Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 94.9% → 100.0%

Time: 5.6s

Alternatives: 9

Speedup: 1.4×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. distribute-lft-out-- N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot -2\right)\\ \mathbf{if}\;y \leq -14200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y -2.0))))
   (if (<= y -14200000.0) t_0 (if (<= y 5.5e+96) (* x (* x 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y * -2.0);
	double tmp;
	if (y <= -14200000.0) {
		tmp = t_0;
	} else if (y <= 5.5e+96) {
		tmp = x * (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y * (-2.0d0))
    if (y <= (-14200000.0d0)) then
        tmp = t_0
    else if (y <= 5.5d+96) then
        tmp = x * (x * 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y * -2.0);
	double tmp;
	if (y <= -14200000.0) {
		tmp = t_0;
	} else if (y <= 5.5e+96) {
		tmp = x * (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y * -2.0)
	tmp = 0
	if y <= -14200000.0:
		tmp = t_0
	elif y <= 5.5e+96:
		tmp = x * (x * 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * -2.0))
	tmp = 0.0
	if (y <= -14200000.0)
		tmp = t_0;
	elseif (y <= 5.5e+96)
		tmp = Float64(x * Float64(x * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y * -2.0);
	tmp = 0.0;
	if (y <= -14200000.0)
		tmp = t_0;
	elseif (y <= 5.5e+96)
		tmp = x * (x * 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -14200000.0], t$95$0, If[LessEqual[y, 5.5e+96], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.42e7 or 5.5000000000000002e96 < y

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.6

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

   5. Simplified 86.6%

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

   if -1.42e7 < y < 5.5000000000000002e96

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.7

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

   5. Simplified 85.7%

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

Alternative 3: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= y -8.5e+208) t_0 (if (<= y 4.2e+152) (* x (* x 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (y <= -8.5e+208) {
		tmp = t_0;
	} else if (y <= 4.2e+152) {
		tmp = x * (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * -x
    if (y <= (-8.5d+208)) then
        tmp = t_0
    else if (y <= 4.2d+152) then
        tmp = x * (x * 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (y <= -8.5e+208) {
		tmp = t_0;
	} else if (y <= 4.2e+152) {
		tmp = x * (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * -x
	tmp = 0
	if y <= -8.5e+208:
		tmp = t_0
	elif y <= 4.2e+152:
		tmp = x * (x * 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(-x))
	tmp = 0.0
	if (y <= -8.5e+208)
		tmp = t_0;
	elseif (y <= 4.2e+152)
		tmp = Float64(x * Float64(x * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * -x;
	tmp = 0.0;
	if (y <= -8.5e+208)
		tmp = t_0;
	elseif (y <= 4.2e+152)
		tmp = x * (x * 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * (-x)), $MachinePrecision]}, If[LessEqual[y, -8.5e+208], t$95$0, If[LessEqual[y, 4.2e+152], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4999999999999992e208 or 4.2000000000000003e152 < y

   1. Initial program 86.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.7

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

   5. Simplified 94.7%

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

   6. Taylor expanded in y around -inf

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot y\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.0

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

   8. Simplified 42.0%

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

   if -8.4999999999999992e208 < y < 4.2000000000000003e152

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.5

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

   5. Simplified 72.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+208}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 5.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.35e-142) (* x -2.0) (* x 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-142) {
		tmp = x * -2.0;
	} 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 <= (-1.35d-142)) then
        tmp = x * (-2.0d0)
    else
        tmp = x * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-142) {
		tmp = x * -2.0;
	} else {
		tmp = x * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-142:
		tmp = x * -2.0
	else:
		tmp = x * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-142)
		tmp = Float64(x * -2.0);
	else
		tmp = Float64(x * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-142)
		tmp = x * -2.0;
	else
		tmp = x * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-142], N[(x * -2.0), $MachinePrecision], N[(x * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e-142

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 36.6

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

   5. Simplified 36.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 6.3

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

   8. Simplified 6.3%

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

   if -1.3499999999999999e-142 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 6.0

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

   8. Simplified 6.0%

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

4. Final simplification 6.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 5.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.35e-142) (- x) (* x 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-142) {
		tmp = -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 <= (-1.35d-142)) then
        tmp = -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 <= -1.35e-142) {
		tmp = -x;
	} else {
		tmp = x * 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-142:
		tmp = -x
	else:
		tmp = x * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-142)
		tmp = Float64(-x);
	else
		tmp = Float64(x * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-142)
		tmp = -x;
	else
		tmp = x * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-142], (-x), N[(x * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e-142

   1. Initial program 92.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 6.3

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

   8. Simplified 6.3%

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

   if -1.3499999999999999e-142 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 6.0

