Average Error: 0.0 → 0.0
Time: 5.2s
Precision: binary64
Cost: 13248
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, x \cdot \mathsf{fma}\left(2, y, x\right)\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma y y (* x (fma 2.0 y x))))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(y, y, (x * fma(2.0, y, x)));
}

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(y, y, Float64(x * fma(2.0, y, x)))
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[(y * y + N[(x * N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original0.0
Target0.0
Herbie0.0
\[x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \]

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{2 \cdot \left(y \cdot x\right) + \left({y}^{2} + {x}^{2}\right)} \]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot \mathsf{fma}\left(2, y, x\right)\right)} \]
    Proof
    (fma.f64 y y (*.f64 x (fma.f64 2 y x))): 0 points increase in error, 0 points decrease in error
    (fma.f64 y y (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 y) x)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 y y (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 2 y) x) (*.f64 x x)))): 1 points increase in error, 0 points decrease in error
    (fma.f64 y y (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 y x))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
    (fma.f64 y y (+.f64 (*.f64 2 (*.f64 y x)) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y y) (+.f64 (*.f64 2 (*.f64 y x)) (pow.f64 x 2)))): 4 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) (+.f64 (*.f64 2 (*.f64 y x)) (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 2 (*.f64 y x)) (pow.f64 x 2)) (pow.f64 y 2))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (*.f64 y x)) (+.f64 (pow.f64 x 2) (pow.f64 y 2)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 2 (*.f64 y x)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 y 2) (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost7104
\[\mathsf{fma}\left(y, y, x \cdot \left(y \cdot 2\right) + x \cdot x\right) \]
Alternative 2
Error0.0
Cost704
\[x \cdot \left(x + y \cdot 2\right) + y \cdot y \]
Alternative 3
Error7.4
Cost580
\[\begin{array}{l} \mathbf{if}\;y \leq 7.693796045868536 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 4
Error1.2
Cost448
\[x \cdot x + y \cdot y \]
Alternative 5
Error7.6
Cost324
\[\begin{array}{l} \mathbf{if}\;y \leq 7.693796045868536 \cdot 10^{-121}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 6
Error27.8
Cost192
\[y \cdot y \]

Error

Reproduce

herbie shell --seed 2022308 
(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)))