Average Error: 1.6 → 1.8
Time: 13.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \sqrt[3]{\left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)} \cdot \left(\sqrt[3]{z \cdot \frac{x}{y}} \cdot \sqrt[3]{z \cdot \frac{x}{y}}\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \sqrt[3]{\left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)} \cdot \left(\sqrt[3]{z \cdot \frac{x}{y}} \cdot \sqrt[3]{z \cdot \frac{x}{y}}\right)\right|
double f(double x, double y, double z) {
        double r1339382 = x;
        double r1339383 = 4.0;
        double r1339384 = r1339382 + r1339383;
        double r1339385 = y;
        double r1339386 = r1339384 / r1339385;
        double r1339387 = r1339382 / r1339385;
        double r1339388 = z;
        double r1339389 = r1339387 * r1339388;
        double r1339390 = r1339386 - r1339389;
        double r1339391 = fabs(r1339390);
        return r1339391;
}
↓
double f(double x, double y, double z) {
        double r1339392 = 4.0;
        double r1339393 = y;
        double r1339394 = r1339392 / r1339393;
        double r1339395 = x;
        double r1339396 = r1339395 / r1339393;
        double r1339397 = r1339394 + r1339396;
        double r1339398 = cbrt(r1339396);
        double r1339399 = r1339398 * r1339398;
        double r1339400 = cbrt(r1339395);
        double r1339401 = cbrt(r1339393);
        double r1339402 = r1339400 / r1339401;
        double r1339403 = z;
        double r1339404 = r1339402 * r1339403;
        double r1339405 = r1339399 * r1339404;
        double r1339406 = cbrt(r1339405);
        double r1339407 = r1339403 * r1339396;
        double r1339408 = cbrt(r1339407);
        double r1339409 = r1339408 * r1339408;
        double r1339410 = r1339406 * r1339409;
        double r1339411 = r1339397 - r1339410;
        double r1339412 = fabs(r1339411);
        return r1339412;
}

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. Initial program 1.6

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

  2. Taylor expanded around 0 1.6

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

  3. Simplified1.6

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

  5. Applied add-cube-cbrt1.8

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

  7. Applied add-cube-cbrt1.8

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

  8. Applied associate-*l*1.8

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

  10. Applied cbrt-div1.8

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

  11. Final simplification1.8

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))