Average Error: 0.1 → 0.1
Time: 2.7s
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 code(double x, double y) {
	return ((double) (((double) (x * y)) * ((double) (1.0 - y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (((double) (((double) (1.0 * ((double) pow(((double) cbrt(1.0)), 3.0)))) + ((double) -(y)))) * x)) * y)) + ((double) (((double) (((double) fma(((double) -(y)), 1.0, y)) * y)) * x))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 2020123 +o rules:numerics
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1 y)))