Average Error: 0.1 → 0.0
Time: 1.7s
Precision: binary64
\[\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x \]
↓
\[\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right) \]
(FPCore (x y z) :precision binary64 (+ (+ (+ (+ (+ x y) y) x) z) x))
↓
(FPCore (x y z) :precision binary64 (fma x 3.0 (fma y 2.0 z)))
double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + z) + x;
}
↓
double code(double x, double y, double z) {
	return fma(x, 3.0, fma(y, 2.0, z));
}

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x + y) + y) + x) + z) + x)
end
↓
function code(x, y, z)
	return fma(x, 3.0, fma(y, 2.0, z))
end

code[x_, y_, z_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] + x), $MachinePrecision]
↓
code[x_, y_, z_] := N[(x * 3.0 + N[(y * 2.0 + z), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(\left(x + y\right) + y\right) + x\right) + z\right) + x
↓
\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right)} \]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, 3, \mathsf{fma}\left(y, 2, z\right)\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))