Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.4% → 100.0%

Time: 6.1s

Alternatives: 3

Speedup: 1.4×

• 27.7% 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, 1.4× 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 96.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 rewrites 100.0%

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

Alternative 2: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot -2\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y -2.0))))
   (if (<= y -2e+96) t_0 (if (<= y 1.7e+26) (* x (* x 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y * -2.0);
	double tmp;
	if (y <= -2e+96) {
		tmp = t_0;
	} else if (y <= 1.7e+26) {
		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 <= (-2d+96)) then
        tmp = t_0
    else if (y <= 1.7d+26) 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 <= -2e+96) {
		tmp = t_0;
	} else if (y <= 1.7e+26) {
		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 <= -2e+96:
		tmp = t_0
	elif y <= 1.7e+26:
		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 <= -2e+96)
		tmp = t_0;
	elseif (y <= 1.7e+26)
		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 <= -2e+96)
		tmp = t_0;
	elseif (y <= 1.7e+26)
		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, -2e+96], t$95$0, If[LessEqual[y, 1.7e+26], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0000000000000001e96 or 1.7000000000000001e26 < y

   1. Initial program 91.3%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   if -2.0000000000000001e96 < y < 1.7000000000000001e26

   1. Initial program 98.8%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

Alternative 3: 57.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]
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 62.9

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

5. Applied rewrites 62.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 2\right)} \]
6. 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 2024216 
(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))))