Average Error: 0.1 → 0.1
Time: 2.4s
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 r23001 = x;
        double r23002 = y;
        double r23003 = r23001 * r23002;
        double r23004 = 1.0;
        double r23005 = r23004 - r23002;
        double r23006 = r23003 * r23005;
        return r23006;
}
↓
double f(double x, double y) {
        double r23007 = 1.0;
        double r23008 = 1.0;
        double r23009 = cbrt(r23008);
        double r23010 = 3.0;
        double r23011 = pow(r23009, r23010);
        double r23012 = r23007 * r23011;
        double r23013 = y;
        double r23014 = -r23013;
        double r23015 = r23012 + r23014;
        double r23016 = x;
        double r23017 = r23015 * r23016;
        double r23018 = r23017 * r23013;
        double r23019 = fma(r23014, r23007, r23013);
        double r23020 = r23019 * r23013;
        double r23021 = r23020 * r23016;
        double r23022 = r23018 + r23021;
        return r23022;
}

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)))