Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
Cost: 13248
\[\left(x + y\right) \cdot \left(x + y\right) \]
↓
\[\mathsf{fma}\left(y, y, x \cdot \mathsf{fma}\left(y, 2, x\right)\right) \]
(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))
↓
(FPCore (x y) :precision binary64 (fma y y (* x (fma y 2.0 x))))
double code(double x, double y) {
	return (x + y) * (x + y);
}
↓
double code(double x, double y) {
	return fma(y, y, (x * fma(y, 2.0, x)));
}

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

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

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

Error

Target

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

Derivation

  1. Initial program 0.0

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

  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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, x \cdot \mathsf{fma}\left(y, 2, x\right)\right)} \]
    Proof
    (fma.f64 y y (*.f64 x (fma.f64 y 2 x))): 0 points increase in error, 0 points decrease in error
    (fma.f64 y y (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y 2) x)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 y y (*.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.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)))): 2 points increase in error, 2 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)))): 3 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<= associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 y 2) (*.f64 2 (*.f64 y x))) (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (*.f64 y x)) (pow.f64 y 2))) (pow.f64 x 2)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (*.f64 y x)) (+.f64 (pow.f64 y 2) (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost704
\[y \cdot \left(y + x\right) + x \cdot \left(y + x\right) \]
Alternative 2
Error20.2
Cost580
\[\begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-136}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 3
Error0.0
Cost448
\[\left(y + x\right) \cdot \left(y + x\right) \]
Alternative 4
Error20.4
Cost324
\[\begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-138}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
Alternative 5
Error28.2
Cost192
\[x \cdot x \]

Error

Reproduce

herbie shell --seed 2022330 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

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