Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.0% → 100.0%

Time: 3.2s

Alternatives: 3

Speedup: 1.4×

• 28.1% 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: 95.0% 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

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 2\\ \mathbf{if}\;x \leq -400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-96}:\\ \;\;\;\;\left(y \cdot x\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) 2.0)))
   (if (<= x -400.0) t_0 (if (<= x 1e-96) (* (* y x) -2.0) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * 2.0;
	double tmp;
	if (x <= -400.0) {
		tmp = t_0;
	} else if (x <= 1e-96) {
		tmp = (y * 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 * x) * 2.0d0
    if (x <= (-400.0d0)) then
        tmp = t_0
    else if (x <= 1d-96) then
        tmp = (y * 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 * x) * 2.0;
	double tmp;
	if (x <= -400.0) {
		tmp = t_0;
	} else if (x <= 1e-96) {
		tmp = (y * x) * -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) * 2.0
	tmp = 0
	if x <= -400.0:
		tmp = t_0
	elif x <= 1e-96:
		tmp = (y * x) * -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * 2.0)
	tmp = 0.0
	if (x <= -400.0)
		tmp = t_0;
	elseif (x <= 1e-96)
		tmp = Float64(Float64(y * x) * -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) * 2.0;
	tmp = 0.0;
	if (x <= -400.0)
		tmp = t_0;
	elseif (x <= 1e-96)
		tmp = (y * x) * -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -400.0], t$95$0, If[LessEqual[x, 1e-96], N[(N[(y * x), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -400 or 9.9999999999999991e-97 < x

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if -400 < x < 9.9999999999999991e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 10^{-96}:\\ \;\;\;\;\left(y \cdot x\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 52.7

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

5. Applied rewrites 52.7%

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

6. Final simplification 52.7%

   \[\leadsto \left(y \cdot x\right) \cdot -2 \]
7. 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 2024255 
(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))))