Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.8% → 99.8%

Time: 6.8s

Alternatives: 7

Speedup: 0.8×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return ((x * x) + ((x * 2.0) * y)) + (y * y)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) + ((x * 2.0) * y)) + (y * y);
end

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot \left(2 + \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (* y y) (+ (* x x) (* y (* x 2.0)))) 5e+301)
   (+ (* x x) (* y (+ y (* x 2.0))))
   (* x (+ x (* y (+ 2.0 (/ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * x) + (y * (x * 2.0)))) <= 5e+301) {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	} else {
		tmp = x * (x + (y * (2.0 + (y / x))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y * y) + ((x * x) + (y * (x * 2.0d0)))) <= 5d+301) then
        tmp = (x * x) + (y * (y + (x * 2.0d0)))
    else
        tmp = x * (x + (y * (2.0d0 + (y / x))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * x) + (y * (x * 2.0)))) <= 5e+301) {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	} else {
		tmp = x * (x + (y * (2.0 + (y / x))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if ((y * y) + ((x * x) + (y * (x * 2.0)))) <= 5e+301:
		tmp = (x * x) + (y * (y + (x * 2.0)))
	else:
		tmp = x * (x + (y * (2.0 + (y / x))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * y) + Float64(Float64(x * x) + Float64(y * Float64(x * 2.0)))) <= 5e+301)
		tmp = Float64(Float64(x * x) + Float64(y * Float64(y + Float64(x * 2.0))));
	else
		tmp = Float64(x * Float64(x + Float64(y * Float64(2.0 + Float64(y / x)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) + ((x * x) + (y * (x * 2.0)))) <= 5e+301)
		tmp = (x * x) + (y * (y + (x * 2.0)));
	else
		tmp = x * (x + (y * (2.0 + (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[N[(N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+301], N[(N[(x * x), $MachinePrecision] + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + N[(y * N[(2.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y)) < 5.0000000000000004e301

   1. Initial program 99.9%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-define 100.0%

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

      4. associate-*l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   if 5.0000000000000004e301 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   1. Initial program 84.8%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ 84.8%

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

      2. +-commutative 84.8%

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

      3. fma-define 84.8%

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

      4. associate-*l* 84.8%

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

   3. Simplified 84.8%

      \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 96.4%

      \[\leadsto x \cdot x + \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 96.4%

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

      2. unpow2 96.4%

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

      3. associate-/l* 92.0%

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

      4. distribute-lft-out 92.0%

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

   7. Simplified 92.0%

      \[\leadsto x \cdot x + \color{blue}{x \cdot \left(y \cdot \left(2 + \frac{y}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + y \cdot \left(2 + \frac{y}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y \cdot \left(2 + \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(x, \mathsf{fma}\left(2, y, x\right), y \cdot y\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma x (fma 2.0 y x) (* y y)))

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(x, fma(2.0, y, x), Float64(y * y))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * N[(2.0 * y + x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. associate-*l* 93.3%

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

   2. *-commutative 93.3%

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

   3. distribute-lft-out 96.9%

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

   4. fma-define 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. fma-define 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(x + y \cdot \left(2 + \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e-158) (* x (+ x (* y (+ 2.0 (/ y x))))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e-158) {
		tmp = x * (x + (y * (2.0 + (y / x))));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d-158)) then
        tmp = x * (x + (y * (2.0d0 + (y / x))))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9e-158) {
		tmp = x * (x + (y * (2.0 + (y / x))));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9e-158:
		tmp = x * (x + (y * (2.0 + (y / x))))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e-158)
		tmp = Float64(x * Float64(x + Float64(y * Float64(2.0 + Float64(y / x)))));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e-158)
		tmp = x * (x + (y * (2.0 + (y / x))));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e-158], N[(x * N[(x + N[(y * N[(2.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e-158

   1. Initial program 92.5%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ 92.5%

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

      2. +-commutative 92.5%

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

      3. fma-define 92.6%

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

      4. associate-*l* 92.6%

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

   3. Simplified 92.6%

      \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 95.9%

      \[\leadsto x \cdot x + \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 95.9%

