Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 93.5% → 99.9%

Time: 5.0s

Alternatives: 5

Speedup: 1.9×

• 43.6% 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: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2 + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+242)
   (fma x x (* y (+ (* x 2.0) y)))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+242) {
		tmp = fma(x, x, (y * ((x * 2.0) + y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+242)
		tmp = fma(x, x, Float64(y * Float64(Float64(x * 2.0) + y)));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

Wolfram

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+242], N[(x * x + N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y)) < 5.0000000000000004e242

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*l* 100.0%

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

      10. distribute-rgt-out 100.0%

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

      11. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 5.0000000000000004e242 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

      2. associate-*l* 88.1%

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

      3. distribute-lft-out 93.3%

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in y around 0 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+242}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (+ (* x x) (* (* x 2.0) y)) (* y y)) 5e+242)
   (+ (* y y) (* x (+ x (* 2.0 y))))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+242) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((((x * x) + ((x * 2.0d0) * y)) + (y * y)) <= 5d+242) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+242) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+242:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) + Float64(Float64(x * 2.0) * y)) + Float64(y * y)) <= 5e+242)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((((x * x) + ((x * 2.0) * y)) + (y * y)) <= 5e+242)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

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+242], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y)) < 5.0000000000000004e242

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-lft-out 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 5.0000000000000004e242 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   1. Initial program 88.1%

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

      1. +-commutative 88.1%

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

      2. associate-*l* 88.1%

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

      3. distribute-lft-out 93.3%

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

   4. Applied egg-rr 93.3%

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

   5. Taylor expanded in y around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot x + \left(x \cdot 2\right) \cdot y\right) + y \cdot y \leq 5 \cdot 10^{+242}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-84}:\\ \;\;\;\;y \cdot \left(x \cdot 2 + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.45e-84) (* y (+ (* x 2.0) y)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.45e-84) {
		tmp = y * ((x * 2.0) + y);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.45d-84) then
        tmp = y * ((x * 2.0d0) + y)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.45e-84) {
		tmp = y * ((x * 2.0) + y);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.45e-84:
		tmp = y * ((x * 2.0) + y)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.45e-84)
		tmp = Float64(y * Float64(Float64(x * 2.0) + y));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.45e-84)
		tmp = y * ((x * 2.0) + y);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.45e-84], N[(y * N[(N[(x * 2.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.44999999999999994e-84

   1. Initial program 97.1%

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

      1. associate-+l+ 97.1%

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

      2. associate-*l* 97.1%

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

      3. *-commutative 97.1%

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

      4. *-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. fma-define 97.1%

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

      7. *-commutative 97.1%

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

      8. *-commutative 97.1%

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

      9. associate-*l* 97.1%

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

      10. distribute-rgt-out 98.9%

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

      11. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 52.3%

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

      1. associate-*r* 52.3%

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

      2. *-commutative 52.3%

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

      3. unpow2 52.3%

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

      4. distribute-rgt-in 54.0%

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

      5. fma-undefine 54.0%

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

   7. Simplified 54.0%

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

      1. fma-undefine 54.0%

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

   9. Applied egg-rr 54.0%

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

   if 3.44999999999999994e-84 < y

   1. Initial program 86.4%

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

      1. associate-+l+ 86.4%

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

      2. associate-*l* 86.4%

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

      3. *-commutative 86.4%

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

      4. *-commutative 86.4%

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

      5. +-commutative 86.4%

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

      6. fma-define 86.4%

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

      7. *-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-*l* 86.4%

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

      10. distribute-rgt-out 93.8%

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

      11. +-commutative 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in x around 0 65.0%

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

      1. associate-*r* 65.0%

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

      2. *-commutative 65.0%

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

      3. unpow2 65.0%

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

      4. distribute-rgt-in 72.5%

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

      5. fma-undefine 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in x around 0 71.8%

      \[\leadsto y \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x \cdot x + y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. +-commutative 93.7%

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

   2. associate-*l* 93.7%

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

   3. distribute-lft-out 96.5%

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

4. Applied egg-rr 96.5%

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

5. Taylor expanded in y around 0 98.0%

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

Alternative 5: 58.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

   1. associate-+l+ 93.7%

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

   2. associate-*l* 93.7%

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

   3. *-commutative 93.7%

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

   4. *-commutative 93.7%

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

   5. +-commutative 93.7%

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

   6. fma-define 93.7%

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

   7. *-commutative 93.7%

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

   8. *-commutative 93.7%

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

   9. associate-*l* 93.7%

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

   10. distribute-rgt-out 97.3%

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

   11. +-commutative 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in x around 0 56.3%

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

   1. associate-*r* 56.3%

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

   2. *-commutative 56.3%

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

   3. unpow2 56.3%

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

   4. distribute-rgt-in 59.8%

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

   5. fma-undefine 59.8%

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

7. Simplified 59.8%

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

8. Taylor expanded in x around 0 59.1%

   \[\leadsto y \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 93.6% 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 2024146 
(FPCore (x y)
  :name "Examples.Basics.ProofTests:f4 from sbv-4.4"
  :precision binary64

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

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