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

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around 0 0.0

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

  3. Applied fma-def_binary640.0

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

  4. Final simplification0.0

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

Reproduce

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