Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.9% → 100.0%

Time: 7.3s

Alternatives: 8

Speedup: 0.8×

• 44.0% 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: 93.9% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + {y}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+306)
   (+ (* x x) (fma y y (* x (* 2.0 y))))
   (+ (* x x) (pow y 2.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+306) {
		tmp = (x * x) + fma(y, y, (x * (2.0 * y)));
	} else {
		tmp = (x * x) + pow(y, 2.0);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+306)
		tmp = Float64(Float64(x * x) + fma(y, y, Float64(x * Float64(2.0 * y))));
	else
		tmp = Float64(Float64(x * x) + (y ^ 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[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+306], N[(N[(x * x), $MachinePrecision] + N[(y * y + N[(x * N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < 4.99999999999999993e306

   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. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 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-def 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%

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

   if 4.99999999999999993e306 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 80.6%

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

      1. associate-+l+ 80.6%

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

      2. *-commutative 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. +-commutative 80.6%

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

      6. fma-def 80.6%

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

      7. *-commutative 80.6%

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

      8. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x \cdot x + \mathsf{fma}\left(y, y, x \cdot \left(2 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + {y}^{2}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + {y}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+306)
   (fma x x (* y (+ (* x 2.0) y)))
   (+ (* x x) (pow y 2.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+306) {
		tmp = fma(x, x, (y * ((x * 2.0) + y)));
	} else {
		tmp = (x * x) + pow(y, 2.0);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+306)
		tmp = fma(x, x, Float64(y * Float64(Float64(x * 2.0) + y)));
	else
		tmp = Float64(Float64(x * x) + (y ^ 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[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+306], N[(x * x + N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < 4.99999999999999993e306

   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. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 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-def 100.0%

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

      7. associate-*l* 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   if 4.99999999999999993e306 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 80.6%

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

      1. associate-+l+ 80.6%

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

      2. *-commutative 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. +-commutative 80.6%

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

      6. fma-def 80.6%

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

      7. *-commutative 80.6%

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

      8. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + {y}^{2}\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+306}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + {y}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+306)
   (+ (* y y) (* x (+ x (* 2.0 y))))
   (+ (* x x) (pow y 2.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+306) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = (x * x) + pow(y, 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 * x) + ((x * 2.0d0) * y)) + (y * y)) <= 5d+306) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    else
        tmp = (x * x) + (y ** 2.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+306) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = (x * x) + Math.pow(y, 2.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+306:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = (x * x) + math.pow(y, 2.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+306)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	else
		tmp = Float64(Float64(x * x) + (y ^ 2.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+306)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	else
		tmp = (x * x) + (y ^ 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[N[(N[(N[(x * x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], 5e+306], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y)) < 4.99999999999999993e306

   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-*r* 100.0%

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

      2. distribute-lft-out 100.0%

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

      3. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

   if 4.99999999999999993e306 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x 2) y)) (*.f64 y y))

   1. Initial program 80.6%

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

      1. associate-+l+ 80.6%

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

      2. *-commutative 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 80.6%

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

      5. +-commutative 80.6%

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

      6. fma-def 80.6%

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

      7. *-commutative 80.6%

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

      8. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in y around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+306}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + {y}^{2}\\ \end{array} \]

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+141}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2 + y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.1e+141) (+ (* y y) (* x (+ x (* 2.0 y)))) (* y (+ (* x 2.0) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+141) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * ((x * 2.0) + y);
	}
	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 <= 5.1d+141) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    else
        tmp = y * ((x * 2.0d0) + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+141) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * ((x * 2.0) + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.1e+141:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = y * ((x * 2.0) + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.1e+141)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	else
		tmp = Float64(y * Float64(Float64(x * 2.0) + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.1e+141)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	else
		tmp = y * ((x * 2.0) + y);
	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, 5.1e+141], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0999999999999997e141

   1. Initial program 93.2%

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

      1. associate-*r* 93.2%

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

      2. distribute-lft-out 97.7%

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

      3. *-commutative 97.7%

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

   3. Applied egg-rr 97.7%

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

   if 5.0999999999999997e141 < y

   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-*r* 83.3%

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

      2. distribute-lft-out 86.1%

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

      3. *-commutative 86.1%

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

   3. Applied egg-rr 86.1%

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

   4. Taylor expanded in x around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. unpow2 83.3%

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

      3. associate-*r* 83.3%

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

      4. *-commutative 83.3%

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

      5. distribute-rgt-out 97.2%

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

      6. *-commutative 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+141}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2 + y\right)\\ \end{array} \]

Alternative 5: 59.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-144}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e-144) (* x x) (* x (+ x (* 2.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-144) {
		tmp = x * x;
	} else {
		tmp = x * (x + (2.0 * y));
	}
	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 <= (-1.15d-144)) then
        tmp = x * x
    else
        tmp = x * (x + (2.0d0 * y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-144) {
		tmp = x * x;
	} else {
		tmp = x * (x + (2.0 * y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.15e-144:
		tmp = x * x
	else:
		tmp = x * (x + (2.0 * y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15e-144)
		tmp = Float64(x * x);
	else
		tmp = Float64(x * Float64(x + Float64(2.0 * y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e-144)
		tmp = x * x;
	else
		tmp = x * (x + (2.0 * y));
	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, -1.15e-144], N[(x * x), $MachinePrecision], N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-144

