Average Error: 0.0 → 0.0
Time: 1.6s
Precision: binary64
\[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
↓
\[y \cdot \left(y + x \cdot 2\right) + {x}^{2} \]
(FPCore (x y) :precision binary64 (+ (+ (* x x) (* (* x 2.0) y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (* y (+ y (* x 2.0))) (pow x 2.0)))
double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
double code(double x, double y) {
	return (y * (y + (x * 2.0))) + pow(x, 2.0);
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
end function
↓
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * (y + (x * 2.0d0))) + (x ** 2.0d0)
end function

public static double code(double x, double y) {
	return ((x * x) + ((x * 2.0) * y)) + (y * y);
}
↓
public static double code(double x, double y) {
	return (y * (y + (x * 2.0))) + Math.pow(x, 2.0);
}

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)
↓
def code(x, y):
	return (y * (y + (x * 2.0))) + math.pow(x, 2.0)

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 Float64(Float64(y * Float64(y + Float64(x * 2.0))) + (x ^ 2.0))
end

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end
↓
function tmp = code(x, y)
	tmp = (y * (y + (x * 2.0))) + (x ^ 2.0);
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[(N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y
↓
y \cdot \left(y + x \cdot 2\right) + {x}^{2}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

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

  4. Applied associate-+r+_binary640.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022130 
(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)))