Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(x + y\right) \cdot \left(z + 1\right) \]
↓
\[y + \mathsf{fma}\left(y, z, \mathsf{fma}\left(x, z, x\right)\right) \]
(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))
↓
(FPCore (x y z) :precision binary64 (+ y (fma y z (fma x z x))))
double code(double x, double y, double z) {
	return (x + y) * (z + 1.0);
}
↓
double code(double x, double y, double z) {
	return y + fma(y, z, fma(x, z, x));
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[\left(x + y\right) \cdot \left(z + 1\right) \]

  2. Taylor expanded in x around 0 0.0

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

  3. Applied fma-def_binary640.0

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

  4. Taylor expanded in x around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))