Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.2% → 100.0%

Time: 3.2s

Alternatives: 6

Speedup: 1.9×

• 43.3% 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: 94.2% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+278)
   (fma x x (* y (+ (* x 2.0) y)))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+278) {
		tmp = fma(x, x, (y * ((x * 2.0) + y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+278)
		tmp = fma(x, x, Float64(y * Float64(Float64(x * 2.0) + y)));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+278], N[(x * x + N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < 5.00000000000000029e278

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. fma-def 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if 5.00000000000000029e278 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 86.1%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ (* x x) (* (* x 2.0) y)) (* y y))))
   (if (<= t_0 5e+278) t_0 (+ (* x x) (* y y)))))

C

double code(double x, double y) {
	double t_0 = ((x * x) + ((x * 2.0) * y)) + (y * y);
	double tmp;
	if (t_0 <= 5e+278) {
		tmp = t_0;
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * x) + ((x * 2.0d0) * y)) + (y * y)
    if (t_0 <= 5d+278) then
        tmp = t_0
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x * x) + ((x * 2.0) * y)) + (y * y);
	double tmp;
	if (t_0 <= 5e+278) {
		tmp = t_0;
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * x) + ((x * 2.0) * y)) + (y * y)
	tmp = 0
	if t_0 <= 5e+278:
		tmp = t_0
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y))
	tmp = 0.0
	if (t_0 <= 5e+278)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x * x) + ((x * 2.0) * y)) + (y * y);
	tmp = 0.0;
	if (t_0 <= 5e+278)
		tmp = t_0;
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+278], t$95$0, N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < 5.00000000000000029e278

   1. Initial program 100.0%

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

   if 5.00000000000000029e278 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 86.1%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+278}:\\ \;\;\;\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e+135) (+ (* y y) (* x (+ x (* 2.0 y)))) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+135) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d+135) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e+135) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+135:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+135)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e+135)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+135], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e135

   1. Initial program 96.7%

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

      1. associate-+l+ 96.7%

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

      2. fma-def 96.7%

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

      3. distribute-rgt-out 98.1%

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

   3. Simplified 98.1%

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

      1. fma-udef 98.1%

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

      2. distribute-rgt-in 96.7%

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

      3. associate-+l+ 96.7%

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

      4. +-commutative 96.7%

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

      5. associate-*l* 96.7%

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

      6. distribute-lft-out 98.6%

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

   5. Applied egg-rr 98.6%

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

   if 1.99999999999999992e135 < y

   1. Initial program 77.5%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
   3. Step-by-step derivation

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+135}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in x around inf 96.8%

   \[\leadsto \color{blue}{{x}^{2}} + y \cdot y \]
3. Step-by-step derivation

   1. unpow2 96.8%

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

4. Simplified 96.8%

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

5. Final simplification 96.8%

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

Alternative 5: 67.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.8e-114) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-114) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d-114) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-114) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e-114:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-114)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-114)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e-114], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998e-114

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. fma-def 97.0%

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

      3. distribute-rgt-out 98.8%

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

   3. Simplified 98.8%

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

      1. fma-udef 98.8%

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

      2. distribute-rgt-in 97.0%

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

      3. associate-+l+ 97.0%

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

      4. +-commutative 97.0%

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

      5. associate-*l* 97.0%

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

      6. distribute-lft-out 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in y around 0 60.4%

      \[\leadsto \color{blue}{{x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 60.4%

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

   8. Simplified 60.4%

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

   if 3.7999999999999998e-114 < y

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0 75.1%

      \[\leadsto \color{blue}{{y}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 75.1%

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

   4. Simplified 75.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 57.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-+l+ 93.7%

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

   2. fma-def 93.7%

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

   3. distribute-rgt-out 97.3%

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

3. Simplified 97.3%

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

   1. fma-udef 97.2%

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

   2. distribute-rgt-in 93.7%

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

   3. associate-+l+ 93.7%

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

   4. +-commutative 93.7%

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

   5. associate-*l* 93.7%

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

   6. distribute-lft-out 96.5%

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

5. Applied egg-rr 96.5%

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

6. Taylor expanded in y around 0 54.0%

   \[\leadsto \color{blue}{{x}^{2}} \]
7. Step-by-step derivation

   1. unpow2 54.0%

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

8. Simplified 54.0%

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

9. Final simplification 54.0%

   \[\leadsto x \cdot x \]

Developer target: 94.2% 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 2023257 
(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)))