Average Error: 0.1 → 0.0
Time: 2.5s
Precision: binary64
\[x \cdot \left(y + z\right) + z \cdot 5 \]
↓
\[\mathsf{fma}\left(5, z, x \cdot \left(z + y\right)\right) \]
(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))
↓
(FPCore (x y z) :precision binary64 (fma 5.0 z (* x (+ z y))))
double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}
↓
double code(double x, double y, double z) {
	return fma(5.0, z, (x * (z + y)));
}

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end
↓
function code(x, y, z)
	return fma(5.0, z, Float64(x * Float64(z + y)))
end

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(5.0 * z + N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x \cdot \left(y + z\right) + z \cdot 5
↓
\mathsf{fma}\left(5, z, x \cdot \left(z + y\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.0
\[\left(x + 5\right) \cdot z + x \cdot y \]

Derivation

  1. Initial program 0.1

    \[x \cdot \left(y + z\right) + z \cdot 5 \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, z \cdot 5\right)} \]

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + 5 \cdot z\right)} \]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(5, z, x \cdot \left(z + y\right)\right)} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(5, z, x \cdot \left(z + y\right)\right) \]

Reproduce

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

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

  (+ (* x (+ y z)) (* z 5.0)))