Average Error: 1.5 → 0.9
Time: 14.8s
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 r27418 = x;
        double r27419 = 4.0;
        double r27420 = r27418 + r27419;
        double r27421 = y;
        double r27422 = r27420 / r27421;
        double r27423 = r27418 / r27421;
        double r27424 = z;
        double r27425 = r27423 * r27424;
        double r27426 = r27422 - r27425;
        double r27427 = fabs(r27426);
        return r27427;
}
↓
double f(double x, double y, double z) {
        double r27428 = x;
        double r27429 = 4.0;
        double r27430 = r27428 + r27429;
        double r27431 = y;
        double r27432 = r27430 / r27431;
        double r27433 = z;
        double r27434 = cbrt(r27433);
        double r27435 = r27434 * r27434;
        double r27436 = cbrt(r27431);
        double r27437 = r27436 * r27436;
        double r27438 = r27435 / r27437;
        double r27439 = r27428 * r27438;
        double r27440 = r27434 / r27436;
        double r27441 = r27439 * r27440;
        double r27442 = r27432 - r27441;
        double r27443 = fabs(r27442);
        return r27443;
}

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 1.5

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  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 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))