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

      2. unpow2 95.9%

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

      3. associate-/l* 92.7%

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

      4. distribute-lft-out 92.7%

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

   7. Simplified 92.7%

      \[\leadsto x \cdot x + \color{blue}{x \cdot \left(y \cdot \left(2 + \frac{y}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out 96.9%

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

   9. Applied egg-rr 96.9%

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

   if -9e-158 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 96.9%

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

   4. Taylor expanded in x around 0 61.3%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} + y \cdot y \]
   5. Step-by-step derivation

      1. associate-*r* 61.3%

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

      2. *-commutative 61.3%

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

      3. *-commutative 61.3%

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

      4. *-commutative 61.3%

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

   6. Simplified 61.3%

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

   7. Taylor expanded in x around inf 57.4%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. unpow2 53.1%

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

      3. associate-*r/ 53.0%

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

      4. distribute-lft-in 56.1%

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

      5. *-commutative 56.1%

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

      6. distribute-lft-in 56.1%

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

      7. *-commutative 56.1%

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

      8. associate-*l* 60.2%

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

   9. Simplified 60.2%

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

   10. Taylor expanded in y around inf 65.2%

       \[\leadsto y \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(x + y \cdot \left(2 + \frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-95}:\\ \;\;\;\;x \cdot x + 2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-95) (+ (* x x) (* 2.0 (* x y))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-95) {
		tmp = (x * x) + (2.0 * (x * y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.35d-95) then
        tmp = (x * x) + (2.0d0 * (x * y))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-95) {
		tmp = (x * x) + (2.0 * (x * y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.35e-95:
		tmp = (x * x) + (2.0 * (x * y))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-95)
		tmp = Float64(Float64(x * x) + Float64(2.0 * Float64(x * y)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e-95)
		tmp = (x * x) + (2.0 * (x * y));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.35e-95], N[(N[(x * x), $MachinePrecision] + N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e-95

   1. Initial program 94.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ 94.2%

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

      2. +-commutative 94.2%

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

      3. fma-define 94.2%

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

      4. associate-*l* 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 67.9%

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

   if 1.35e-95 < y

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 96.3%

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

   4. Taylor expanded in x around 0 72.9%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} + y \cdot y \]
   5. Step-by-step derivation

      1. associate-*r* 72.9%

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

      2. *-commutative 72.9%

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

      3. *-commutative 72.9%

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

      4. *-commutative 72.9%

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

   6. Simplified 72.9%

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

   7. Taylor expanded in x around inf 66.5%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in 59.3%

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

      2. unpow2 59.3%

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

      3. associate-*r/ 59.2%

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

      4. distribute-lft-in 62.8%

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

      5. *-commutative 62.8%

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

      6. distribute-lft-in 62.8%

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

      7. *-commutative 62.8%

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

      8. associate-*l* 69.6%

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

   9. Simplified 69.6%

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

   10. Taylor expanded in y around inf 78.2%

       \[\leadsto y \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-95}:\\ \;\;\;\;x \cdot x + 2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.5e-96) (* x (+ x (* 2.0 y))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-96) {
		tmp = x * (x + (2.0 * y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.5d-96) then
        tmp = x * (x + (2.0d0 * y))
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.5e-96) {
		tmp = x * (x + (2.0 * y));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.5e-96:
		tmp = x * (x + (2.0 * y))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.5e-96)
		tmp = Float64(x * Float64(x + Float64(2.0 * y)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.5e-96)
		tmp = x * (x + (2.0 * y));
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.5e-96], N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.50000000000000001e-96

   1. Initial program 94.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ 94.2%

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

      2. +-commutative 94.2%

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

      3. fma-define 94.2%

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

      4. associate-*l* 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 67.9%

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

   6. Taylor expanded in x around 0 70.8%

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

   if 6.50000000000000001e-96 < y

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 96.3%

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

   4. Taylor expanded in x around 0 72.9%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} + y \cdot y \]
   5. Step-by-step derivation

