Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 92.8% → 99.7%

Time: 3.1s

Alternatives: 4

Speedup: 0.8×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.8% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1e+190) (* x x) (+ (* x x) (* y (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e+190) {
		tmp = x * x;
	} else {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d+190)) then
        tmp = x * x
    else
        tmp = (x * x) + (y * (y + (x * 2.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1e+190) {
		tmp = x * x;
	} else {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1e+190:
		tmp = x * x
	else:
		tmp = (x * x) + (y * (y + (x * 2.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1e+190)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(x * x) + Float64(y * Float64(y + Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+190)
		tmp = x * x;
	else
		tmp = (x * x) + (y * (y + (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1e+190], N[(x * x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e190

   1. Initial program 78.1%

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

      1. associate-+l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 81.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 81.3%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 100.0%

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

   if -1.0000000000000001e190 < x

   1. Initial program 92.4%

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

      1. associate-+l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+190}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-76}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e-76) (* x x) (* y (+ y (* x 2.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-76) {
		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 <= (-2.25d-76)) 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 <= -2.25e-76) {
		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 <= -2.25e-76:
		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 <= -2.25e-76)
		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 <= -2.25e-76)
		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[LessEqual[x, -2.25e-76], 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 < -2.25e-76

   1. Initial program 83.7%

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

      1. associate-+l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 93.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 93.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 80.3%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 80.3%

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

   if -2.25e-76 < x

   1. Initial program 94.1%

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

      1. associate-+l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0

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

      1. fma-define N/A

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

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(2, x \cdot \left(y \cdot \color{blue}{1}\right), {y}^{2}\right) \]

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. fma-define N/A

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

      9. associate-*l* N/A

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

      10. *-lft-identity N/A

          \[\leadsto \left(2 \cdot \frac{x}{y}\right) \cdot {y}^{2} + 1 \cdot \color{blue}{{y}^{2}} \]

      11. distribute-rgt-in N/A

          \[\leadsto {y}^{2} \cdot \color{blue}{\left(2 \cdot \frac{x}{y} + 1\right)} \]

      12. +-commutative N/A

          \[\leadsto {y}^{2} \cdot \left(1 + \color{blue}{2 \cdot \frac{x}{y}}\right) \]

      13. unpow2 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\color{blue}{1} + 2 \cdot \frac{x}{y}\right) \]

      14. associate-*l* N/A

          \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(1 + 2 \cdot \frac{x}{y}\right)\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + 2 \cdot \frac{x}{y}\right)\right)}\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot 1 + \color{blue}{y \cdot \left(2 \cdot \frac{x}{y}\right)}\right)\right) \]

      17. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y + \color{blue}{y} \cdot \left(2 \cdot \frac{x}{y}\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y + y \cdot \frac{2 \cdot x}{\color{blue}{y}}\right)\right) \]

      19. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y + \frac{y \cdot \left(2 \cdot x\right)}{\color{blue}{y}}\right)\right) \]

      20. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y + \frac{y}{y} \cdot \color{blue}{\left(2 \cdot x\right)}\right)\right) \]

      21. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y + 1 \cdot \left(\color{blue}{2} \cdot x\right)\right)\right) \]

      22. *-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(y + 2 \cdot \color{blue}{x}\right)\right) \]

      23. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{\left(2 \cdot x\right)}\right)\right) \]

      24. *-lowering-*.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{*.f64}\left(2, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 68.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-76}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-76}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -2.25e-76) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-76) {
		tmp = x * x;
	} else {
		tmp = y * 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 <= (-2.25d-76)) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-76) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.25e-76:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.25e-76)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.25e-76)
		tmp = x * x;
	else
		tmp = y * 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, -2.25e-76], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.25e-76

   1. Initial program 83.7%

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

      1. associate-+l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 93.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 93.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 80.3%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 80.3%

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

   if -2.25e-76 < x

   1. Initial program 94.1%

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

      1. associate-+l+ N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   7. Simplified 68.3%

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

Alternative 4: 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 90.6%

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

   1. associate-+l+ N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\left(x \cdot 2\right) \cdot y + y \cdot y\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\left(x \cdot 2\right) \cdot y} + y \cdot y\right)\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot 2 + y\right)}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(x \cdot 2\right), \color{blue}{y}\right)\right)\right) \]

   7. *-lowering-*.f64 96.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right)\right)\right) \]

3. Simplified 96.1%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 59.7%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

7. Simplified 59.7%

   \[\leadsto \color{blue}{x \cdot x} \]
8. Add Preprocessing

Developer Target 1: 92.9% 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 2024138 
(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)))