r26494 = cbrt(r26489);
        double r26495 = r26494 * r26494;
        double r26496 = r26493 / r26495;
        double r26497 = r26486 * r26496;
        double r26498 = r26492 / r26494;
        double r26499 = r26497 * r26498;
        double r26500 = r26490 - r26499;
        double r26501 = fabs(r26500);
        return r26501;
}

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 1.5

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

  3. Applied div-inv1.5

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

  4. Applied associate-*l*3.4

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

  5. Simplified3.4

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

  7. Applied add-cube-cbrt3.7

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

  8. Applied add-cube-cbrt3.7

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

  9. Applied times-frac3.7

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

  10. Applied associate-*r*0.9

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

  11. Final simplification0.9

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))