   1. Initial program 90.3%

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

      1. associate-+l+ 90.3%

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

      2. *-commutative 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-*l* 90.3%

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

      5. +-commutative 90.3%

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

      6. fma-def 90.3%

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

      7. *-commutative 90.3%

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

      8. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around inf 69.0%

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

      1. unpow2 69.0%

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

   6. Applied egg-rr 69.0%

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

   if -1.15e-144 < x

   1. Initial program 92.6%

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

      1. associate-+l+ 92.6%

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

      2. *-commutative 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-*l* 92.6%

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

      5. +-commutative 92.6%

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

      6. fma-def 92.6%

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

      7. *-commutative 92.6%

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

      8. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in x around inf 52.0%

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

      1. +-commutative 52.0%

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

      2. unpow2 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-*r* 52.0%

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

      5. *-commutative 52.0%

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

      6. distribute-rgt-in 56.3%

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

      7. *-commutative 56.3%

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

   6. Simplified 56.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-144}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2 \cdot y\right)\\ \end{array} \]

Alternative 6: 87.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-110}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2 + y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-110) (* x x) (* y (+ (* x 2.0) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-110) {
		tmp = x * x;
	} else {
		tmp = y * ((x * 2.0) + y);
	}
	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 <= (-3.5d-110)) then
        tmp = x * x
    else
        tmp = y * ((x * 2.0d0) + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-110) {
		tmp = x * x;
	} else {
		tmp = y * ((x * 2.0) + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.5e-110:
		tmp = x * x
	else:
		tmp = y * ((x * 2.0) + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-110)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(Float64(x * 2.0) + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-110)
		tmp = x * x;
	else
		tmp = y * ((x * 2.0) + y);
	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, -3.5e-110], N[(x * x), $MachinePrecision], N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999974e-110

   1. Initial program 89.5%

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

      1. associate-+l+ 89.5%

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

      2. *-commutative 89.5%

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

      3. *-commutative 89.5%

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

      4. associate-*l* 89.5%

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

      5. +-commutative 89.5%

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

      6. fma-def 89.5%

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

      7. *-commutative 89.5%

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

      8. *-commutative 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around inf 74.4%

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

      1. unpow2 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -3.49999999999999974e-110 < 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-*r* 92.9%

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

      2. distribute-lft-out 97.0%

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

      3. *-commutative 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

      2. unpow2 59.7%

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

      3. associate-*r* 59.7%

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

      4. *-commutative 59.7%

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

      5. distribute-rgt-out 62.7%

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

      6. *-commutative 62.7%

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

   6. Simplified 62.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-110}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2 + y\right)\\ \end{array} \]

Alternative 7: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-291) (* x x) (* (* x 2.0) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-291) {
		tmp = x * x;
	} else {
		tmp = (x * 2.0) * y;
	}
	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 <= (-5.5d-291)) then
        tmp = x * x
    else
        tmp = (x * 2.0d0) * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-291) {
		tmp = x * x;
	} else {
		tmp = (x * 2.0) * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.5e-291:
		tmp = x * x
	else:
		tmp = (x * 2.0) * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-291)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(x * 2.0) * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e-291)
		tmp = x * x;
	else
		tmp = (x * 2.0) * y;
	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, -5.5e-291], N[(x * x), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5000000000000002e-291

   1. Initial program 92.5%

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

      1. associate-+l+ 92.5%

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

      2. *-commutative 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*l* 92.5%

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

      5. +-commutative 92.5%

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

      6. fma-def 92.5%

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

      7. *-commutative 92.5%

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

      8. *-commutative 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around inf 59.8%

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

      1. unpow2 59.8%

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

   6. Applied egg-rr 59.8%

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

   if -5.5000000000000002e-291 < x

   1. Initial program 91.1%

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

      1. associate-+l+ 91.1%

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

      2. *-commutative 91.1%

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

      3. *-commutative 91.1%

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

      4. associate-*l* 91.1%

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

      5. +-commutative 91.1%

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

      6. fma-def 91.2%

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

      7. *-commutative 91.2%

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

      8. *-commutative 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. +-commutative 56.8%

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

      2. unpow2 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-*r* 56.8%

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

      5. *-commutative 56.8%

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

      6. distribute-rgt-in 61.9%

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

      7. *-commutative 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in x around 0 17.3%

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

      1. associate-*r* 17.3%

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

      2. *-commutative 17.3%

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

      3. *-commutative 17.3%

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

      4. *-commutative 17.3%

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

   9. Simplified 17.3%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot y\\ \end{array} \]

Alternative 8: 57.0% 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 91.8%

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

   1. associate-+l+ 91.8%

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

   2. *-commutative 91.8%

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

   3. *-commutative 91.8%

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

   4. associate-*l* 91.8%

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

   5. +-commutative 91.8%

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

   6. fma-def 91.8%

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

   7. *-commutative 91.8%

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

   8. *-commutative 91.8%

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

3. Simplified 91.8%

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

4. Taylor expanded in x around inf 61.4%

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

   1. unpow2 61.4%

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

6. Applied egg-rr 61.4%

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

7. Final simplification 61.4%

   \[\leadsto x \cdot x \]

Developer target: 93.9% 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 2023336 
(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)))