Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.4% → 100.0%

Time: 6.3s

Alternatives: 4

Speedup: 1.3×

• 27.2% 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: 95.4% 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

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

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 96.9%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x - x \cdot y\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x - y\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x - y\right)}\right)\right) \]

   4. --lowering--.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x -2.0))))
   (if (<= y -0.28) t_0 (if (<= y 2.5e-7) (* x (* 2.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -0.28) {
		tmp = t_0;
	} else if (y <= 2.5e-7) {
		tmp = x * (2.0 * x);
	} 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 * (-2.0d0))
    if (y <= (-0.28d0)) then
        tmp = t_0
    else if (y <= 2.5d-7) then
        tmp = x * (2.0d0 * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -0.28) {
		tmp = t_0;
	} else if (y <= 2.5e-7) {
		tmp = x * (2.0 * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * -2.0)
	tmp = 0
	if y <= -0.28:
		tmp = t_0
	elif y <= 2.5e-7:
		tmp = x * (2.0 * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * -2.0))
	tmp = 0.0
	if (y <= -0.28)
		tmp = t_0;
	elseif (y <= 2.5e-7)
		tmp = Float64(x * Float64(2.0 * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * -2.0);
	tmp = 0.0;
	if (y <= -0.28)
		tmp = t_0;
	elseif (y <= 2.5e-7)
		tmp = x * (2.0 * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.28], t$95$0, If[LessEqual[y, 2.5e-7], N[(x * N[(2.0 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.28000000000000003 or 2.49999999999999989e-7 < y

   1. Initial program 93.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x - x \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x - y\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x - y\right)}\right)\right) \]

      4. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   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

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-2 \cdot x\right)}\right) \]

      4. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(-2, \color{blue}{x}\right)\right) \]

   7. Simplified 86.3%

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

   if -0.28000000000000003 < y < 2.49999999999999989e-7

   1. Initial program 100.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x - x \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x - y\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x - y\right)}\right)\right) \]

      4. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 88.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 88.0%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot x\right), \color{blue}{x}\right) \]

      3. *-lowering-*.f64 88.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), x\right) \]

   9. Applied egg-rr 88.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;y \leq -0.215:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x -2.0))))
   (if (<= y -0.215) t_0 (if (<= y 8.5e-7) (* 2.0 (* x x)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -0.215) {
		tmp = t_0;
	} else if (y <= 8.5e-7) {
		tmp = 2.0 * (x * x);
	} 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 * (-2.0d0))
    if (y <= (-0.215d0)) then
        tmp = t_0
    else if (y <= 8.5d-7) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -0.215) {
		tmp = t_0;
	} else if (y <= 8.5e-7) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * -2.0)
	tmp = 0
	if y <= -0.215:
		tmp = t_0
	elif y <= 8.5e-7:
		tmp = 2.0 * (x * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * -2.0))
	tmp = 0.0
	if (y <= -0.215)
		tmp = t_0;
	elseif (y <= 8.5e-7)
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * -2.0);
	tmp = 0.0;
	if (y <= -0.215)
		tmp = t_0;
	elseif (y <= 8.5e-7)
		tmp = 2.0 * (x * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.215], t$95$0, If[LessEqual[y, 8.5e-7], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.214999999999999997 or 8.50000000000000014e-7 < y

   1. Initial program 93.8%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x - x \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x - y\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x - y\right)}\right)\right) \]

      4. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   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

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-2 \cdot x\right)}\right) \]

      4. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(-2, \color{blue}{x}\right)\right) \]

   7. Simplified 86.3%

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

   if -0.214999999999999997 < y < 8.50000000000000014e-7

   1. Initial program 100.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x - x \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x - y\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x - y\right)}\right)\right) \]

      4. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 88.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 88.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.215:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x - x \cdot y\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x - y\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x - y\right)}\right)\right) \]

   4. --lowering--.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 55.1%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

7. Simplified 55.1%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot x\right)} \]
8. 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 2024158 
(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))))