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

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

Alternatives: 8

Speedup: 1.2×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 99.9% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.1 \cdot 10^{-54}:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{elif}\;y \cdot y \leq 3.9 \cdot 10^{-28} \lor \neg \left(y \cdot y \leq 2.6 \cdot 10^{+156}\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1.1e-54)
   (+ (* x x) (* x 2.0))
   (if (or (<= (* y y) 3.9e-28) (not (<= (* y y) 2.6e+156))) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.1e-54) {
		tmp = (x * x) + (x * 2.0);
	} else if (((y * y) <= 3.9e-28) || !((y * y) <= 2.6e+156)) {
		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 ((y * y) <= 1.1d-54) then
        tmp = (x * x) + (x * 2.0d0)
    else if (((y * y) <= 3.9d-28) .or. (.not. ((y * y) <= 2.6d+156))) 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 ((y * y) <= 1.1e-54) {
		tmp = (x * x) + (x * 2.0);
	} else if (((y * y) <= 3.9e-28) || !((y * y) <= 2.6e+156)) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.1e-54:
		tmp = (x * x) + (x * 2.0)
	elif ((y * y) <= 3.9e-28) or not ((y * y) <= 2.6e+156):
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.1e-54)
		tmp = Float64(Float64(x * x) + Float64(x * 2.0));
	elseif ((Float64(y * y) <= 3.9e-28) || !(Float64(y * y) <= 2.6e+156))
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.1e-54)
		tmp = (x * x) + (x * 2.0);
	elseif (((y * y) <= 3.9e-28) || ~(((y * y) <= 2.6e+156)))
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.1e-54], N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(y * y), $MachinePrecision], 3.9e-28], N[Not[LessEqual[N[(y * y), $MachinePrecision], 2.6e+156]], $MachinePrecision]], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 1.1e-54

   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 73.1%

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

      1. unpow2 73.1%

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

      2. distribute-lft-in 61.3%

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

      3. *-rgt-identity 61.3%

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

      4. associate-*l* 74.1%

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

      5. distribute-lft-out 74.1%

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

      6. associate-*r/ 74.1%

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

      7. metadata-eval 74.1%

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

      8. unpow2 74.1%

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

      9. unpow2 74.1%

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

      10. associate-/l* 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in x around inf 95.3%

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

      1. distribute-lft-in 95.3%

         \[\leadsto \color{blue}{x \cdot x + x \cdot 2} \]

   8. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{x \cdot x + x \cdot 2} \]

   if 1.1e-54 < (*.f64 y y) < 3.89999999999999999e-28 or 2.60000000000000019e156 < (*.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 90.5%

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

      1. unpow2 90.5%

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

   5. Simplified 90.5%

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

   if 3.89999999999999999e-28 < (*.f64 y y) < 2.60000000000000019e156

   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 84.8%

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

      1. unpow2 84.8%

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

      2. distribute-lft-in 58.8%

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

      3. *-rgt-identity 58.8%

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

      4. associate-*l* 85.3%

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

      5. distribute-lft-out 85.3%

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

      6. associate-*r/ 85.3%

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

      7. metadata-eval 85.3%

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

      8. unpow2 85.3%

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

      9. unpow2 85.3%

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

      10. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

   6. Taylor expanded in x around inf 62.7%

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

   7. Taylor expanded in x around inf 62.8%

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

      1. unpow2 62.8%

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

   9. Simplified 62.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.1 \cdot 10^{-54}:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{elif}\;y \cdot y \leq 3.9 \cdot 10^{-28} \lor \neg \left(y \cdot y \leq 2.6 \cdot 10^{+156}\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-54}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;y \cdot y \leq 5.4 \cdot 10^{-32} \lor \neg \left(y \cdot y \leq 2.6 \cdot 10^{+156}\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-54)
   (* x (+ x 2.0))
   (if (or (<= (* y y) 5.4e-32) (not (<= (* y y) 2.6e+156))) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-54) {
		tmp = x * (x + 2.0);
	} else if (((y * y) <= 5.4e-32) || !((y * y) <= 2.6e+156)) {
		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 ((y * y) <= 1d-54) then
        tmp = x * (x + 2.0d0)
    else if (((y * y) <= 5.4d-32) .or. (.not. ((y * y) <= 2.6d+156))) 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 ((y * y) <= 1e-54) {
		tmp = x * (x + 2.0);
	} else if (((y * y) <= 5.4e-32) || !((y * y) <= 2.6e+156)) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e-54:
		tmp = x * (x + 2.0)
	elif ((y * y) <= 5.4e-32) or not ((y * y) <= 2.6e+156):
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-54)
		tmp = Float64(x * Float64(x + 2.0));
	elseif ((Float64(y * y) <= 5.4e-32) || !(Float64(y * y) <= 2.6e+156))
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1e-54)
		tmp = x * (x + 2.0);
	elseif (((y * y) <= 5.4e-32) || ~(((y * y) <= 2.6e+156)))
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-54], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(y * y), $MachinePrecision], 5.4e-32], N[Not[LessEqual[N[(y * y), $MachinePrecision], 2.6e+156]], $MachinePrecision]], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 1e-54

