Average Error: 0.1 → 0.1
Time: 1.7s
Precision: binary64
\[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
↓
\[\mathsf{fma}\left(y \cdot \left(y \cdot x\right), -1, y \cdot x\right) \]
(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))
↓
(FPCore (x y) :precision binary64 (fma (* y (* y x)) -1.0 (* y x)))
double code(double x, double y) {
	return (x * y) * (1.0 - y);
}
↓
double code(double x, double y) {
	return fma((y * (y * x)), -1.0, (y * x));
}

function code(x, y)
	return Float64(Float64(x * y) * Float64(1.0 - y))
end
↓
function code(x, y)
	return fma(Float64(y * Float64(y * x)), -1.0, Float64(y * x))
end

code[x_, y_] := N[(N[(x * y), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(y * x), $MachinePrecision]), $MachinePrecision]

\left(x \cdot y\right) \cdot \left(1 - y\right)
↓
\mathsf{fma}\left(y \cdot \left(y \cdot x\right), -1, y \cdot x\right)

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. Taylor expanded in y around 0 5.2

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

  3. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022170 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))