Average Error: 0.1 → 0.1
Time: 7.8s
Precision: binary64
\[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
↓
\[\mathsf{fma}\left(y, x, \left(y \cdot x\right) \cdot \left(-y\right)\right) \]
\left(x \cdot y\right) \cdot \left(1 - y\right)
↓
\mathsf{fma}\left(y, x, \left(y \cdot x\right) \cdot \left(-y\right)\right)

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))
↓
(FPCore (x y) :precision binary64 (fma y x (* (* y x) (- y))))
double code(double x, double y) {
	return (x * y) * (1.0 - y);
}
↓
double code(double x, double y) {
	return fma(y, x, ((y * x) * -y));
}

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. Applied egg-rr1.1

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

  3. Applied egg-rr1.2

    \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot y}\right)}^{2} \cdot \left(\sqrt[3]{x \cdot y} \cdot \left(1 - y\right)\right)} \]

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, x, \left(y \cdot x\right) \cdot \left(-y\right)\right) \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))