Average Error: 0.1 → 0.1
Time: 2.5s
Precision: 64
\[\left(x \cdot y\right) \cdot \left(1 - y\right)\]
↓
\[\left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x\]
\left(x \cdot y\right) \cdot \left(1 - y\right)
↓
\left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x
double f(double x, double y) {
        double r29685 = x;
        double r29686 = y;
        double r29687 = r29685 * r29686;
        double r29688 = 1.0;
        double r29689 = r29688 - r29686;
        double r29690 = r29687 * r29689;
        return r29690;
}
↓
double f(double x, double y) {
        double r29691 = 1.0;
        double r29692 = 1.0;
        double r29693 = cbrt(r29692);
        double r29694 = 3.0;
        double r29695 = pow(r29693, r29694);
        double r29696 = r29691 * r29695;
        double r29697 = y;
        double r29698 = -r29697;
        double r29699 = r29696 + r29698;
        double r29700 = x;
        double r29701 = r29699 * r29700;
        double r29702 = r29701 * r29697;
        double r29703 = fma(r29698, r29691, r29697);
        double r29704 = r29703 * r29697;
        double r29705 = r29704 * r29700;
        double r29706 = r29702 + r29705;
        return r29706;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(x \cdot y\right) \cdot \left(1 - y\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.5

    \[\leadsto \left(x \cdot y\right) \cdot \left(1 - \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right)\]

  4. Applied add-cube-cbrt0.5

    \[\leadsto \left(x \cdot y\right) \cdot \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)\]

  5. Applied prod-diff0.5

    \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{y}, \sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\right)}\]

  6. Applied distribute-lft-in0.5

    \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \left(x \cdot y\right) \cdot \mathsf{fma}\left(-\sqrt[3]{y}, \sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}\]

  7. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y} + \left(x \cdot y\right) \cdot \mathsf{fma}\left(-\sqrt[3]{y}, \sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\]

  8. Simplified0.1

    \[\leadsto \left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \color{blue}{\left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x}\]

  9. Final simplification0.1

    \[\leadsto \left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1 y)))