Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.0% → 98.8%

Time: 4.0s

Alternatives: 5

Speedup: 1.0×

• 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: 94.0% 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, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+233}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e+233) (+ (* x (+ x (* y 2.0))) (* y y)) (* y (+ y (* x 2.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+233) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.5d+233) then
        tmp = (x * (x + (y * 2.0d0))) + (y * y)
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.5e+233) {
		tmp = (x * (x + (y * 2.0))) + (y * y);
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.5e+233:
		tmp = (x * (x + (y * 2.0))) + (y * y)
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.5e+233)
		tmp = Float64(Float64(x * Float64(x + Float64(y * 2.0))) + Float64(y * y));
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5e+233)
		tmp = (x * (x + (y * 2.0))) + (y * y);
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.5e+233], 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 < 9.5000000000000001e233

   1. Initial program 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-*r* 94.6%

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

      3. distribute-lft-out 97.9%

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

      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 9.5000000000000001e233 < y

   1. Initial program 73.3%

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

      1. +-commutative 73.3%

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

      2. associate-*r* 73.3%

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

      3. distribute-lft-out 73.3%

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

      4. *-commutative 73.3%

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

   3. Applied egg-rr 73.3%

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

   4. Taylor expanded in x around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. unpow2 73.3%

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

      3. associate-*r* 73.3%

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

      4. *-commutative 73.3%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+233}:\\ \;\;\;\;x \cdot \left(x + y \cdot 2\right) + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x * x) + (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[(N[(x * x), $MachinePrecision] + N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. +-commutative 93.3%

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

   3. fma-def 93.4%

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

   4. associate-*l* 93.4%

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

3. Simplified 93.4%

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

4. Taylor expanded in y around inf 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-16} \lor \neg \left(x \leq -1.45 \cdot 10^{-24}\right) \land x \leq -1.26 \cdot 10^{-107}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.3e-16) (and (not (<= x -1.45e-24)) (<= x -1.26e-107)))
   (* x x)
   (* y (+ y (* x 2.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -2.3e-16) || (!(x <= -1.45e-24) && (x <= -1.26e-107))) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.3d-16)) .or. (.not. (x <= (-1.45d-24))) .and. (x <= (-1.26d-107))) then
        tmp = x * x
    else
        tmp = y * (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.3e-16) || (!(x <= -1.45e-24) && (x <= -1.26e-107))) {
		tmp = x * x;
	} else {
		tmp = y * (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -2.3e-16) or (not (x <= -1.45e-24) and (x <= -1.26e-107)):
		tmp = x * x
	else:
		tmp = y * (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -2.3e-16) || (!(x <= -1.45e-24) && (x <= -1.26e-107)))
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.3e-16) || (~((x <= -1.45e-24)) && (x <= -1.26e-107)))
		tmp = x * x;
	else
		tmp = y * (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[x, -2.3e-16], And[N[Not[LessEqual[x, -1.45e-24]], $MachinePrecision], LessEqual[x, -1.26e-107]]], N[(x * x), $MachinePrecision], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999999e-16 or -1.4499999999999999e-24 < x < -1.2599999999999999e-107

   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. +-commutative 90.5%

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

      3. fma-def 90.5%

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

      4. associate-*l* 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in x around inf 73.2%

      \[\leadsto \color{blue}{{x}^{2}} \]
   5. Step-by-step derivation

      1. pow2 73.2%

         \[\leadsto \color{blue}{x \cdot x} \]

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{x \cdot x} \]

   if -2.2999999999999999e-16 < x < -1.4499999999999999e-24 or -1.2599999999999999e-107 < x

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. associate-*r* 94.5%

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

      3. distribute-lft-out 97.2%

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

      4. *-commutative 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around 0 67.9%

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

      1. +-commutative 67.9%

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

      2. unpow2 67.9%

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

      3. associate-*r* 67.9%

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

      4. *-commutative 67.9%

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

      5. distribute-rgt-out 70.7%

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

      6. *-commutative 70.7%

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

   6. Simplified 70.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-16} \lor \neg \left(x \leq -1.45 \cdot 10^{-24}\right) \land x \leq -1.26 \cdot 10^{-107}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]

Alternative 4: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-306}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.35e-306) (* x x) (* y (* x 2.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.35e-306) {
		tmp = x * x;
	} else {
		tmp = 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 <= (-2.35d-306)) then
        tmp = x * x
    else
        tmp = 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 <= -2.35e-306) {
		tmp = x * x;
	} else {
		tmp = y * (x * 2.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.35e-306:
		tmp = x * x
	else:
		tmp = y * (x * 2.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.35e-306)
		tmp = Float64(x * x);
	else
		tmp = 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 <= -2.35e-306)
		tmp = x * x;
	else
		tmp = 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, -2.35e-306], N[(x * x), $MachinePrecision], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e-306

   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. associate-+l+ 93.8%

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

      2. +-commutative 93.8%

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

      3. fma-def 93.9%

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

      4. associate-*l* 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in x around inf 55.5%

      \[\leadsto \color{blue}{{x}^{2}} \]
   5. Step-by-step derivation

      1. pow2 55.5%

         \[\leadsto \color{blue}{x \cdot x} \]

   6. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{x \cdot x} \]

   if -2.3500000000000001e-306 < x

   1. Initial program 92.9%

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

      1. associate-+l+ 92.9%

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

      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 92.9%

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

      4. associate-*l* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in y around 0 48.8%

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

   5. Taylor expanded in x around 0 16.1%

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

      1. associate-*r* 16.1%

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

      2. *-commutative 16.1%

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

      3. *-commutative 16.1%

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

      4. *-commutative 16.1%

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

   7. Simplified 16.1%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-306}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \]

Alternative 5: 57.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. +-commutative 93.3%

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

   3. fma-def 93.4%

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

   4. associate-*l* 93.4%

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

3. Simplified 93.4%

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

4. Taylor expanded in x around inf 54.4%

   \[\leadsto \color{blue}{{x}^{2}} \]
5. Step-by-step derivation

   1. pow2 54.4%

      \[\leadsto \color{blue}{x \cdot x} \]

6. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{x \cdot x} \]

7. Final simplification 54.4%

   \[\leadsto x \cdot x \]

Developer target: 94.0% 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 2023308 
(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)))