Average Error: 0.0 → 0.0

Time: 4.6s

Precision: binary64

Cost: 7104

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]

↓

\[\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y + y \cdot y\right) \]

(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))

↓

(FPCore (x y) :precision binary64 (fma x x (+ (* (* x 2.0) y) (* y y))))

double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}

↓

double code(double x, double y) {
	return fma(x, x, (((x * 2.0) * y) + (y * y)));
}

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
end

↓

function code(x, y)
	return fma(x, x, Float64(Float64(Float64(x * 2.0) * y) + Float64(y * y)))
end

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

↓

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

\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y

↓

\mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y + y \cdot y\right)

Target

Original	0.0
Target	0.0
Herbie	0.0

\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \]

Derivation

Initial program 0.0
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(x, 2, y\right)\right)} \]
Proof
(fma.f64 x x (*.f64 y (fma.f64 x 2 y))): 0 points increase in error, 0 points decrease in error
(fma.f64 x x (*.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 2) y)))): 0 points increase in error, 0 points decrease in error
(fma.f64 x x (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 x 2) y) (*.f64 y y)))): 0 points increase in error, 1 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (+.f64 (*.f64 (*.f64 x 2) y) (*.f64 y y)))): 4 points increase in error, 0 points decrease in error
(Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))): 0 points increase in error, 0 points decrease in error
Applied egg-rr0.0
\[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(x \cdot 2\right) \cdot y + y \cdot y}\right) \]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(x, x, \left(x \cdot 2\right) \cdot y + y \cdot y\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	704

\[y \cdot y + x \cdot \left(x + 2 \cdot y\right) \]

Alternative 2
Error	1.3
Cost	448

\[y \cdot y + x \cdot x \]

Alternative 3
Error	9.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 2.550945691437359 \cdot 10^{-62}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 4
Error	27.6
Cost	192

\[y \cdot y \]

Reproduce

herbie shell --seed 2022311 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :herbie-target
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

  (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))