      1. associate-*r* 72.9%

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

      2. *-commutative 72.9%

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

      3. *-commutative 72.9%

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

      4. *-commutative 72.9%

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

   6. Simplified 72.9%

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

   7. Taylor expanded in x around inf 66.5%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in 59.3%

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

      2. unpow2 59.3%

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

      3. associate-*r/ 59.2%

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

      4. distribute-lft-in 62.8%

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

      5. *-commutative 62.8%

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

      6. distribute-lft-in 62.8%

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

      7. *-commutative 62.8%

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

      8. associate-*l* 69.6%

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

   9. Simplified 69.6%

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

   10. Taylor expanded in y around inf 78.2%

       \[\leadsto y \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.38 \cdot 10^{-95}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.38e-95) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.38e-95) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.38d-95) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.38e-95) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.38e-95:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.38e-95)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.38e-95)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.38e-95], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.38e-95

   1. Initial program 94.2%

      \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l+ 94.2%

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

      2. +-commutative 94.2%

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

      3. fma-define 94.2%

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

      4. associate-*l* 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 95.5%

      \[\leadsto x \cdot x + \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 95.5%

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

      2. unpow2 95.5%

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

      3. associate-/l* 94.3%

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

      4. distribute-lft-out 94.3%

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

   7. Simplified 94.3%

      \[\leadsto x \cdot x + \color{blue}{x \cdot \left(y \cdot \left(2 + \frac{y}{x}\right)\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in x around inf 70.3%

       \[\leadsto x \cdot \color{blue}{x} \]

   if 1.38e-95 < y

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 96.3%

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

   4. Taylor expanded in x around 0 72.9%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} + y \cdot y \]
   5. Step-by-step derivation

      1. associate-*r* 72.9%

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

      2. *-commutative 72.9%

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

      3. *-commutative 72.9%

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

      4. *-commutative 72.9%

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

   6. Simplified 72.9%

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

   7. Taylor expanded in x around inf 66.5%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in 59.3%

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

      2. unpow2 59.3%

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

      3. associate-*r/ 59.2%

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

      4. distribute-lft-in 62.8%

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

      5. *-commutative 62.8%

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

      6. distribute-lft-in 62.8%

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

      7. *-commutative 62.8%

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

      8. associate-*l* 69.6%

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

   9. Simplified 69.6%

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

   10. Taylor expanded in y around inf 78.2%

       \[\leadsto y \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.38 \cdot 10^{-95}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 4.3× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* x x))

C

assert(x < y);
double code(double x, double y) {
	return x * x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x * x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x * x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x * x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x * x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 93.3%

   \[\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. associate-+l+ 93.3%

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

   2. +-commutative 93.3%

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

   3. fma-define 93.3%

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

   4. associate-*l* 93.3%

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

3. Simplified 93.3%

   \[\leadsto \color{blue}{x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 92.1%

   \[\leadsto x \cdot x + \color{blue}{x \cdot \left(2 \cdot y + \frac{{y}^{2}}{x}\right)} \]
6. Step-by-step derivation

   1. *-commutative 92.1%

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

   2. unpow2 92.1%

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

   3. associate-/l* 90.1%

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

   4. distribute-lft-out 90.1%

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

7. Simplified 90.1%

   \[\leadsto x \cdot x + \color{blue}{x \cdot \left(y \cdot \left(2 + \frac{y}{x}\right)\right)} \]
8. Step-by-step derivation

   1. distribute-lft-out 93.6%

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

9. Applied egg-rr 93.6%

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

10. Taylor expanded in x around inf 58.5%

    \[\leadsto x \cdot \color{blue}{x} \]

11. Final simplification 58.5%

    \[\leadsto x \cdot x \]
12. Add Preprocessing

Developer target: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x + \left(y \cdot y + \left(x \cdot y\right) \cdot 2\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * x) + ((y * y) + ((x * y) * 2.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) + ((x * y) * 2.0));
end

Wolfram

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

Reproduce

herbie shell --seed 2024115 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (+ (* x x) (+ (* y y) (* (* x y) 2.0)))

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