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

   8. Simplified 6.0%

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

4. Final simplification 6.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 5.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.35e-142) (- x) x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-142) {
		tmp = -x;
	} else {
		tmp = 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 <= (-1.35d-142)) then
        tmp = -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-142) {
		tmp = -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-142:
		tmp = -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-142)
		tmp = Float64(-x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-142)
		tmp = -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-142], (-x), x]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e-142

   1. Initial program 92.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.8

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

   5. Simplified 71.8%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 6.3

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

   8. Simplified 6.3%

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

   if -1.3499999999999999e-142 < x

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot x} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 3.3

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

   8. Simplified 3.3%

      \[\leadsto \color{blue}{-x} \]
   9. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + x\right)}\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{0} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      5. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{x}^{3}}\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      6. cube-neg N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      8. sqr-pow N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      12. sqr-neg N/A

          \[\leadsto \frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      13. pow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      14. sqr-pow N/A

          \[\leadsto \frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      15. +-lft-identity N/A

          \[\leadsto \frac{\color{blue}{0 + {x}^{3}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{{0}^{3}} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

      17. flip3-+ N/A

          \[\leadsto \color{blue}{0 + x} \]

      18. +-lft-identity 6.0

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

   10. Applied egg-rr 6.0%

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

Alternative 7: 27.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 53.5

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

5. Simplified 53.5%

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

6. Taylor expanded in y around -inf

   \[\leadsto x \cdot \color{blue}{\left(-1 \cdot y\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 25.2

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

8. Simplified 25.2%

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

9. Final simplification 25.2%

   \[\leadsto y \cdot \left(-x\right) \]
10. Add Preprocessing

Alternative 8: 14.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 53.5

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

5. Simplified 53.5%

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

6. Taylor expanded in y around -inf

   \[\leadsto x \cdot \color{blue}{\left(-1 \cdot y\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 25.2

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

8. Simplified 25.2%

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

   1. distribute-rgt-neg-out N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x \cdot y} \]

   3. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(x \cdot y\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {\left(x \cdot y\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   5. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({\left(x \cdot y\right)}^{3}\right)}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   6. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x \cdot y\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   7. distribute-rgt-neg-out N/A

      \[\leadsto \frac{{\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{{\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   10. sqr-pow N/A

       \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   11. unpow-prod-down N/A

       \[\leadsto \frac{{x}^{3} \cdot \color{blue}{{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{{x}^{3} \cdot {\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   13. lift-neg.f64 N/A

       \[\leadsto \frac{{x}^{3} \cdot {\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   14. sqr-neg N/A

       \[\leadsto \frac{{x}^{3} \cdot {\color{blue}{\left(y \cdot y\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   15. pow-prod-down N/A

       \[\leadsto \frac{{x}^{3} \cdot \color{blue}{\left({y}^{\left(\frac{3}{2}\right)} \cdot {y}^{\left(\frac{3}{2}\right)}\right)}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   16. sqr-pow N/A

       \[\leadsto \frac{{x}^{3} \cdot \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   17. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{3}}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{3}}{\color{blue}{0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)\right)} \]

   19. +-lft-identity N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{3}}{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + 0 \cdot \left(x \cdot y\right)}} \]

   20. distribute-rgt-out N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{3}}{\color{blue}{\left(x \cdot y\right) \cdot \left(x \cdot y + 0\right)}} \]

   21. +-commutative N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{3}}{\left(x \cdot y\right) \cdot \color{blue}{\left(0 + x \cdot y\right)}} \]

   22. +-lft-identity N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{3}}{\left(x \cdot y\right) \cdot \color{blue}{\left(x \cdot y\right)}} \]

10. Applied egg-rr 13.2%

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

11. Final simplification 13.2%

    \[\leadsto x \cdot y \]
12. Add Preprocessing

Alternative 9: 4.3% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 60.4

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

5. Simplified 60.4%

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

6. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot x} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 4.4

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

8. Simplified 4.4%

   \[\leadsto \color{blue}{-x} \]
9. Step-by-step derivation

   1. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + x\right)}\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}}\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{0} + {x}^{3}\right)\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   5. +-lft-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{x}^{3}}\right)}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   6. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   8. sqr-pow N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   12. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   13. pow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   14. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   15. +-lft-identity N/A

       \[\leadsto \frac{\color{blue}{0 + {x}^{3}}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{{0}^{3}} + {x}^{3}}{0 \cdot 0 + \left(x \cdot x - 0 \cdot x\right)} \]

   17. flip3-+ N/A

       \[\leadsto \color{blue}{0 + x} \]

   18. +-lft-identity 4.3

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

10. Applied egg-rr 4.3%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

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

  (* 2.0 (- (* x x) (* x y))))