Examples.Basics.ProofTests:f4 from sbv-4.4

Percentage Accurate: 94.1% → 99.9%

Time: 10.3s

Alternatives: 7

Speedup: 1.9×

• 42.9% 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: 94.1% 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, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) \leq 10^{+260}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= (+ (* y y) (+ (* x x) (* y (* x 2.0)))) 1e+260)
   (+ (* x x) (* y (+ y (* x 2.0))))
   (+ (* y y) (* x x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * x) + (y * (x * 2.0)))) <= 1e+260) {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	} else {
		tmp = (y * y) + (x * x);
	}
	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 * y) + ((x * x) + (y * (x * 2.0d0)))) <= 1d+260) then
        tmp = (x * x) + (y * (y + (x * 2.0d0)))
    else
        tmp = (y * y) + (x * x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * x) + (y * (x * 2.0)))) <= 1e+260) {
		tmp = (x * x) + (y * (y + (x * 2.0)));
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if ((y * y) + ((x * x) + (y * (x * 2.0)))) <= 1e+260:
		tmp = (x * x) + (y * (y + (x * 2.0)))
	else:
		tmp = (y * y) + (x * x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * y) + Float64(Float64(x * x) + Float64(y * Float64(x * 2.0)))) <= 1e+260)
		tmp = Float64(Float64(x * x) + Float64(y * Float64(y + Float64(x * 2.0))));
	else
		tmp = Float64(Float64(y * y) + Float64(x * x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) + ((x * x) + (y * (x * 2.0)))) <= 1e+260)
		tmp = (x * x) + (y * (y + (x * 2.0)));
	else
		tmp = (y * y) + (x * x);
	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[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+260], N[(N[(x * x), $MachinePrecision] + N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * x), $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)) < 1.00000000000000007e260

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. distribute-rgt-out 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-rgt-neg-out 99.8%

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

      6. unsub-neg 99.8%

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

      7. fmm-undef 99.8%

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

      8. distribute-rgt-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 1.00000000000000007e260 < (+.f64 (+.f64 (*.f64 x x) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) (*.f64 y y))

   1. Initial program 83.5%

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

      1. associate-+l+ 83.5%

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

      2. distribute-rgt-out 92.9%

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

      3. *-commutative 92.9%

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

      4. remove-double-neg 92.9%

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

      5. distribute-rgt-neg-out 92.9%

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

      6. unsub-neg 92.9%

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

      7. fmm-undef 92.9%

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

      8. distribute-rgt-neg-out 92.9%

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

      9. remove-double-neg 92.9%

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

      10. *-commutative 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in y around 0 92.9%

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

      1. unpow2 92.9%

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

   7. Applied egg-rr 92.9%

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

   8. Taylor expanded in y around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot x + y \cdot \left(x \cdot 2\right)\right) \leq 10^{+260}:\\ \;\;\;\;x \cdot x + y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(x, Float64(x + Float64(2.0 * y)), Float64(y * y))
end

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. associate-*l* 91.7%

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

   2. *-commutative 91.7%

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

   3. distribute-lft-out 95.2%

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

   4. fma-define 99.9%

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

   5. +-commutative 99.9%

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

   6. *-commutative 99.9%

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

   7. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+114}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e+114) (+ (* y y) (* x (+ x (* 2.0 y)))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.25e+114) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} 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 <= 2.25d+114) then
        tmp = (y * y) + (x * (x + (2.0d0 * y)))
    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 <= 2.25e+114) {
		tmp = (y * y) + (x * (x + (2.0 * y)));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.25e+114:
		tmp = (y * y) + (x * (x + (2.0 * y)))
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.25e+114)
		tmp = Float64(Float64(y * y) + Float64(x * Float64(x + Float64(2.0 * y))));
	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 <= 2.25e+114)
		tmp = (y * y) + (x * (x + (2.0 * y)));
	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, 2.25e+114], N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25e114

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0 97.9%

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

   if 2.25e114 < y

   1. Initial program 78.4%

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

   3. Taylor expanded in x around 0 84.3%

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

   4. Taylor expanded in x around 0 76.7%

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

   5. Taylor expanded in y around 0 92.4%

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

   6. Taylor expanded in y around inf 98.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+114}:\\ \;\;\;\;y \cdot y + x \cdot \left(x + 2 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8e-88) (* y (+ y (* x 2.0))) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8e-88) {
		tmp = y * (y + (x * 2.0));
	} 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 <= 8d-88) then
        tmp = y * (y + (x * 2.0d0))
    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 <= 8e-88) {
		tmp = y * (y + (x * 2.0));
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8e-88:
		tmp = y * (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 <= 8e-88)
		tmp = Float64(y * Float64(y + Float64(x * 2.0)));
	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 <= 8e-88)
		tmp = y * (y + (x * 2.0));
	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, 8e-88], N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999947e-88

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 97.4%

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

   4. Taylor expanded in x around 0 44.7%

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

   5. Taylor expanded in y around 0 47.2%

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

   if 7.99999999999999947e-88 < y

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 91.6%

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

   4. Taylor expanded in x around 0 70.3%

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

   5. Taylor expanded in y around 0 78.7%

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

   6. Taylor expanded in y around inf 79.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(y + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e-301) (* 2.0 (* x y)) (* y y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-301) {
		tmp = 2.0 * (x * y);
	} 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 <= (-1.05d-301)) then
        tmp = 2.0d0 * (x * y)
    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 <= -1.05e-301) {
		tmp = 2.0 * (x * y);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.05e-301:
		tmp = 2.0 * (x * y)
	else:
		tmp = y * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.05e-301)
		tmp = Float64(2.0 * Float64(x * y));
	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 <= -1.05e-301)
		tmp = 2.0 * (x * y);
	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, -1.05e-301], N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0499999999999999e-301

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 96.2%

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

   4. Taylor expanded in x around 0 52.6%

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

   5. Taylor expanded in x around inf 13.8%

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

      1. *-commutative 13.8%

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

   7. Simplified 13.8%

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

   if -1.0499999999999999e-301 < y

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 94.4%

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

   4. Taylor expanded in x around 0 55.5%

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

   5. Taylor expanded in y around 0 61.0%

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

   6. Taylor expanded in y around inf 61.6%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-301}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. associate-+l+ 91.7%

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

   2. distribute-rgt-out 96.4%

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

   3. *-commutative 96.4%

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

   4. remove-double-neg 96.4%

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

   5. distribute-rgt-neg-out 96.4%

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

   6. unsub-neg 96.4%

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

   7. fmm-undef 96.4%

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

   8. distribute-rgt-neg-out 96.4%

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

   9. remove-double-neg 96.4%

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

   10. *-commutative 96.4%

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

3. Simplified 96.4%

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

5. Taylor expanded in y around 0 96.4%

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

   1. unpow2 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in y around inf 98.3%

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

Alternative 7: 56.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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 = y * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in x around 0 95.2%

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

4. Taylor expanded in x around 0 54.2%

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

5. Taylor expanded in y around 0 58.9%

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

6. Taylor expanded in y around inf 59.0%

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

Developer Target 1: 94.1% 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 2024172 
(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)))