Average Error: 0.0 → 0.0
Time: 9.6s
Precision: binary64
\[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
↓
\[\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, y \cdot x\right)\right) \]
(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))
↓
(FPCore (x y z t a b) :precision binary64 (fma t z (fma a b (* y x))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return fma(t, z, fma(a, b, (y * x)));
}

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

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(t * z + N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Taylor expanded in y around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))