Average Error: 0.1 → 0.1
Time: 5.4s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot z \]
↓
\[\mathsf{fma}\left(z, z \cdot y, x\right) \]
(FPCore (x y z) :precision binary64 (+ x (* (* y z) z)))
↓
(FPCore (x y z) :precision binary64 (fma z (* z y) x))
double code(double x, double y, double z) {
	return x + ((y * z) * z);
}
↓
double code(double x, double y, double z) {
	return fma(z, (z * y), x);
}

function code(x, y, z)
	return Float64(x + Float64(Float64(y * z) * z))
end
↓
function code(x, y, z)
	return fma(z, Float64(z * y), x)
end

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

x + \left(y \cdot z\right) \cdot z
↓
\mathsf{fma}\left(z, z \cdot y, x\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x + \left(y \cdot z\right) \cdot z \]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))