Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.3% → 100.0%

Time: 4.8s

Alternatives: 6

Speedup: 1.3×

• 28.3% 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: 95.3% 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, 1.3× speedup

Math

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

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(x * Float64(2.0 * Float64(x - y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. metadata-eval 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. +-commutative 100.0%

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

   8. sub-neg 100.0%

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

7. Simplified 100.0%

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

Alternative 2: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-18} \lor \neg \left(y \leq 6500000000\right):\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.05e-18) (not (<= y 6500000000.0)))
   (* -2.0 (* x y))
   (* 2.0 (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.05e-18) || !(y <= 6500000000.0)) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = 2.0 * (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 <= (-1.05d-18)) .or. (.not. (y <= 6500000000.0d0))) then
        tmp = (-2.0d0) * (x * y)
    else
        tmp = 2.0d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.05e-18) || !(y <= 6500000000.0)) {
		tmp = -2.0 * (x * y);
	} else {
		tmp = 2.0 * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.05e-18) or not (y <= 6500000000.0):
		tmp = -2.0 * (x * y)
	else:
		tmp = 2.0 * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.05e-18) || !(y <= 6500000000.0))
		tmp = Float64(-2.0 * Float64(x * y));
	else
		tmp = Float64(2.0 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.05e-18) || ~((y <= 6500000000.0)))
		tmp = -2.0 * (x * y);
	else
		tmp = 2.0 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.05e-18], N[Not[LessEqual[y, 6500000000.0]], $MachinePrecision]], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e-18 or 6.5e9 < y

   1. Initial program 91.4%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.5%

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

   if -1.05e-18 < y < 6.5e9

   1. Initial program 100.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 89.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-18} \lor \neg \left(y \leq 6500000000\right):\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{elif}\;y \leq 68000000:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e-15)
   (* x (* y -2.0))
   (if (<= y 68000000.0) (* x (* x 2.0)) (* -2.0 (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e-15) {
		tmp = x * (y * -2.0);
	} else if (y <= 68000000.0) {
		tmp = x * (x * 2.0);
	} else {
		tmp = -2.0 * (x * 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 <= (-3.8d-15)) then
        tmp = x * (y * (-2.0d0))
    else if (y <= 68000000.0d0) then
        tmp = x * (x * 2.0d0)
    else
        tmp = (-2.0d0) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e-15) {
		tmp = x * (y * -2.0);
	} else if (y <= 68000000.0) {
		tmp = x * (x * 2.0);
	} else {
		tmp = -2.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e-15:
		tmp = x * (y * -2.0)
	elif y <= 68000000.0:
		tmp = x * (x * 2.0)
	else:
		tmp = -2.0 * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e-15)
		tmp = Float64(x * Float64(y * -2.0));
	elseif (y <= 68000000.0)
		tmp = Float64(x * Float64(x * 2.0));
	else
		tmp = Float64(-2.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.8e-15)
		tmp = x * (y * -2.0);
	elseif (y <= 68000000.0)
		tmp = x * (x * 2.0);
	else
		tmp = -2.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e-15], N[(x * N[(y * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 68000000.0], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000002e-15

   1. Initial program 95.2%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 84.5%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

      3. associate-*r* 84.5%

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

   10. Simplified 84.5%

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

   if -3.8000000000000002e-15 < y < 6.8e7

   1. Initial program 100.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 89.7%

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

   if 6.8e7 < y

   1. Initial program 87.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{elif}\;y \leq 68000000:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{elif}\;y \leq 75000000:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e-15)
   (* x (* y -2.0))
   (if (<= y 75000000.0) (* 2.0 (* x x)) (* -2.0 (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e-15) {
		tmp = x * (y * -2.0);
	} else if (y <= 75000000.0) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = -2.0 * (x * 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 <= (-9d-15)) then
        tmp = x * (y * (-2.0d0))
    else if (y <= 75000000.0d0) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = (-2.0d0) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e-15) {
		tmp = x * (y * -2.0);
	} else if (y <= 75000000.0) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = -2.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e-15:
		tmp = x * (y * -2.0)
	elif y <= 75000000.0:
		tmp = 2.0 * (x * x)
	else:
		tmp = -2.0 * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e-15)
		tmp = Float64(x * Float64(y * -2.0));
	elseif (y <= 75000000.0)
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = Float64(-2.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e-15)
		tmp = x * (y * -2.0);
	elseif (y <= 75000000.0)
		tmp = 2.0 * (x * x);
	else
		tmp = -2.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e-15], N[(x * N[(y * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 75000000.0], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999995e-15

   1. Initial program 95.2%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 84.5%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

      3. associate-*r* 84.5%

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

   10. Simplified 84.5%

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

   if -8.9999999999999995e-15 < y < 7.5e7

   1. Initial program 100.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 89.7%

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

   if 7.5e7 < y

   1. Initial program 87.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(y \cdot -2\right)\\ \mathbf{elif}\;y \leq 75000000:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

Alternative 6: 56.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 56.0%

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

Developer Target 1: 100.0% accurate, 1.3× 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 2024186 
(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))))