Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.2% → 98.8%

Time: 4.3s

Alternatives: 6

Speedup: 1.0×

• 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: 93.2% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ x \cdot x + {y}^{2} \end{array} \]

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-+l+ 90.6%

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

   2. associate-*l* 90.6%

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

   3. *-commutative 90.6%

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

   4. *-commutative 90.6%

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

   5. +-commutative 90.6%

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

   6. fma-def 90.6%

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

   7. *-commutative 90.6%

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

   8. *-commutative 90.6%

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

3. Simplified 90.6%

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

5. Taylor expanded in y around inf 99.6%

   \[\leadsto x \cdot x + \color{blue}{{y}^{2}} \]

6. Final simplification 99.6%

   \[\leadsto x \cdot x + {y}^{2} \]
7. Add Preprocessing

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e+214) (+ (* x (+ x (* y 2.0))) (* y y)) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.3e+214) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.3d+214) then
        tmp = (x * (x + (y * 2.0d0))) + (y * y)
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e+214) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.3e+214:
		tmp = (x * (x + (y * 2.0))) + (y * y)
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.3e+214)
		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

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e+214)
		tmp = (x * (x + (y * 2.0))) + (y * y);
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.3e+214], 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 < 4.29999999999999983e214

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. associate-*r* 92.3%

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

      3. distribute-lft-out 96.6%

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

      4. *-commutative 96.6%

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

   4. Applied egg-rr 96.6%

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

   if 4.29999999999999983e214 < y

   1. Initial program 73.9%

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

      1. +-commutative 73.9%

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

      2. associate-*r* 73.9%

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

      3. distribute-lft-out 82.6%

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

      4. *-commutative 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in x around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. associate-*r* 73.9%

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

      4. *-commutative 73.9%

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

      5. distribute-rgt-out 91.3%

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

      6. *-commutative 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + x \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-13) (+ (* x x) (* x (* y 2.0))) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-13) {
		tmp = (x * x) + (x * (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.9d-13) then
        tmp = (x * x) + (x * (y * 2.0d0))
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-13) {
		tmp = (x * x) + (x * (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.9e-13:
		tmp = (x * x) + (x * (y * 2.0))
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-13)
		tmp = Float64(Float64(x * x) + Float64(x * Float64(y * 2.0)));
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-13)
		tmp = (x * x) + (x * (y * 2.0));
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.9e-13], N[(N[(x * x), $MachinePrecision] + 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 < 3.90000000000000004e-13

   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. associate-*l* 93.3%

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

      3. *-commutative 93.3%

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

      4. *-commutative 93.3%

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

      5. +-commutative 93.3%

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

      6. fma-def 93.3%

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

      7. *-commutative 93.3%

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

      8. *-commutative 93.3%

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

   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 y around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. *-commutative 62.2%

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

      3. associate-*r* 62.2%

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

   7. Simplified 62.2%

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

   if 3.90000000000000004e-13 < y

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. associate-*r* 84.2%

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

      3. distribute-lft-out 92.1%

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

      4. *-commutative 92.1%

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

   4. Applied egg-rr 92.1%

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

   5. Taylor expanded in x around 0 68.8%

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

      1. +-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. associate-*r* 68.8%

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

      4. *-commutative 68.8%

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

      5. distribute-rgt-out 76.7%

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

      6. *-commutative 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + x \cdot \left(y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. +-commutative 90.6%

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

   2. associate-*r* 90.6%

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

   3. distribute-lft-out 95.3%

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

   4. *-commutative 95.3%

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

4. Applied egg-rr 95.3%

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

5. Taylor expanded in x around 0 50.5%

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

   1. +-commutative 50.5%

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

   2. unpow2 50.5%

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

   3. associate-*r* 50.5%

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

   4. *-commutative 50.5%

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

   5. distribute-rgt-out 55.2%

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

   6. *-commutative 55.2%

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

7. Simplified 55.2%

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

8. Final simplification 55.2%

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

Alternative 5: 15.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-+l+ 90.6%

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

   2. associate-*l* 90.6%

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

   3. *-commutative 90.6%

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

   4. *-commutative 90.6%

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

   5. +-commutative 90.6%

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

   6. fma-def 90.6%

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

   7. *-commutative 90.6%

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

   8. *-commutative 90.6%

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

3. Simplified 90.6%

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

5. Taylor expanded in y around 0 52.1%

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

   1. associate-*r* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*r* 52.5%

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

7. Simplified 52.5%

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

8. Taylor expanded in x around 0 12.0%

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

9. Final simplification 12.0%

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

Alternative 6: 15.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-+l+ 90.6%

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

   2. associate-*l* 90.6%

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

   3. *-commutative 90.6%

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

   4. *-commutative 90.6%

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

   5. +-commutative 90.6%

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

   6. fma-def 90.6%

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

   7. *-commutative 90.6%

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

   8. *-commutative 90.6%

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

3. Simplified 90.6%

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

5. Taylor expanded in y around 0 52.1%

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

   1. associate-*r* 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*r* 52.5%

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

7. Simplified 52.5%

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

8. Taylor expanded in x around 0 12.0%

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

   1. associate-*r* 12.0%

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

   2. *-commutative 12.0%

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

   3. associate-*r* 12.4%

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

10. Simplified 12.4%

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

11. Final simplification 12.4%

    \[\leadsto x \cdot \left(y \cdot 2\right) \]
12. 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 2024010 
(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)))