Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 8

Speedup: 1.5×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 59.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot 2 + x \cdot x\right) \leq 0.01:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* y y) (+ (* x 2.0) (* x x))) 0.01) (* x 2.0) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * 2.0) + (x * x))) <= 0.01) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y * y) + ((x * 2.0d0) + (x * x))) <= 0.01d0) then
        tmp = x * 2.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * 2.0) + (x * x))) <= 0.01) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) + ((x * 2.0) + (x * x))) <= 0.01:
		tmp = x * 2.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * y) + Float64(Float64(x * 2.0) + Float64(x * x))) <= 0.01)
		tmp = Float64(x * 2.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) + ((x * 2.0) + (x * x))) <= 0.01)
		tmp = x * 2.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * y), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.01], N[(x * 2.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) (*.f64 y y)) < 0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot 2 \]
   7. Step-by-step derivation

   8. Applied rewrites 76.3%

      \[\leadsto x \cdot 2 \]

   if 0.0100000000000000002 < (+.f64 (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) (*.f64 y y))

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot 2 + x \cdot x\right) \leq 0.01:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 0.01) (fma 2.0 x (* y y)) (fma x x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 0.01) {
		tmp = fma(2.0, x, (y * y));
	} else {
		tmp = fma(x, x, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 0.01)
		tmp = fma(2.0, x, Float64(y * y));
	else
		tmp = fma(x, x, Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 0.01], N[(2.0 * x + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 0.0100000000000000002 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

Alternative 4: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 1e+62) (fma 2.0 x (* y y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 1e+62) {
		tmp = fma(2.0, x, (y * y));
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 1e+62)
		tmp = fma(2.0, x, Float64(y * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 1e+62], N[(2.0 * x + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 1.00000000000000004e62

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 1.00000000000000004e62 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

Alternative 5: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 5e+33) (* y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 5e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * 2.0d0) + (x * x)) <= 5d+33) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 5e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * 2.0) + (x * x)) <= 5e+33:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 5e+33)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * 2.0) + (x * x)) <= 5e+33)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 5e+33], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 4.99999999999999973e33

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if 4.99999999999999973e33 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

Alternative 6: 84.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1.5e+90) (* x (+ x 2.0)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.5e+90) {
		tmp = x * (x + 2.0);
	} 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 * y) <= 1.5d+90) then
        tmp = x * (x + 2.0d0)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.5e+90) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.5e+90:
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.5e+90)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.5e+90)
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.5e+90], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.49999999999999989e90

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if 1.49999999999999989e90 < (*.f64 y y)

   1. Initial program 100.0%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 8: 41.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
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. lower-*.f64 43.6

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

5. Applied rewrites 43.6%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

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

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