Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.1% → 99.9%

Time: 6.4s

Alternatives: 5

Speedup: 1.9×

• 44.1% 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: 94.1% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) + y \cdot y \leq 5 \cdot 10^{+249}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* y (* x 2.0))) (* y y)) 5e+249)
   (+ (* x x) (* y (+ y (* x 2.0))))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((((x * x) + (y * (x * 2.0))) + (y * y)) <= 5e+249) {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	} 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) :: tmp
    if ((((x * x) + (y * (x * 2.0d0))) + (y * y)) <= 5d+249) then
        tmp = (x * x) + (y * (y + (x * 2.0d0)))
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((((x * x) + (y * (x * 2.0))) + (y * y)) <= 5e+249) {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (((x * x) + (y * (x * 2.0))) + (y * y)) <= 5e+249:
		tmp = (x * x) + (y * (y + (x * 2.0)))
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(y * Float64(x * 2.0))) + Float64(y * y)) <= 5e+249)
		tmp = Float64(Float64(x * x) + Float64(y * Float64(y + Float64(x * 2.0))));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((((x * x) + (y * (x * 2.0))) + (y * y)) <= 5e+249)
		tmp = (x * x) + (y * (y + (x * 2.0)));
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+249], N[(N[(x * x), $MachinePrecision] + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $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 #s(literal 2 binary64)) y)) (*.f64 y y)) < 4.9999999999999996e249

   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. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-define 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

   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 4.9999999999999996e249 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   1. Initial program 91.9%

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

      1. associate-+l+ 91.9%

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

      2. associate-*l* 91.9%

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

      3. *-commutative 91.9%

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

      4. *-commutative 91.9%

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

      5. +-commutative 91.9%

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

      6. fma-define 91.9%

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

      7. *-commutative 91.9%

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

      8. *-commutative 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around 0 95.2%

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

   6. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) + y \cdot y \leq 5 \cdot 10^{+249}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. associate-+l+ 96.1%

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

   2. associate-*l* 96.1%

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

   3. *-commutative 96.1%

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

   4. *-commutative 96.1%

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

   5. +-commutative 96.1%

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

   6. fma-define 96.1%

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

   7. *-commutative 96.1%

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

   8. *-commutative 96.1%

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

3. Simplified 96.1%

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

   1. +-commutative 96.1%

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

   2. fma-undefine 96.1%

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

   3. associate-*r* 96.1%

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

   4. associate-+r+ 96.1%

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

   5. +-commutative 96.1%

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

   6. fma-define 96.1%

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

   7. +-commutative 96.1%

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

   8. associate-*r* 96.1%

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

   9. distribute-lft-out 98.4%

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

   10. *-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 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 96.1%

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

   1. associate-+l+ 96.1%

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

   2. associate-*l* 96.1%

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

   3. *-commutative 96.1%

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

   4. *-commutative 96.1%

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

   5. +-commutative 96.1%

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

   6. fma-define 96.1%

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

   7. *-commutative 96.1%

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

   8. *-commutative 96.1%

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

3. Simplified 96.1%

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

5. Taylor expanded in y around 0 97.6%

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

6. Taylor expanded in y around inf 98.1%

   \[\leadsto x \cdot x + y \cdot \color{blue}{y} \]
7. Add Preprocessing

Alternative 4: 15.5% 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 96.1%

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

   1. associate-+l+ 96.1%

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

   2. associate-*l* 96.1%

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

   3. *-commutative 96.1%

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

   4. *-commutative 96.1%

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

   5. +-commutative 96.1%

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

   6. fma-define 96.1%

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

   7. *-commutative 96.1%

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

   8. *-commutative 96.1%

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

3. Simplified 96.1%

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

5. Taylor expanded in y around 0 51.0%

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

   1. *-commutative 51.0%

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

   2. associate-*r* 51.0%

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

   3. *-commutative 51.0%

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

7. Simplified 51.0%

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

8. Taylor expanded in x around 0 11.3%

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

   1. associate-*r* 11.7%

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

10. Simplified 11.7%

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

11. Taylor expanded in x around 0 11.3%

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

    1. associate-*r* 11.7%

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

    2. *-commutative 11.7%

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

    3. associate-*r* 11.3%

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

13. Simplified 11.3%

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

14. Final simplification 11.3%

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

Alternative 5: 15.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. associate-+l+ 96.1%

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

   2. associate-*l* 96.1%

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

   3. *-commutative 96.1%

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

   4. *-commutative 96.1%

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

   5. +-commutative 96.1%

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

   6. fma-define 96.1%

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

   7. *-commutative 96.1%

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

   8. *-commutative 96.1%

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

3. Simplified 96.1%

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

5. Taylor expanded in y around 0 51.0%

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

   1. *-commutative 51.0%

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

   2. associate-*r* 51.0%

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

   3. *-commutative 51.0%

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

7. Simplified 51.0%

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

8. Taylor expanded in x around 0 11.3%

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

9. Final simplification 11.3%

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

Developer Target 1: 94.1% 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 2024139 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* x x) (+ (* y y) (* (* x y) 2))))

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