Average Error: 0.1 → 0.1
Time: 3.1s
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 r30128 = x;
        double r30129 = y;
        double r30130 = r30128 * r30129;
        double r30131 = 1.0;
        double r30132 = r30131 - r30129;
        double r30133 = r30130 * r30132;
        return r30133;
}
↓
double f(double x, double y) {
        double r30134 = 1.0;
        double r30135 = 1.0;
        double r30136 = cbrt(r30135);
        double r30137 = 3.0;
        double r30138 = pow(r30136, r30137);
        double r30139 = r30134 * r30138;
        double r30140 = y;
        double r30141 = -r30140;
        double r30142 = r30139 + r30141;
        double r30143 = x;
        double r30144 = r30142 * r30143;
        double r30145 = r30144 * r30140;
        double r30146 = fma(r30141, r30134, r30140);
        double r30147 = r30146 * r30140;
        double r30148 = r30147 * r30143;
        double r30149 = r30145 + r30148;
        return r30149;
}

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 2020001 +o rules:numerics
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1 y)))