   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 73.1%

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

      1. unpow2 73.1%

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

      2. distribute-lft-in 61.3%

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

      3. *-rgt-identity 61.3%

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

      4. associate-*l* 74.1%

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

      5. distribute-lft-out 74.1%

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

      6. associate-*r/ 74.1%

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

      7. metadata-eval 74.1%

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

      8. unpow2 74.1%

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

      9. unpow2 74.1%

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

      10. associate-/l* 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in x around inf 95.3%

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

   if 1e-54 < (*.f64 y y) < 5.39999999999999962e-32 or 2.60000000000000019e156 < (*.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 90.5%

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

      1. unpow2 90.5%

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

   5. Simplified 90.5%

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

   if 5.39999999999999962e-32 < (*.f64 y y) < 2.60000000000000019e156

   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 84.8%

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

      1. unpow2 84.8%

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

      2. distribute-lft-in 58.8%

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

      3. *-rgt-identity 58.8%

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

      4. associate-*l* 85.3%

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

      5. distribute-lft-out 85.3%

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

      6. associate-*r/ 85.3%

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

      7. metadata-eval 85.3%

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

      8. unpow2 85.3%

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

      9. unpow2 85.3%

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

      10. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

   6. Taylor expanded in x around inf 62.7%

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

   7. Taylor expanded in x around inf 62.8%

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

      1. unpow2 62.8%

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

   9. Simplified 62.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-54}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{elif}\;y \cdot y \leq 5.4 \cdot 10^{-32} \lor \neg \left(y \cdot y \leq 2.6 \cdot 10^{+156}\right):\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-117}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-197}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 18000000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+35)
   (* x x)
   (if (<= x -2.4e-117)
     (* y y)
     (if (<= x -2.7e-197)
       (* x 2.0)
       (if (<= x 18000000000000.0) (* y y) (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+35) {
		tmp = x * x;
	} else if (x <= -2.4e-117) {
		tmp = y * y;
	} else if (x <= -2.7e-197) {
		tmp = x * 2.0;
	} else if (x <= 18000000000000.0) {
		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 <= (-1.45d+35)) then
        tmp = x * x
    else if (x <= (-2.4d-117)) then
        tmp = y * y
    else if (x <= (-2.7d-197)) then
        tmp = x * 2.0d0
    else if (x <= 18000000000000.0d0) 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 <= -1.45e+35) {
		tmp = x * x;
	} else if (x <= -2.4e-117) {
		tmp = y * y;
	} else if (x <= -2.7e-197) {
		tmp = x * 2.0;
	} else if (x <= 18000000000000.0) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e+35:
		tmp = x * x
	elif x <= -2.4e-117:
		tmp = y * y
	elif x <= -2.7e-197:
		tmp = x * 2.0
	elif x <= 18000000000000.0:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+35)
		tmp = Float64(x * x);
	elseif (x <= -2.4e-117)
		tmp = Float64(y * y);
	elseif (x <= -2.7e-197)
		tmp = Float64(x * 2.0);
	elseif (x <= 18000000000000.0)
		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 <= -1.45e+35)
		tmp = x * x;
	elseif (x <= -2.4e-117)
		tmp = y * y;
	elseif (x <= -2.7e-197)
		tmp = x * 2.0;
	elseif (x <= 18000000000000.0)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+35], N[(x * x), $MachinePrecision], If[LessEqual[x, -2.4e-117], N[(y * y), $MachinePrecision], If[LessEqual[x, -2.7e-197], N[(x * 2.0), $MachinePrecision], If[LessEqual[x, 18000000000000.0], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999997e35 or 1.8e13 < 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 86.7%

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

      1. unpow2 86.7%

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

      2. distribute-lft-in 60.1%

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

      3. *-rgt-identity 60.1%

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

      4. associate-*l* 86.7%

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

      5. distribute-lft-out 86.7%

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

      6. associate-*r/ 86.7%

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

      7. metadata-eval 86.7%

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

      8. unpow2 86.7%

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

      9. unpow2 86.7%

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

      10. associate-/l* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 89.5%

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

   7. Taylor expanded in x around inf 89.5%

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

      1. unpow2 89.5%

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

   9. Simplified 89.5%

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

   if -1.44999999999999997e35 < x < -2.40000000000000014e-117 or -2.70000000000000017e-197 < x < 1.8e13

