Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.3% → 98.7%

Time: 5.4s

Alternatives: 5

Speedup: 0.8×

• 44.6% 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: 93.3% 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

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot \left(x + y \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1e+181)
   (fma y y (* x (+ x (* y 2.0))))
   (fma x x (* y (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1e+181) {
		tmp = fma(y, y, (x * (x + (y * 2.0))));
	} else {
		tmp = fma(x, x, (y * (y + (x * 2.0))));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1e+181)
		tmp = fma(y, y, Float64(x * Float64(x + Float64(y * 2.0))));
	else
		tmp = fma(x, x, Float64(y * Float64(y + Float64(x * 2.0))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1e+181], N[(y * y + N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999992e180

   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+ 92.0%

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

      2. associate-*l* 92.0%

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

      3. *-commutative 92.0%

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

      4. *-commutative 92.0%

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

      5. +-commutative 92.0%

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

      6. fma-define 92.0%

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

      7. *-commutative 92.0%

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

      8. *-commutative 92.0%

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

   3. Simplified 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-undefine 92.0%

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

      3. associate-*r* 92.0%

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

      4. associate-+r+ 91.9%

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

      5. +-commutative 91.9%

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

      6. fma-define 92.0%

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

      7. +-commutative 92.0%

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

      8. associate-*r* 92.0%

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

      9. distribute-lft-out 96.9%

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

      10. *-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 9.9999999999999992e180 < y

   1. Initial program 78.1%

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

      1. associate-+l+ 78.1%

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

      2. associate-*l* 78.1%

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

      3. *-commutative 78.1%

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

      4. *-commutative 78.1%

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

      5. +-commutative 78.1%

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

      6. fma-define 78.1%

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

      7. *-commutative 78.1%

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

      8. *-commutative 78.1%

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

      9. associate-*l* 78.1%

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

      10. distribute-rgt-out 96.9%

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

      11. +-commutative 96.9%

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

   3. Simplified 96.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot \left(x + y \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+244}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+244) (* x (+ x (* y 2.0))) (fma x x (* y (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+244) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = fma(x, x, (y * (y + (x * 2.0))));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+244)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = fma(x, x, Float64(y * Float64(y + Float64(x * 2.0))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.4e+244], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999988e244

   1. Initial program 84.6%

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

      1. associate-+l+ 84.6%

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

      2. associate-*l* 84.6%

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

      3. *-commutative 84.6%

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

      4. *-commutative 84.6%

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

      5. +-commutative 84.6%

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

      6. fma-define 84.6%

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

      7. *-commutative 84.6%

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

      8. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in y around 0 84.6%

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

   6. Taylor expanded in x around 0 100.0%

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

   if -2.39999999999999988e244 < x

   1. Initial program 90.5%

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

      1. associate-+l+ 90.5%

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

      2. associate-*l* 90.5%

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

      3. *-commutative 90.5%

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

      4. *-commutative 90.5%

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

      5. +-commutative 90.5%

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

      6. fma-define 90.5%

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

      7. *-commutative 90.5%

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

      8. *-commutative 90.5%

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

      9. associate-*l* 90.5%

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

      10. distribute-rgt-out 95.9%

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

      11. +-commutative 95.9%

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

   3. Simplified 95.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+244}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e+203) (* x (+ x (* y 2.0))) (+ (* x x) (* y (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.3e+203) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = (x * x) + (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 (x <= (-1.3d+203)) then
        tmp = x * (x + (y * 2.0d0))
    else
        tmp = (x * x) + (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 (x <= -1.3e+203) {
		tmp = x * (x + (y * 2.0));
	} else {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.3e+203:
		tmp = x * (x + (y * 2.0))
	else:
		tmp = (x * x) + (y * (y + (x * 2.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.3e+203)
		tmp = Float64(x * Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(Float64(x * x) + 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 (x <= -1.3e+203)
		tmp = x * (x + (y * 2.0));
	else
		tmp = (x * x) + (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[x, -1.3e+203], N[(x * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2999999999999999e203

   1. Initial program 83.3%

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

      1. associate-+l+ 83.3%

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

      2. associate-*l* 83.3%

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

      3. *-commutative 83.3%

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

      4. *-commutative 83.3%

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

      5. +-commutative 83.3%

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

      6. fma-define 83.3%

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

      7. *-commutative 83.3%

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

      8. *-commutative 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in y around 0 83.3%

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

   6. Taylor expanded in x around 0 95.8%

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

   if -1.2999999999999999e203 < x

   1. Initial program 90.9%

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

      1. associate-+l+ 90.9%

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

      2. associate-*l* 90.9%

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

      3. *-commutative 90.9%

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

      4. *-commutative 90.9%

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

      5. +-commutative 90.9%

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

      6. fma-define 90.9%

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

      7. *-commutative 90.9%

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

      8. *-commutative 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in y around 0 96.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.7% 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 90.2%

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

   1. associate-+l+ 90.2%

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

   2. associate-*l* 90.2%

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

   3. *-commutative 90.2%

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

   4. *-commutative 90.2%

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

   5. +-commutative 90.2%

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

   6. fma-define 90.2%

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

   7. *-commutative 90.2%

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

   8. *-commutative 90.2%

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

3. Simplified 90.2%

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

5. Taylor expanded in y around 0 50.3%

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

6. Taylor expanded in x around 0 55.4%

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

7. Final simplification 55.4%

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

Alternative 5: 15.1% 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 90.2%

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

   1. associate-+l+ 90.2%

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

   2. associate-*l* 90.2%

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

   3. *-commutative 90.2%

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

   4. *-commutative 90.2%

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

   5. +-commutative 90.2%

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

   6. fma-define 90.2%

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

   7. *-commutative 90.2%

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

   8. *-commutative 90.2%

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

3. Simplified 90.2%

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

5. Taylor expanded in y around 0 50.3%

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

6. Taylor expanded in x around 0 11.0%

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

7. Final simplification 11.0%

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

Developer target: 93.3% 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 2024059 
(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)))