Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.2% → 100.0%

Time: 7.1s

Alternatives: 6

Speedup: 0.8×

• 44.7% 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: 93.2% 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

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0005:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \left(1 + \frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 0.0005)
   (+ (* x (+ x (* y 2.0))) (* y y))
   (* (pow y 2.0) (+ 1.0 (* (/ x y) (+ 2.0 (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.0005) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = pow(y, 2.0) * (1.0 + ((x / y) * (2.0 + (x / 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 <= 0.0005d0) then
        tmp = (x * (x + (y * 2.0d0))) + (y * y)
    else
        tmp = (y ** 2.0d0) * (1.0d0 + ((x / y) * (2.0d0 + (x / y))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0005) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = Math.pow(y, 2.0) * (1.0 + ((x / y) * (2.0 + (x / y))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.0005:
		tmp = (x * (x + (y * 2.0))) + (y * y)
	else:
		tmp = math.pow(y, 2.0) * (1.0 + ((x / y) * (2.0 + (x / y))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.0005)
		tmp = Float64(Float64(x * Float64(x + Float64(y * 2.0))) + Float64(y * y));
	else
		tmp = Float64((y ^ 2.0) * Float64(1.0 + Float64(Float64(x / y) * Float64(2.0 + Float64(x / y)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0005)
		tmp = (x * (x + (y * 2.0))) + (y * y);
	else
		tmp = (y ^ 2.0) * (1.0 + ((x / y) * (2.0 + (x / 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, 0.0005], N[(N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[Power[y, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-4

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-*l* 94.6%

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

      3. distribute-lft-out 97.3%

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

   4. Applied egg-rr 97.3%

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

   if 5.0000000000000001e-4 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in y around inf 81.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{{x}^{2}}{{y}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 81.7%

         \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{\color{blue}{x \cdot x}}{{y}^{2}}\right)\right) \]

      2. unpow2 81.7%

         \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{x \cdot x}{\color{blue}{y \cdot y}}\right)\right) \]

      3. times-frac 99.9%

         \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right)\right) \]

      4. distribute-rgt-out 99.9%

         \[\leadsto {y}^{2} \cdot \left(1 + \color{blue}{\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)}\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + \frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0005:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot \left(1 + \frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e+132) (+ (* x (+ x (* y 2.0))) (* y y)) (pow y 2.0)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e+132) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = pow(y, 2.0);
	}
	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 <= 8.2d+132) then
        tmp = (x * (x + (y * 2.0d0))) + (y * y)
    else
        tmp = y ** 2.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e+132) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = Math.pow(y, 2.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e+132:
		tmp = (x * (x + (y * 2.0))) + (y * y)
	else:
		tmp = math.pow(y, 2.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e+132)
		tmp = Float64(Float64(x * Float64(x + Float64(y * 2.0))) + Float64(y * y));
	else
		tmp = y ^ 2.0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e+132)
		tmp = (x * (x + (y * 2.0))) + (y * y);
	else
		tmp = y ^ 2.0;
	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, 8.2e+132], N[(N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[Power[y, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.19999999999999983e132

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. associate-*l* 93.5%

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

      3. distribute-lft-out 97.6%

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

   4. Applied egg-rr 97.6%

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

   if 8.19999999999999983e132 < y

   1. Initial program 77.5%

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

   3. Taylor expanded in x around 0 95.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+204}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e+204) (+ (* x (+ x (* y 2.0))) (* y y)) (* y (+ y (* x 2.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+204) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * (y + (x * 2.0));
	}
	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 <= 1.2d+204) then
        tmp = (x * (x + (y * 2.0d0))) + (y * y)
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+204) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.2e+204:
		tmp = (x * (x + (y * 2.0))) + (y * y)
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.2e+204)
		tmp = Float64(Float64(x * Float64(x + Float64(y * 2.0))) + Float64(y * y));
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e+204)
		tmp = (x * (x + (y * 2.0))) + (y * y);
	else
		tmp = y * (y + (x * 2.0));
	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, 1.2e+204], N[(N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e204

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*l* 92.2%

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

      3. distribute-lft-out 97.8%

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

   4. Applied egg-rr 97.8%

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

   if 1.2e204 < y

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0 80.0%

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

      1. associate-*r* 80.0%

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

      2. distribute-rgt-out 96.0%

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

   5. Applied egg-rr 96.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+204}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.5e-128) (* x (+ x (* y 2.0))) (* y (+ y (* x 2.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-128) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	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 <= 3.5d-128) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-128) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.5e-128:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-128)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-128)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * (y + (x * 2.0));
	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, 3.5e-128], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e-128

   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 y around inf 46.7%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{{x}^{2}}{{y}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 46.7%

         \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{\color{blue}{x \cdot x}}{{y}^{2}}\right)\right) \]

      2. unpow2 46.7%

         \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{x \cdot x}{\color{blue}{y \cdot y}}\right)\right) \]

      3. times-frac 63.1%

         \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right)\right) \]

      4. distribute-rgt-out 66.2%

         \[\leadsto {y}^{2} \cdot \left(1 + \color{blue}{\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)}\right) \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + \frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative 66.2%

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right) + 1\right)} \]

      2. flip3-+ 47.1%

         \[\leadsto {y}^{2} \cdot \color{blue}{\frac{{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)}^{3} + {1}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)}} \]

