Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.4% → 98.7%

Time: 4.6s

Alternatives: 5

Speedup: 1.0×

• 43.7% 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: 93.4% 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: 98.7% 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 93.0%

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

   1. associate-+l+ 92.9%

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

   2. +-commutative 92.9%

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

   3. fma-def 93.0%

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

   4. associate-*l* 93.0%

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

3. Simplified 93.0%

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

4. Taylor expanded in y around inf 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+247}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e+247) (+ (* x (+ x (* y 2.0))) (* y y)) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e+247) {
		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 <= 4.8d+247) 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 <= 4.8e+247) {
		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 <= 4.8e+247:
		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 <= 4.8e+247)
		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 <= 4.8e+247)
		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, 4.8e+247], 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 < 4.8e247

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. associate-*r* 93.8%

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

      3. distribute-lft-out 97.9%

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

      4. *-commutative 97.9%

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

   3. Applied egg-rr 97.9%

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

   if 4.8e247 < y

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-*r* 75.0%

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

      3. distribute-lft-out 75.0%

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

      4. *-commutative 75.0%

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

   3. Applied egg-rr 75.0%

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

   4. Taylor expanded in x around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*r* 75.0%

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

      4. *-commutative 75.0%

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

      5. distribute-rgt-out 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+247}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 3: 68.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e-100) (* x (+ x (* y 2.0))) (* y (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e-100) {
		tmp = 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 (x <= (-6d-100)) then
        tmp = 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 (x <= -6e-100) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e-100:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e-100)
		tmp = Float64(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 (x <= -6e-100)
		tmp = x * (x + (y * 2.0));
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e-100], 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 x < -6.0000000000000001e-100

   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. +-commutative 91.9%

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

      3. fma-def 91.9%

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

      4. associate-*l* 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. *-commutative 70.6%

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

      4. *-commutative 70.6%

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

   6. Simplified 70.6%

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

      1. fma-def 70.6%

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

      2. associate-*r* 70.6%

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

      3. *-commutative 70.6%

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

   8. Applied egg-rr 70.6%

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

      1. fma-udef 70.6%

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

      2. distribute-lft-out 74.6%

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

   10. Applied egg-rr 74.6%

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

   if -6.0000000000000001e-100 < x

   1. Initial program 93.6%

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

      1. +-commutative 93.6%

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

      2. associate-*r* 93.6%

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

      3. distribute-lft-out 97.4%

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

      4. *-commutative 97.4%

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

   3. Applied egg-rr 97.4%

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

   4. Taylor expanded in x around 0 67.0%

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

      1. +-commutative 67.0%

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

      2. unpow2 67.0%

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

      3. associate-*r* 67.0%

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

      4. *-commutative 67.0%

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

      5. distribute-rgt-out 69.6%

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

      6. *-commutative 69.6%

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

   6. Simplified 69.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 4: 56.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * (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 * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. associate-+l+ 92.9%

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

   2. +-commutative 92.9%

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

   3. fma-def 93.0%

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

   4. associate-*l* 93.0%

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

3. Simplified 93.0%

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

4. Taylor expanded in y around 0 54.1%

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

   1. associate-*r* 54.1%

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

   2. *-commutative 54.1%

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

   3. *-commutative 54.1%

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

   4. *-commutative 54.1%

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

6. Simplified 54.1%

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

   1. fma-def 54.1%

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

   2. associate-*r* 54.1%

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

   3. *-commutative 54.1%

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

8. Applied egg-rr 54.1%

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

   1. fma-udef 54.1%

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

   2. distribute-lft-out 58.0%

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

10. Applied egg-rr 58.0%

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

11. Final simplification 58.0%

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

Alternative 5: 14.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. associate-+l+ 92.9%

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

   2. +-commutative 92.9%

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

   3. fma-def 93.0%

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

   4. associate-*l* 93.0%

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

3. Simplified 93.0%

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

4. Taylor expanded in y around 0 54.1%

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

   1. associate-*r* 54.1%

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

   2. *-commutative 54.1%

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

   3. *-commutative 54.1%

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

   4. *-commutative 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in x around 0 18.2%

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

   1. *-commutative 18.2%

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

   2. associate-*r* 18.2%

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

   3. *-commutative 18.2%

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

   4. associate-*r* 18.2%

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

9. Simplified 18.2%

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

10. Final simplification 18.2%

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

Developer target: 93.4% 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 2023310 
(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)))