Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.0% → 98.3%

Time: 4.8s

Alternatives: 6

Speedup: 1.1×

• 43.7% 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: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot \mathsf{fma}\left(2, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e+92) (fma y y (* x (fma 2.0 y x))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.8e+92) {
		tmp = fma(y, y, (x * fma(2.0, y, x)));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.8e+92)
		tmp = fma(y, y, Float64(x * fma(2.0, y, x)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.8e+92], N[(y * y + N[(x * N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000009e92

   1. Initial program 95.7%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-out N/A

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

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

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 4.80000000000000009e92 < y

   1. Initial program 83.3%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 94.1

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

   5. Simplified 94.1%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+240}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \mathsf{fma}\left(x, 2, y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+240) (* x x) (fma x x (* y (fma x 2.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+240) {
		tmp = x * x;
	} else {
		tmp = fma(x, x, (y * fma(x, 2.0, y)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+240)
		tmp = Float64(x * x);
	else
		tmp = fma(x, x, Float64(y * fma(x, 2.0, y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.2e+240], N[(x * x), $MachinePrecision], N[(x * x + N[(y * N[(x * 2.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999998e240

   1. Initial program 81.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if -4.1999999999999998e240 < x

   1. Initial program 94.2%

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

      1. associate-+l+ N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. distribute-rgt-out N/A

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

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

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

      5. accelerator-lowering-fma.f64 97.5

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

   4. Applied egg-rr 97.5%

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

Alternative 3: 86.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-56) (fma x x (* y (* x 2.0))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-56) {
		tmp = fma(x, x, (y * (x * 2.0)));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-56)
		tmp = fma(x, x, Float64(y * Float64(x * 2.0)));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.8e-56], N[(x * x + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8000000000000002e-56

   1. Initial program 95.1%

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

      1. associate-+l+ N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. distribute-rgt-out N/A

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

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

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

      5. accelerator-lowering-fma.f64 96.8

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 67.8

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

   7. Simplified 67.8%

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

   if 3.8000000000000002e-56 < y

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(x \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-56) (* x (fma 2.0 y x)) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-56) {
		tmp = x * fma(2.0, y, x);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-56)
		tmp = Float64(x * fma(2.0, y, x));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.45e-56], N[(x * N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.44999999999999996e-56

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. accelerator-lowering-fma.f64 71.0

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

   5. Simplified 71.0%

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

   if 1.44999999999999996e-56 < y

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

Alternative 5: 86.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-65}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.6e-65) (* x x) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-65) {
		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 (y <= 1.6d-65) 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 (y <= 1.6e-65) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.6e-65:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-65)
		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 (y <= 1.6e-65)
		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[y, 1.6e-65], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e-65

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 70.7

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

   5. Simplified 70.7%

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

   if 1.6e-65 < y

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.1

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

   5. Simplified 86.1%

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

Alternative 6: 57.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 57.9

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

5. Simplified 57.9%

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

Developer Target 1: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024204 
(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)))