   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 60.8%

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

      1. unpow2 60.8%

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

   5. Simplified 60.8%

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

   if -2.40000000000000014e-117 < x < -2.70000000000000017e-197

   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 41.7%

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

      1. unpow2 41.7%

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

      2. distribute-lft-in 41.7%

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

      3. *-rgt-identity 41.7%

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

      4. associate-*l* 64.0%

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

      5. distribute-lft-out 64.0%

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

      6. associate-*r/ 64.0%

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

      7. metadata-eval 64.0%

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

      8. unpow2 64.0%

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

      9. unpow2 64.0%

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

      10. associate-/l* 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in x around inf 68.7%

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

   7. Taylor expanded in x around 0 68.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+35}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-117}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-197}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;x \leq 18000000000000:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-141}:\\ \;\;\;\;x \cdot x + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-141) (+ (* x x) (* x 2.0)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-141) {
		tmp = (x * x) + (x * 2.0);
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d-141) then
        tmp = (x * x) + (x * 2.0d0)
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-141) {
		tmp = (x * x) + (x * 2.0);
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-141:
		tmp = (x * x) + (x * 2.0)
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-141)
		tmp = Float64(Float64(x * x) + Float64(x * 2.0));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-141)
		tmp = (x * x) + (x * 2.0);
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-141], N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999999e-141

   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 72.8%

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

      1. unpow2 72.8%

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

      2. distribute-lft-in 60.2%

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

      3. *-rgt-identity 60.2%

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

      4. associate-*l* 73.9%

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

      5. distribute-lft-out 73.9%

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

      6. associate-*r/ 73.9%

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

      7. metadata-eval 73.9%

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

      8. unpow2 73.9%

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

      9. unpow2 73.9%

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

      10. associate-/l* 74.1%

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

   5. Simplified 74.1%

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

   6. Taylor expanded in x around inf 98.8%

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

      1. distribute-lft-in 98.8%

         \[\leadsto \color{blue}{x \cdot x + x \cdot 2} \]

   8. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{x \cdot x + x \cdot 2} \]

   if 4.9999999999999999e-141 < (*.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 98.0%

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

      1. unpow2 98.0%

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

   5. Simplified 98.0%

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

Alternative 6: 59.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 2.0))) (* x x) (* x 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.0) || !(x <= 2.0)) {
		tmp = x * x;
	} else {
		tmp = x * 2.0;
	}
	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)) .or. (.not. (x <= 2.0d0))) then
        tmp = x * x
    else
        tmp = x * 2.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (x <= -2.0) or not (x <= 2.0):
		tmp = x * x
	else:
		tmp = x * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 2.0))
		tmp = Float64(x * x);
	else
		tmp = Float64(x * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 2.0)))
		tmp = x * x;
	else
		tmp = x * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(x * x), $MachinePrecision], N[(x * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 2 < 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 87.8%

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

      1. unpow2 87.8%

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

      2. distribute-lft-in 63.4%

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

      3. *-rgt-identity 63.4%

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

      4. associate-*l* 87.7%

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

      5. distribute-lft-out 87.7%

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

      6. associate-*r/ 87.7%

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

      7. metadata-eval 87.7%

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

      8. unpow2 87.7%

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

      9. unpow2 87.7%

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

      10. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 84.9%

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

   7. Taylor expanded in x around inf 83.4%

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

      1. unpow2 83.4%

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

   9. Simplified 83.4%

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

   if -2 < x < 2

   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 45.4%

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

      1. unpow2 45.4%

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

      2. distribute-lft-in 45.5%

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

      3. *-rgt-identity 45.5%

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

      4. associate-*l* 60.4%

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

      5. distribute-lft-out 60.4%

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

      6. associate-*r/ 60.4%

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

      7. metadata-eval 60.4%

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

      8. unpow2 60.4%

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

      9. unpow2 60.4%

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

      10. associate-/l* 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around inf 47.4%

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

   7. Taylor expanded in x around 0 45.8%

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

4. Final simplification 63.8%

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

Alternative 7: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(x * N[(x + 2.0), $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. distribute-lft-out 100.0%

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

   2. *-commutative 100.0%

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

   3. +-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 20.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 2.0), $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 65.8%

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

   1. unpow2 65.8%

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

   2. distribute-lft-in 54.1%

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

   3. *-rgt-identity 54.1%

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

   4. associate-*l* 73.5%

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

   5. distribute-lft-out 73.5%

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

   6. associate-*r/ 73.5%

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

   7. metadata-eval 73.5%

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

   8. unpow2 73.5%

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

   9. unpow2 73.5%

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

   10. associate-/l* 79.5%

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

5. Simplified 79.5%

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

6. Taylor expanded in x around inf 65.4%

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

7. Taylor expanded in x around 0 25.4%

   \[\leadsto \color{blue}{2 \cdot x} \]

8. Final simplification 25.4%

   \[\leadsto x \cdot 2 \]
9. Add Preprocessing

Developer target: 99.9% 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 2024097 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :alt
  (+ (* y y) (+ (* 2.0 x) (* x x)))

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