      3. pow3 47.1%

         \[\leadsto {y}^{2} \cdot \frac{\color{blue}{\left(\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)} + {1}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

      4. metadata-eval 47.1%

         \[\leadsto {y}^{2} \cdot \frac{\left(\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \color{blue}{1}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

      5. +-commutative 47.1%

         \[\leadsto {y}^{2} \cdot \frac{\color{blue}{1 + \left(\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

      6. pow3 47.1%

         \[\leadsto {y}^{2} \cdot \frac{1 + \color{blue}{{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)}^{3}}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

      7. associate-*l/ 47.1%

         \[\leadsto {y}^{2} \cdot \frac{1 + {\color{blue}{\left(\frac{x \cdot \left(2 + \frac{x}{y}\right)}{y}\right)}}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

      8. associate-/l* 45.9%

         \[\leadsto {y}^{2} \cdot \frac{1 + {\color{blue}{\left(x \cdot \frac{2 + \frac{x}{y}}{y}\right)}}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

   7. Applied egg-rr 45.9%

      \[\leadsto {y}^{2} \cdot \color{blue}{\frac{1 + {\left(x \cdot \frac{2 + \frac{x}{y}}{y}\right)}^{3}}{{\left(x \cdot \frac{2 + \frac{x}{y}}{y}\right)}^{2} + \left(1 - x \cdot \frac{2 + \frac{x}{y}}{y}\right)}} \]

   8. Taylor expanded in y around 0 65.1%

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

      1. associate-*r* 65.1%

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

      2. +-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*r* 65.1%

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

      6. distribute-lft-in 68.2%

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

   10. Simplified 68.2%

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

   if 3.5e-128 < y

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0 68.1%

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

      1. associate-*r* 68.1%

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

      2. distribute-rgt-out 72.3%

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

   5. Applied egg-rr 72.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 + (y * 2.0d0))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x * (x + (y * 2.0))

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x * (x + (y * 2.0));
end

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in y around inf 61.0%

   \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{{x}^{2}}{{y}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 61.0%

      \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{\color{blue}{x \cdot x}}{{y}^{2}}\right)\right) \]

   2. unpow2 61.0%

      \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \frac{x \cdot x}{\color{blue}{y \cdot y}}\right)\right) \]

   3. times-frac 76.3%

      \[\leadsto {y}^{2} \cdot \left(1 + \left(2 \cdot \frac{x}{y} + \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right)\right) \]

   4. distribute-rgt-out 78.7%

      \[\leadsto {y}^{2} \cdot \left(1 + \color{blue}{\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)}\right) \]

5. Simplified 78.7%

   \[\leadsto \color{blue}{{y}^{2} \cdot \left(1 + \frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 78.7%

      \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right) + 1\right)} \]

   2. flip3-+ 59.3%

      \[\leadsto {y}^{2} \cdot \color{blue}{\frac{{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)}^{3} + {1}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)}} \]

   3. pow3 59.3%

      \[\leadsto {y}^{2} \cdot \frac{\color{blue}{\left(\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)} + {1}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

   4. metadata-eval 59.3%

      \[\leadsto {y}^{2} \cdot \frac{\left(\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \color{blue}{1}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

   5. +-commutative 59.3%

      \[\leadsto {y}^{2} \cdot \frac{\color{blue}{1 + \left(\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

   6. pow3 59.3%

      \[\leadsto {y}^{2} \cdot \frac{1 + \color{blue}{{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right)}^{3}}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

   7. associate-*l/ 59.3%

      \[\leadsto {y}^{2} \cdot \frac{1 + {\color{blue}{\left(\frac{x \cdot \left(2 + \frac{x}{y}\right)}{y}\right)}}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

   8. associate-/l* 58.5%

      \[\leadsto {y}^{2} \cdot \frac{1 + {\color{blue}{\left(x \cdot \frac{2 + \frac{x}{y}}{y}\right)}}^{3}}{\left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) + \left(1 \cdot 1 - \left(\frac{x}{y} \cdot \left(2 + \frac{x}{y}\right)\right) \cdot 1\right)} \]

7. Applied egg-rr 58.5%

   \[\leadsto {y}^{2} \cdot \color{blue}{\frac{1 + {\left(x \cdot \frac{2 + \frac{x}{y}}{y}\right)}^{3}}{{\left(x \cdot \frac{2 + \frac{x}{y}}{y}\right)}^{2} + \left(1 - x \cdot \frac{2 + \frac{x}{y}}{y}\right)}} \]

8. Taylor expanded in y around 0 51.7%

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

   1. associate-*r* 51.7%

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

   2. +-commutative 51.7%

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

   3. unpow2 51.7%

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

   4. *-commutative 51.7%

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

   5. associate-*r* 51.7%

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

   6. distribute-lft-in 57.2%

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

10. Simplified 57.2%

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

11. Final simplification 57.2%

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

Alternative 6: 14.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return 2.0 * (y * 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 = 2.0d0 * (y * x)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return 2.0 * (y * x)

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 2.0 * (y * x);
end

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in x around 0 54.7%

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

4. Taylor expanded in x around inf 16.0%

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

5. Final simplification 16.0%

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

Developer target: 93.2% 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 2024085 
(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)))